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RRT, 6.KǘZRԕ- =u.nMRԹ"G)u%KB]B SbTyu.nH6.hYAswJl]J0. ԕ- =u.nMRԹ"@غg)u*^<غ7t$P6.hYAswJl]J0. )M*Zz\қR-mswBE SbJ7tԩx [b\БCغg)u*^<غ7t$QJl]IRгPغ/akl]K(jRT, 6.KǘZRԕ- =u.nMRԹ"Tغg)u*^<غ7t$PRT, 6.KǘZRSbJ7tԩx [b\Бjl]IRгPغ/akl]K,JRT, 6.KǘZRԕ- =u.nMRԹ"ũu%KB]B SbTyu.nHl]IRгPغ/akl]K,6.hYAswJl]J0. 5M*Zz\қR-mswBESbJ7tԩx [b\Б`*l]IRгPغ/akl]K, M*Zz\қR-mswBEu%KB]B SbTyu.nH6.hYAswJl]J0. %M*Zz\қR-mswBF SbJ7tԩx [b\Бغg)u*^<غ7t$`6.hYAswJl]J0. RRT, 6.KǘZRԕ- =u.nMRԹ#)u%KB]B SbTyu.nH6.hYAswJl]J0. ԕ- =u.nMRԹ#غg)u*^<غ7t$6.hYAswJl]J0. (JRT, 6.KǘZR)M*Zz\қR-mswBJԕ- =u.nMRԹ% SbJ7tԩx [b\ВSbJ7tԩx [b\Вu%KB]B SbTyu.nICغg)u*^<غ7t$6.hYAswJl]J0. ( M*Zz\қR-mswBJSbJ7tԩx [b\Бl]IRгPغ/akl]K4JRT, 6.KǘZRԕ- =u.nMRԹ#Eu%KB]B SbTyu.nHъl]IRгPغ/akl]K46.hYAswJl]J0. 5M*Zz\қR-mswBFSbJ7tԩx [b\Б*l]IRгPغ/akl]K4 M*Zz\қR-mswBFu%KB]B SbTyu.nH6.hYAswJl]J0. %M*Zz\қR-mswBG SbJ7tԩx [b\Бغg)u*^<غ7t$p6.hYAswJl]J0. 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RRT, 6.KǘZR ԕ- =u.nMRԹC)u%KB]B SbTyu.nP6.hYAswJl]J0. ԕ- =u.nMRԹCغg)u*^<غ7t(6.hYAswJl]J0. (JRT, 6.KǘZR)M*Zz\қR-mswBԕ- =u.nMRԹE SbJ7tԩx [b\ТSbJ7tԩx [b\Тu%KB]B SbTyu.nQCغg)u*^<غ7t(6.hYAswJl]J0. 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(bRT, 6.KǘZRRRT, 6.KǘZR5M*Zz\қR-mswBԕ- =u.nMRԹeu%KB]B SbTyu.nY@Jl]IRгPغ/akl]KPRT, 6.KǘZRSbJ7tԩx [b\бTغg)u*^<غ7t,h6.hYAswJl]J0. -M*Zz\қR-mswBƌSbJ7tԩx [b\б)u%KB]B SbTyu.nXѪl]IRгPغ/akl]K4rRT, 6.KǘZRSbJ7tԩx [b\бJl]IRгPغ/akl]K4 M*Zz\қR-mswB)u%KB]B SbTyu.nX*l]IRгPغ/akl]K8RRT, 6.KǘZRԕ- =u.nMRԹc)u%KB]B SbTyu.nXغg)u*^<غ7t,p6.hYAswJl]J0. 9M*Zz\қR-mswBu%KB]B SbTyu.nX%6.hYAswJl]J0. ԕ- =u.nMRԹ`*l]IRгPغ/akl]K%M*Zz\қR-mswBE)u%KB]B SbTyu.nX6.hYAswJl]J0. ԕ- =u.nMRԹ`%)u%KB]B SbTyu.nX6.hYAswJl]J0. ԕ- =u.nMRԹ` *l]IRгPغ/akl]Kԕ- =u.nMRԹ` jl]IRгPغ/akl]Kԕ- =u.nMRԹ`BTغg)u*^<غ7t,RRT, 6.KǘZR SbJ7tԩx [b\а!l]IRгPغ/akl]KRRT, 6.KǘZR SbJ7tԩx [b\а!l]IRгPغ/akl]Kԕ- =u.nMRԹ`@Jl]IRгPغ/akl]Kԕ- =u.nMRԹ`jl]IRгPغ/akl]K%M*Zz\қR-mswB)u%KB]B SbTyu.nX6.hYAswJl]J0. ԕ- =u.nMRԹ`e)u%KB]B SbTyu.nX6.hYAswJl]J0. ԕ- =u.nMRԹ``*l]IRгPغ/akl]Kԕ- =u.nMRԹ``jl]IRгPغ/akl]K%M*Zz\қR-mswCSbJ7tԩx [b\"غg)u*^<غ7t0H6.hYAswJl]J0. 1M*Zz\қR-mswCԕ- =u.nMRԹFu%KB]B SbTyu.n`l]IRгPغ/akl]K$M*Zz\қR-mswC)u%KB]B SbTyu.n`56.hYAswJl]J0. ԕ- =u.nMRԹu%KB]B SbTyu.n`Jl]IRгPغ/akl]K(ZRT, 6.KǘZR ԕ- =u.nMRԹSbJ7tԩx [b\CTغg)u*^<غ7t0P6.hYAswJl]J0. ԕ- =u.nMRԹغg)u*^<غ7t0PRT, 6.KǘZR SbJ7tԩx [b\bTغg)u*^<غ7t0X6.hYAswJl]J0. -M*Zz\қR-mswCSbJ7tԩx [b\e)u%KB]B SbTyu.n`l]IRгPغ/akl]K,rRT, 6.KǘZR SbJ7tԩx [b\`Jl]IRгPغ/akl]K, M*Zz\қR-mswC)u%KB]B SbTyu.n`*l]IRгPغ/akl]K0RRT, 6.KǘZR ԕ- =u.nMRԹ)u%KB]B SbTyu.n`”غg)u*^<غ7t0`6.hYAswJl]J0. 9M*Zz\қR-mswCu%KB]B SbTyu.n`%6.hYAswJl]J0. ԕ- =u.nMRԹ)u%KB]B SbTyu.naBTغg)u*^<غ7t0Jl]IRгPغ/akl]KP6.hYAswJl]J0. (bRT, 6.KǘZRRRT, 6.KǘZR5M*Zz\қR-mswC ԕ- =u.nMRԹu%KB]B SbTyu.na@Jl]IRгPغ/akl]KPRT, 6.KǘZR SbJ7tԩx [b\Tغg)u*^<غ7t0h6.hYAswJl]J0. -M*Zz\қR-mswCSbJ7tԩx [b\)u%KB]B SbTyu.n`Ѫl]IRгPغ/akl]K4rRT, 6.KǘZR SbJ7tԩx [b\Jl]IRгPغ/akl]K4 M*Zz\қR-mswC)u%KB]B SbTyu.n`*l]IRгPغ/akl]K8RRT, 6.KǘZRԕ- =u.nMRԹ)u%KB]B SbTyu.n`غg)u*^<غ7t0p6.hYAswJl]J0. 9M*Zz\қR-mswCu%KB]B SbTyu.n`%6.hYAswJl]J0. ԕ- =u.nMRԹ*l]IRгPغ/akl]K%M*Zz\қR-mswCE)u%KB]B SbTyu.n`6.hYAswJl]J0. ԕ- =u.nMRԹ%)u%KB]B SbTyu.n`6.hYAswJl]J0. ԕ- =u.nMRԹ *l]IRгPغ/akl]Kԕ- =u.nMRԹ jl]IRгPغ/akl]Kԕ- =u.nMRԹBTغg)u*^<غ7t0RRT, 6.KǘZR SbJ7tԩx [b\!l]IRгPغ/akl]KRRT, 6.KǘZR SbJ7tԩx [b\!l]IRгPغ/akl]Kԕ- =u.nMRԹ@Jl]IRгPغ/akl]Kԕ- =u.nMRԹjl]IRгPغ/akl]K%M*Zz\қR-mswC)u%KB]B SbTyu.n`6.hYAswJl]J0. ԕ- =u.nMRԹe)u%KB]B SbTyu.n`6.hYAswJl]J0. ԕ- =u.nMRԹ`*l]IRгPغ/akl]Kԕ- =u.nMRԹ`jl]IRгPغ/akl]K(JRT, 6.KǘZR%M*Zz\қR-mswE ԕ- =u.nMRԹSbJ7tԩx [b\BF)u%KB]B SbTyu.n%)u%KB]B SbTyu.n#Tغg)u*^<غ7tPl]IRгPغ/akl]K(H RT, 6.KǘZRԕ- =u.nMRԹu%KB]B SbTyu.nJl]IRгPغ/akl]K(P6.hYAswJl]J0.(RRT, 6.KǘZR-M*Zz\қR-mswE ԕ- =u.nMRԹԕ- =u.nMRԹ SbJ7tԩx [b\Bغg)u*^<غ7tP6.hYAswJl]J0.RRT, 6.KǘZRԕ- =u.nMRԹ)u%KB]B SbTyu.n6.hYAswJl]J0.ԕ- =u.nMRԹغg)u*^<غ7t4JRT, 6.KǘZRSbJ7tԩx [b\jl]IRгPغ/akl]K1M*Zz\қR-mswCJSbJ7tԩx [b\Ѫl]IRгPغ/akl]K9M*Zz\қR-mswC@Tغg)u*^<غ7t4 M*Zz\қR-mswC@غg)u*^<غ7t8JRT, 6.KǘZR SbJ7tԩx [b\jl]IRгPغ/akl]K1M*Zz\қR-mswCSbJ7tԩx [b\l]IRгPغ/akl]K9M*Zz\қR-mswCTغg)u*^<غ7t8 M*Zz\қR-mswCغg)u*^<غ7t%M*Zz\қR-msw@"غg)u*^<غ7t-M*Zz\қR-msw@#غg)u*^<غ7tRRT, 6.KǘZRFu%KB]B SbTyu.nrRT, 6.KǘZR@Tغg)u*^<غ7tԕ- =u.nMRԹ56.hYAswJl]J0. SbJ7tԩx [b\6.hYAswJl]J0. SbJ7tԩx [b\6.hYAswJl]J0.ԕ- =u.nMRԹ!l]IRгPغ/akl]Kԕ- =u.nMRԹ 6.hYAswJl]J0.)u%KB]B SbTyu.n M*Zz\қR-msw@bTغg)u*^<غ7tl]IRгPغ/akl]KDL T, 6.KǘZR&x*&h*Zz\қR-mshOّ5g6dMzhQ@ T*!w!R" x\'6.KǘZR, 6.KǘZغ/akJ@'Ț3B2&?gf4j(v` @`*ِ)._DDқR-)Zz\қR-` @0)`'Ț3B2&?gf4j(v` @`*ِ)._DDқR-)Zz\қR-` @0)kSbTy@P,hDdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١0JdUq;JV7tԩx [b\БBغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١,?IH*v8`BxthYAswJl]J0. 1M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1c6IZه$`3 =u.nMRԹ"SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fXBS$lÎ(X3 =u.nMRԹ"u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf0 iMAVa JV7tԩx [b\БCغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١PMAVa$i_:R, 6.KǘZRSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fQCS$lË,JV7tԩx [b\Б@Jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyً(h dUr'JV7tԩx [b\Б@jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyٔ$h& A0å+B]B SbTyu.nH6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! 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DA 7tԩx O١$$h;fE aҕg)u*^<غ7t(P6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١5?IH*v8"ҕg)u*^<غ7t(XP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vc9?IH*v9C å+B]B SbTyu.nP6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١,jdUr ++ =u.nMRԹCغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١%?IH*v8BJгPغ/akl]K0 M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1cƧ6IZه$?f)Zz\қR-mswBSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fpdUr 3JV7tԩx [b\Тu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf0Q@IMAVa JV7tԩx [b\Т)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf,`!)MAVa,:R, 6.KǘZR-M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1D6IZه,`/:R, 6.KǘZR1M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1FOl 8hthYAswJl]J0. (6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t(h6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$IMAVaĔ$_7 =u.nMRԹC@غ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١(P& A0JV7tԩx [b\Сjl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyً4 ?IH*v8BJaҕg)u*^<غ7t(rP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vc$lÔ8h:R, 6.KǘZRfA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$h& A0℁(V)Zz\қR-mswB SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fP& A0ō;JV7tԩx [b\С”غ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١8rdUq@@CG0JгPغ/akl]K8ZP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vb Ol ,H0/^)Zz\қR-mswB SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fP iMAVa 4W/ =u.nMRԹCSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fPiMAVaĎ$W/ =u.nMRԹCu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(X& A0E(O/ =u.nMRԹC)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf( jdUq8JV7tԩx [b\С*l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyي$$h;fPEJгPغ/akl]K8 M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1bDOl $ `^)Zz\қR-mswBu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf0hѩMAVa ;+ =u.nMRԹ@*l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]K%M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1`C$lÊ,_3 =u.nMRԹ@"غ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$ ZdUrE0JгPغ/akl]K-M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1JOl ,p!XthYAswJl]J0. 81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١4rdUq`caҕg)u*^<غ7t(rP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vb $$h;fH!?:R, 6.KǘZRu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf0p)MAVa (X:R, 6.KǘZR)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(h dUqB; =u.nMRԹ@ jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyف Ol 8P? =u.nMRԹ@Jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]K%M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1BĎOl 8P:R, 6.KǘZR SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fXрiMAVač(w3 =u.nMRԹ@Bغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١0JdUqe +JV7tԩx [b\Р!l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyي4RdUqc@V)Zz\қR-mswBSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fhÓ$lÎ(Pv)Zz\қR-mswBu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8p& A0GJгPغ/akl]K9M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1c6IZه4P3 =u.nMRԹ@@*l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyي$$h;fPJгPغ/akl]K81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t( JP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46v`@"S$lÉ0)Zz\қR-mswB)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf, ?IH*v9CJгPغ/akl]K-M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;0b@$lÔ8i_)Zz\қR-mswB)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(X& A0Ōҕg)u*^<غ7t( 6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t,H6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١PMAVaČ(g/ =u.nMRԹbu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(H& A0Daҕg)u*^<غ7t,P6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t,X6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١,h& A0JV7tԩx [b\бJl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyً$$h;fh !:R, 6.KǘZRSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fP#S$lÊ/JV7tԩx [b\бl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]K8JP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vc6IZه(p?XthYAswJl]J0. )M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1CG Ol 4h?:R, 6.KǘZR81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١$h& A0Ŋ+P:R, 6.KǘZR81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١0PIMAVa 4OthYAswJl]J0. 81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١,RdUq@ҕg)u*^<غ7t,ZP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vb $$h;fq@ CJV7tԩx [b\аl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyً0bdUqJ'X:R, 6.KǘZR81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١RdUq"3å+B]B SbTyu.nX6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$ ?IH*v8JгPغ/akl]KRP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46v`e $$h;fhр@?ҕg)u*^<غ7t,jP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46v`CDOl ,pC?<:R, 6.KǘZRSbO?fDקɞrّ5;1008@*ِ)._DDқR-?f 4& A0(N)Zz\қR-mswBTغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$X“$lÊ(HBpJгPغ/akl]K81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١4`$& A07 =u.nMRԹغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t0P6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١PiMAVaō,X:R, 6.KǘZR 81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١$PIMAVaƍ(N)Zz\қR-mswC)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf,`& A0Pf)Zz\қR-mswCu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(`& A0ㄍå+B]B SbTyu.n`6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١8$& A0, =u.nMRԹغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١0 ?IH*v8@`#ŕJV7tԩx [b\`jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyً8P& A0G WthYAswJl]J0. 81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١P& A0E(WthYAswJl]J0. (bP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vb Ol $ C? å+B]B SbTyu.naE)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(p IMAVa $_:R, 6.KǘZR5M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1BOl ,aBxthYAswJl]J0. (rP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46v`b@S$lÍ$_)Zz\қR-mswC SbO?fDקɞrّ5;1008@*ِ)._DDқR-?f `& A0 ' =u.nMRԹ)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf4 & A0ㅔ( =u.nMRԹu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(P& A0↎+P:R, 6.KǘZR SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fpdUr 3JV7tԩx [b\Tغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١8X& A0 (`CJгPغ/akl]K4RP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46ve )?IH*v8 ҕg)u*^<غ7t0h6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$`)MAVaā+ =u.nMRԹ@Tغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١(jdUq% 3P:R, 6.KǘZR SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fP!)MAVa(XJгPغ/akl]K4 M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1B)?IH*v8@ å+B]B SbTyu.n`6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١(X& A0J0_xthYAswJl]J0. 9M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1@COl (XBҕg)u*^<غ7t0p P"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vb(JdUr(n)Zz\қR-mswC)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgfPb$lÍ(`7 =u.nMRԹغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١4& A0@Eå+B]B SbTyu.n` P"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vc$lÔ8h:R, 6.KǘZRSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fX"$lÍ'JV7tԩx [b\Jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyٍQIMAVa(/n)Zz\қR-mswCEu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf,HMAVa+JV7tԩx [b\l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyي5?IH*v8BŁJгPغ/akl]KRP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46v`E$$h;fB<:R, 6.KǘZRSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fpщMAVaŊҕg)u*^<غ7t0rP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vbDOl 0X0@8thYAswJl]J0. SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fiMAVaĀJгPغ/akl]K81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١$1iMAVa $P:R, 6.KǘZR81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$lÔ(X_:R, 6.KǘZR81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١0 ?IH*v8@+ =u.nMRԹ`jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyٔ$bdUq å+B]B SbTyu.n*l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]K(H6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١PIMAVaƁ å+B]B SbTyu.nCTغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١8 `$& A0@čPthYAswJl]J0.-M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-mswE SbO?fDקɞrّ5;1008@*ِ)._DDқR-?fpdUr 3JV7tԩx [b\E)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8 ?IH*v9Cå+B]B SbTyu.n衪l]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]K(rP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vc$lÔ8h:R, 6.KǘZRSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fpdUr 3JV7tԩx [b\@Jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]K( M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-mswCDu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8 ?IH*v9Cå+B]B SbTyu.nh6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t46.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t4M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-mswC@غ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t4 M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-mswCu' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8 ?IH*v9Cå+B]B SbTyu.np6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t86.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t8M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-mswCغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t8 M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-msw@"Tغ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t)M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-msw@"غ>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t1M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-msw@%)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8 ?IH*v9Cå+B]B SbTyu.njP"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vc$lÔ8h:R, 6.KǘZRG)u' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8 ?IH*v9Cå+B]B SbTyu.nM?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-msw@ Jl]@'Ț3B2&?gf8FO{2" ׅq舛SbTyَOl PᢆJгPغ/akl]KSbO?fDקɞrّ5;1008@*ِ)._DDқR-?fpdUr 3JV7tԩx [b\6.DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fE aҕg)u*^<غ7t%M?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1iMAVa4PV)Zz\қR-mswN81~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١( ?IH*v8#JгPغ/akl]KD(DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١8 ?IH*v9@%JгPغ/akl]KD Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyَ-?IH*v9@b pJгPغ/akl]KD Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyي8X& A0E+ =u.nMRԹI(QJ&h?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1`IMAVaƁå+B]B SbTyu.mg5D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(PMAVaƍ(o/ =u.nMRԹI(P481~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١4H& A0 (W/ =u.nMRԹI(PO?fDקɞrّ5;1008@*ِ)._DDқR-?f XdUq#G å+B]B SbTyu.mgfA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١4p& A0 4_/ =u.nMRԹI(ZO?fDקɞrّ5;1008@*ِ)._DDқR-?fP)MAVa ;JV7tԩx [b\$,JO?fDקɞrّ5;1008@*ِ)._DDқR-?f`MAVaŁaҕg)u*^<غ63 fA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١(XdUr ҕg)u*^<غ63 fA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١(`)MAVa(Pn)Zz\қR-mshOّ5g6dMzqE''f  %D.d! DA 7tԩx O١$ rdUq"E aҕg)u*^<غ63 Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyٍ5?IH*v9BEå+B]B SbTyu.mgfA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;fP?:R, 6.KǘZR&x`j&h?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1 IMAVa /JV7tԩx [b\$1D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(RdUqb@FJгPغ/akl]KD Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyً$$h;fpJгPغ/akl]KD Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyي$$h;fheJгPغ/akl]KD Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyف Ol 0QCGJгPغ/akl]KD Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyي(p& A0Jå+B]B SbTyu.mgRO?fDקɞrّ5;1008@*ِ)._DDқR-?f`& A0@@XthYAswJl]J0.͢LCDdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١-?IH*v8BJгPغ/akl]KD↎Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyي0 ?IH*v8… å+B]B SbTyu.mgfA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١,IIMAVa ;JV7tԩx [b\$4 D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf8?IH*v8`C@XthYAswJl]J0.͢LC@L"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vcOl 4XXthYAswJl]J0.͢LCL"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46vb Ol 4hV)Zz\қR-mshOّ5g6dMzqE''f  %D.d! DA 7tԩx O١0)MAVaå+B]B SbTyu.mgQ3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyٍQIMAVaå+B]B SbTyu.mgfA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١0 hdUqC@F+ =u.nMRԹIP481~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١0h dUq "Jå+B]B SbTyu.mg9D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf,`)MAVaƁ +JV7tԩx [b\$(D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(4& A0ㅋW/ =u.nMRԹIPO?fDקɞrّ5;1008@*ِ)._DDқR-?fX'6IZه(q_:R, 6.KǘZR&y@j&h?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;1b$$h;fPaҕg)u*^<غ63ƨDdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١1?IH*v9BFaҕg)u*^<غ63Ɓ(DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١PIMAVa (w/ =u.nMRԹI1D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf(piMAVaŀW/ =u.nMRԹIRO?fDקɞrّ5;1008@*ِ)._DDқR-?fP`)MAVaā å+B]B SbTyu.mgQ3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyي)?IH*v9@"Ɖҕg)u*^<غ63G(DdMz! ^q3QI كQ Bk8DM)u*^Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١4p dUqƁJгPغ/akl]KD j&h?pc^&yCfDקpLR(v`TBfBD|N=wJl]J0;2'6IZه4P?V)Zz\қR-msh< D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf($lË$JV7tԩx [b\$fA>Oّ5g6dMzqE''f  %D.d! DA 7tԩx O١$$h;f`JгPغ/akl]KDBL"2&?L̉Oَ(QS䨅̄!H5rz"&غ/ai46v``b$lÍ8/V)Zz\қR-msh<ZO?fDקɞrّ5;1008@*ِ)._DDқR-?faBS$lÉ( _:R, 6.KǘZR&x481~̉OOّ5g6dMzqE''f  %D.d! DA 7tԩx O١0X dUqG å+B]B SbTyu.mgQ3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyً4JdUqe8w3 =u.nMRԹI9D' |"k9lȚ ON ?J]B^/Ǣ"nMRCgf,`4& A0J'JV7tԩx [b\$Q3@ 'Ț3B2&?gf8FO{2" ׅq舛SbTyً$& A0Ċå+B]B SbTyu.mg(DdMz! ^q3QI كQ Bk8DM)u*^!h6lhǝH*]VψBsͧMbh@lkֳC̡]˶mAMun՛\j;\_9mgVÈ+"5,a 42̚5o^#׬cHq?kYrO $@P#4 7NسkνËOӫ_Ͼ˟OOϿ(h& 6F(Vhe0 #X""~h"+")b6(+(8"*$-X$$PJy>Y#|c\Fc):9Z馍l #cYeIRg0*ImJ _ȒEhhS.裐F*a;䨅 k:~SZj_H6_H6_H6./usr/lpp/essl.html.en_US/html/essl.htmlf.en_U Guide and Reference

Guide and Reference

Engineering and Scientific
Subroutine Library for AIX
Guide and Reference

SA22-7272-01

Program Number: 5765-C42


Notes!



Second Edition (October 1998)

This edition applies to Version 3 Release 1.1 of the IBM* Engineering and Scientific Subroutine Library (ESSL) for Advanced Interactive Executive (AIX)* licensed program, program number 5765-C42 and to all subsequent releases and modifications until otherwise indicated by new editions. Significant changes or additions to the text and illustrations are indicated by a vertical line (|) to the left of the change.

In this document, ESSL refers to the above version of ESSL for AIX. Changes are periodically made to the information herein.

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\#2K uߢ_=8,_Oŋ.:*Vۗ^ˢ& E{b0MEEns y^FBWhԨJd%v ]oȲG{=oM'%E\w5~{;w;Xgx#$X2DL0!4iy5m  sʌѣR!4h!4hM4w-Bh系!]Mr،㢔 W~&ȉFe{b0MEEns y^FfFFfG|Y5;ȳ5; wŖhMh[\tQX]^˞Չ[7\w׻sd3w\r~;OY%+{;xdD#{b0MEEns+0QѩQѣ,΍:4wfth,Q,Fbb2K"hO!]y9)<\tP ~/[p½>UKdD#5{b0MEEnso+0Q\Ԩ\3]s6f}rߑfkyalWb2K^˞{7"`Y=1|ͷ9<\yFr3]R3],vٚYe; w w-.:(,.^jek_gwC bܹ2K{;xc[秬nY\tS;r[7Bx ~cw0Or،㢩vD#:4a򚊋6^W`<9QѣR3G|Y;fth"FXG|Yb;dE(X-?_gwC bsS Wx#$+_!z|nȉFiH=1|ͷ9@+0QҐ5*4R}rߑiH|oXuFIqMXu!NvXu"$a]y5m y^F\ԨҋiEQsoȴ瞹lv7Xu~2Kj[w[?_jD#>SQQfۜg(TiJR|ZRzm)Coȴ>wu'߯b}ޜ\tU}on|}Q"(j*,s\yFrF@ )FaJ4w(\@0<ͶxU75%E+|vb.Nx$aa򚊋6y^BjT!|H@yޙ;Ȑ:oĄzۻߨ㢚>~?~ڱ>~A0H$`={F)ma`W(T#5*#;#;`[ V'lFIqCoN6__jon|}QDKQ# Hf)"%`F#hȏ@"`u2#椈$b0]AfpRDL1x.FDzpSRDL1x.FDzpcRDL1x.FDzp I0Ha`I0Ha`f$b0]AfCRDL1x.FDzp)"& SQQfۜ(*+0jLn=Y0K@+чj*,s\ ~ "` WTY0 gI5%~zzm0K@+чj*,sW`<1 Guide and Reference

Guide and Reference

Special Notices

References in this publication to IBM products, programs, or services do not imply that IBM intends to make these available in all countries in which IBM operates. Any reference to an IBM product, program, or service is not intended to state or imply that only IBM's product, program, or service may be used. Any functionally equivalent product, program, or service that does not infringe any of IBM's intellectual property rights may be used instead of the IBM product, program, or service. Evaluation and verification of operation in conjunction with other products, except those expressly designated by IBM, are the user's responsibility.

IBM may have patents or pending patent applications covering subject matter in this document. The furnishing of this document does not give you any license to these patents. You can send license inquiries, in writing, to:

IBM Director of Licensing
IBM Corporation
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Licensees of this program who wish to have information about it for the purpose of enabling: (i) the exchange of information between independently created programs and other programs (including this one) and (ii) the mutual use of the information which has been exchanged, should contact:

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Such information may be available, subject to appropriate terms and conditions, including in some cases, payment of fee.

Trademarks

The following terms, denoted by an asterisk (*), are trademarks of the IBM Corporation in the United States or other countries or both: 
AIX
IBM
IBMLink
PowerPC Architecture


POWER2 Architecture
RS/6000
SP

Other company, product, and service names, which may be denoted by a double asterisk (**), may be trademarks or service marks of others.

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Guide and Reference

Programming Interfaces

This Engineering and Scientific Subroutine Library (ESSL) Version 3 Guide and Reference manual is intended to help the customer to do application programming. This ESSL Version 3 Guide and Reference manual documents General-use Programming Interface and Associated Guidance Information provided by ESSL Version 3.

General-use programming interfaces allow the customer to write programs that obtain the services of ESSL Version 3.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ending kz~SZjeH6eH6eH6./usr/lpp/essl.html.en_US/html/essl006.htmlen_U Guide and Reference

Guide and Reference

About This Book

The Engineering and Scientific Subroutine Library (ESSL) for AIX is a set of high-performance mathematical subroutines. ESSL is provided as five run-time libraries, running on RS/6000* POWER, PowerPC, POWER2, and POWER3 processors. ESSL can be used with Fortran, C, C++, and Programming Language/I (PL/I) programs operating under the AIX operating system.

This book is a guide and reference manual for using ESSL in doing application programming. It includes:

  • An overview of ESSL and guidance information for designing, coding, and processing your program, as well as migrating existing programs, and diagnosing problems

  • Reference information for coding each ESSL calling sequence

This book is written for a wide class of ESSL users: scientists, mathematicians, engineers, statisticians, computer scientists, and system programmers. It assumes a basic knowledge of mathematics in the areas of ESSL computation. It also assumes that users are familiar with Fortran, C, C++, and PL/I programming.

How to Use This Book

Front Matter consists of the Table of Contents, Special Notices, and Preface. Use these to find or interpret information in the book.

Part 1. "Guide Information" provides guidance information for using ESSL. It covers the user-oriented tasks of learning, designing, coding, migrating, processing, and diagnosing. Use the following chapters when performing any of these tasks:

  • Chapter 1, "Learning about ESSL" gives an introduction to ESSL, providing highlights and general information. Read this chapter first to determine the aspects of ESSL you want to use.

  • Chapter 2, "Designing Your Program" provides ESSL-specific information that helps you design your program. Read this chapter before designing your program.

  • Chapter 3, "Setting Up Your Data Structures" describes all types of data structures, such as vectors, matrices, and sequences. Use this information when designing and coding your program.

  • Chapter 4, "Coding Your Program" tells you how to code your scalar and array data, how to code calls to ESSL in Fortran, C, C++, and PL/I programs, and how to do the coding necessary to handle errors. Use this information when coding your program.

  • Chapter 5, "Processing Your Program" describes how to process your program under your particular operating system on your hardware. Use this information after you have coded your program and are ready to run it.

  • Chapter 6, "Migrating Your Programs" explains all aspects of migration to ESSL, to this version of ESSL, to different processors, and to future releases and future processors. Read this chapter before starting to design your program.

  • Chapter 7, "Handling Problems" provides diagnostic procedures for analyzing all ESSL problems. When you encounter a problem, use the symptom indexes at the beginning of this chapter to guide you to the appropriate diagnostic procedure.

Part 2. "Reference Information" provides reference information you need to code the ESSL calling sequences. It covers each of the mathematical areas of ESSL, and the utility subroutines. Each chapter begins with an introduction, followed by the subroutine descriptions. Each introduction applies to all the subroutines in that chapter and is especially important in planning your use of the subroutines and avoiding problems. To understand the information in the subroutine descriptions, see "How to Interpret the Subroutine Descriptions". Use the appropriate chapter when coding your program:

  • Chapter 8, "Linear Algebra Subprograms"

  • Chapter 9, "Matrix Operations"

  • Chapter 10, "Linear Algebraic Equations"

  • Chapter 11, "Eigensystem Analysis"

  • Chapter 12, "Fourier Transforms, Convolutions and Correlations, and Related Computations"

  • Chapter 13, "Sorting and Searching"

  • Chapter 14, "Interpolation"

  • Chapter 15, "Numerical Quadrature"

  • Chapter 16, "Random Number Generation"

  • Chapter 17, "Utilities"

Appendix A. Basic Linear Algebra Subprograms provides a list of the Level 1, 2, and 3 Basic Linear Algebra Subprograms (BLAS) included in ESSL.

Appendix B. LAPACK provides a list of the LAPACK subroutines included in ESSL.

Glossary contains definitions of terms used in this book.

Bibliography provides information about publications related to ESSL. Use it when you need more information than this book provides.

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How to Find a Subroutine Description

If you want to locate a subroutine description and you know the subroutine name, you can use the "Looking for a Subroutine?", following the Table of Contents. You can also find them listed individually or under the entry "subroutines, ESSL" in the Index.

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Where to Find Related Publications

If you have a question about the SP, PSSP, or a related product, the following online information resources make it easy to find the information you are looking for:

  • If you have installed the RS/6000 SP Resource Center available with Parallel System Support Programs (PSSP) Version 3 Release 1 or later, you can access the SP Resource Center by issuing the command:

    /usr/lpp/ssp/bin/resource_center

    If you have the SP Resource Center on CD ROM, see the readme.txt file for information on how to run it.

  • Access the RS/6000 Web site at:

    http://www.rs6000.ibm.com

A list of all ESSL publications, as well as related programming and hardware publications, are listed in the bibliography. Also included is a list of math background publications you may find helpful, along with the necessary information for ordering them from independent sources. See "Bibliography".

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How to Look Up a Bibliography Reference

Special references are made throughout this book to mathematical background publications and software libraries, available through IBM, publishers, or other companies. All of these are described in detail in the bibliography. A reference to one of these is made by using a bracketed number. The number refers to the item listed under that number in the bibliography. For example, reference [1] cites the first item listed in the bibliography.

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Special Terms

Standard data processing and mathematical terms are used in this book. Terminology is generally consistent with that used for Fortran. See the Glossary for definitions of terms used in this book.

Short and Long Precision

Because ESSL can be used with more than one programming language, the terms short precision and long precision are used in place of the Fortran terms single precision and double precision.

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Subroutines and Subprograms

An ESSL subroutine is a named sequence of instructions within the ESSL product library whose execution is invoked by a call. A subroutine can be called in one or more user programs and at one or more times within each program. The ESSL subroutines are referred to as subprograms in the area of linear algebra subprograms. The term subprograms is used because it is consistent with the BLAS. Many of the linear algebra subprograms correspond to the BLAS; these are listed in Appendix A. "Basic Linear Algebra Subprograms (BLAS)".

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How to Interpret the Subroutine Names with a Prefix Underscore

A name specified in this book with an underscore (_) prefix, such as _GEMUL, refers to all the versions of the subroutine with that name. To get the entire list of subroutines that name refers to, substitute the first letter for each version of the subroutine. For example, _GEMUL, refers to all versions of the matrix multiplication subroutine: SGEMUL, DGEMUL, CGEMUL, and ZGEMUL. You do not use the underscore in coding the names of the ESSL subroutines in your program. You code a complete name, such as SGEMUL. For details about these names, see "The Variety of Mathematical Functions".

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Abbreviated Names

The abbreviated names used in this book are defined below.
Short Name Full Name
AIX Advanced Interactive Executive
BLAS Basic Linear Algebra Subprograms
ESSL IBM Engineering and Scientific Subroutine Library
HTML Hypertext Markup Language
LAPACK Linear Algebra Package
PL/I Programming Language/I
POWER, POWER2, POWER3, and PowerPC processors RS/6000 processors
SL MATH Subroutine Library--Mathematics
SMP Symmetric Multi-Processing
SSP Scientific Subroutine Package

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Fonts

This book uses a variety of special fonts to distinguish between many mathematical and programming items. These are defined below.
Special Font Example Description
Italic with no subscripts m, inc1x, aux, iopt Calling sequence argument or mathematical variable
Italic with subscripts x1, amn, xj1,j2 Element of a vector, matrix, or sequence
Bold italic lowercase x, y, z Vector or sequence
Bold italic uppercase A, B, C Matrix
Gothic uppercase A, B, C, AGB

IM=ISMAX(4,X,2)

Array

Fortran statement

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Special Notations and Conventions

This section explains the special notations and conventions used in this book to describe various types of data.

Scalar Data

Following are the special notations used in the examples in this book for scalar data items. These notations are used to simplify the examples, and they do not imply usage of any precision. For a definition of scalar data in Fortran, C, C++, and PL/I, see "Coding Your Program".
Data Item Example Description
Character item 'T' Character(s) in single quotation marks
Hexadecimal string X'97FA00C1' String of 4-bit hexadecimal characters
Logical item .TRUE. .FALSE. True or false logical value, as indicated
Integer data 1 Number with no decimal point
Real data 1.6 Number with a decimal point
Complex data (1.0,-2.9) Real part followed by the imaginary part 
Continuation


     _
1.6666


Continue the last digit
(1.6666666... and so forth)

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Vectors

A vector is represented as a single row or column of subscripted elements enclosed in square brackets. The subscripts refer to the element positions within the vector:



Figure ESYGR8 not displayed.


For a definition of vector, see "Vectors".

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Matrices

A matrix is represented as a block of elements enclosed in square brackets. Subscripts refer to the row and column positions, respectively:



Figure ESYGR9 not displayed.


For a definition of matrix, see "Matrices".

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Sequences

Sequences are used in the areas of sorting, searching, Fourier transforms, convolutions, and correlations. For a definition of sequences, see "Sequences".

One-Dimensional Sequences

A one-dimensional sequence is represented as a series of elements enclosed in parentheses. Subscripts refer to the element position within the sequence:

(x1, x2, x3, ..., xn)

Two-Dimensional Sequences

A two-dimensional sequence is represented as a series of columns of elements. (They are represented in the same way as a matrix without the square brackets.) Subscripts refer to the element positions within the first and second dimensions, respectively:



Figure ESYGR10 not displayed.


Three-Dimensional Sequences

A three-dimensional sequence is represented as a series of blocks of elements. Subscripts refer to the elements positions within the first, second, and third dimensions, respectively:



Figure ESYGR11 not displayed.


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Arrays

Arrays contain vectors, matrices, or sequences. For a definition of array, see "How Do You Set Up Your Arrays?".

One-Dimensional Arrays

A one-dimensional array is represented as a single row of numeric elements enclosed in parentheses: 

                     (1.0, 2.0, 3.0, 4.0, 5.0)

Elements not significant to the computation are usually not shown in the array. One dot appears for each element not shown. In the following array, five elements are significant to the computation, and two elements not used in the computation exist between each of the elements shown: 

        (1.0, . , . ,2.0, . , . ,3.0, . , . ,4.0, . , . ,5.0)

This notation is used to show vector elements inside an array.

Two-Dimensional Arrays

A two-dimensional array is represented as a block of numeric elements enclosed in square brackets: 

                     *                      *
                     | 1.0  11.0  5.0  25.0 |
                     | 2.0  12.0  6.0  26.0 |
                     | 3.0  13.0  7.0  27.0 |
                     | 4.0  14.0  8.0  28.0 |
                     *                      *

Elements not significant to the computation are usually not shown in the array. One dot appears for each element not shown. The following array contains three rows and two columns not used in the computation: 

                    *                           *
                    | .    .   .    .    .    . |
                    | .    .   .    .    .    . |
                    | .   1.0  2.0  5.0  4.0  . |
                    | .   2.0  3.0  6.0  3.0  . |
                    | .   3.0  4.0  7.0  2.0  . |
                    | .   4.0  5.0  8.0  1.0  . |
                    | .    .    .    .    .   . |
                    *                           *

This notation is used to show matrix elements inside an array.

Three-Dimensional Arrays

A three-dimensional array is represented as a series of blocks of elements separated by ellipses. Each block appears like a two-dimensional array: 


*                      *   *                          *       *                         *
| 1.0  11.0  5.0  25.0 |   | 10.0  111.0  15.0  125.0 |       | 100.0  11.0  15.0  25.0 |
| 2.0  12.0  6.0  26.0 |   | 20.0  112.0  16.0  126.0 |  ...  | 200.0  12.0  16.0  26.0 |
| 3.0  13.0  7.0  27.0 |   | 30.0  113.0  17.0  127.0 |       | 300.0  13.0  17.0  27.0 |
| 4.0  14.0  8.0  28.0 |   | 40.0  114.0  18.0  128.0 |       | 400.0  14.0  18.0  28.0 |
*                      *   *                          *       *                         *

Elements not significant to the computation are usually not shown in the array. One dot appears for each element not shown, just as for two-dimensional arrays.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  k6~SZjE1 K61 K61 K6E./usr/lpp/essl.html.en_US/html/essl02.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui lU~SZj'mH6mH6mH6./usr/lpp/essl.html.en_US/html/essl020.htmlen_U  "<>AT/=ILRe 1BDEGNPVainorst23;FMOScdghlmx#&.4CHUXbfpuy,-05689Yv()7:_jw!k|+Qq{}*Kx&xaFu`g:x8>xpM{3:>xpM{3:>xpM vIpp<8Y/qݦgR\9iwiԗAÅh {3:>xpM^LΤr,Aw3. 4Pfu%Ð}f6fu%Ð}f{3:>xpE vIpp<8YqݦgR\9hE;K  vIpp<8Y[wiԗAÅ({3:>xpEqݦgR\9h=;K m;K *fu%Ð}f{3:>xpE^LΤr,`wiԗAÅ,{3:>xpE3. 4X*fu%Ð}fLΤr,Р=;K 6fu%Ð}f vIpp<8YBqݦgR\9hPU;K ^LΤr,РM;K oqݦgR\9hP*fu%Ð}fqݦgR\9h {3:>xpALN*IUV\6U,Ye6k0T$Rej`RΈK4l%fSlIfauJe[)2ʶJҫX&ʥ,:iRs )6DX\5YڥY."UU*V-*Ue)YRx{3k0TE[Y*LYVJ(cB3"hQ)f-3?piт)>ߺSR}?x72c3s+R}tF8ק͢,".f;7iF~NIӽEQYh  J(Xe@iT C <q@o)т)>zwUb4JU*%6yn|R}tFހ\Sh3yт)>ߺSR}|P*$Q)t\r-3;3ϮIҝ"ǧ{nCUCqhph~:;Ly}t`O7~=;):$t"LV W ܆㡸wL>0E'PJt`O):[NScn8if;R}tFI)ձt JY fi|t7viF~NIӽ:ZGKpiۊt`#KqN1~]"( :0E'ߏNNVetYVXaېмt7vҭ,5tlr-W%DX*$Ys1N:Zf;wsg]"( :0E'ߏNi|r-im):[NSn8if;R}tFޠVemP,Stt9Nӷj:[ viF~NIӽ:Z9N.ЈDŽ.Ј@t|r-3yт)>ߺSR}qNSTM,MyNTm6b-_KLyn|R}tFېzwKU) -:[ <q@o)т)>zwKU)eU*mlh)` &c|>Zfyт)>ߺSR}qNKpj{t9Nc>0E'PJt`OX^kh@Ir-3;3ϮIҝ"ǧ{nC{ @5=qeaېn8c>0E'PJt`O6ʫ"S(z÷!єIlB,5:`B" nCK㡸wL>0E'PJt`Ov܀hshL& &GcV&$dh]TNdopn83>0E'PJt`O`Y,. U fi|t7vNXD *+;rn83;3ϮIҝ"ǧzft 9! &ɢI!uՉ3:@1.a$B%U :[;Lå?sϮIҝ"ǧ{'DX,"Ve Dnt6RtE%f^" ̴lStKV)qӴ @II0p@(av A/B[[m dРk]t}5bbLΐ FFR7DhDs${ A/B[KqiI z"@5[0!M`YʶefZkf)8:[NS@)qӴwL>0E'PJt`O / c+܆:;pjz <q@o)т)>zwRtE%f^$MVM ̋l3Xv4:;FSe'DX,"Ve"@5[6ʫ"S(z÷!1~[-3<q@o)т)>zwaۇKr }͡2l &GcV&$dh"a$B%U :[;Lå?sϮIҝ"ǧ{'M *efZkb-/S@47@43i|r-n)t@43;3ϮIҝ"ǧzft 9! &ɠk]t}5bbLΐ FFivK4"UQ9 å:[<q@o)т)>zwRt*nSh-)1~[-3<q@o)т)>zwaۇKr }͡2lXBﱫftb24n,ЉUDH&0n;R}tFI˷2lV`b-WKi+h~9Nt@.Q":Zf;wsg]"( :0E'ߏNOB0 AC0i|t7v((`@9aӴV8if-y}t`O7~=;):@HUMUv-LP6d[a÷!qSYyZn-% Й6MXBﱫftb24KY̐M8- n-c|>Zfyт)>ߺSR}L AC3܆:;B0 AC0in8if-y}t`O7~=;):ay2Zʶ܆:;@Jb /2TAP0å^ 9! &ɠk]t}5bbLΐ FFvK4"UQ9 -åwL>0E'PJt`O阞((`@9aېCqh@9((f ;EcawF~NIӽ򓦄fEU+Kj`B*+;rn8),RC 䆘YRh nCx:;B0 AC0i+@4p@(e((`@9aېn84PPsE hqӴ!pF0E'PJt`O阞ېCqn OB  OLå?sϮIҝ"ǧ{'K. -Nʶ܆:;Lyn|R}tFޙ t7v (3S0n;R}tFIlB,5:`Boc㶖\6Z?q-ϖ}t`O7~=; nC{ wF~NIӽSnlYuWnʶ܆:;Lyn|R}tF--S'XKW),l^ocwF~NIӽl^oc㶊*b--SKTj:[DX*$Yvyn|R}tF޴t\r-3yт)>ߺSR}|饗0!XKU)1~[-3<q@o)т)>zwXjNi?sϮIҝ"ǧ{'DXieLV)r-3;3ϮIҝ"ǧzB@)qt3yт)>ߺSR}|饉LV)qte6Rt*,MfMa  3;3ϮIҝ"ǧz"Lͦ1NS f;R}tFIMʤJ$jSi?q-ϖ}t`O7~=;ʗi|t7vc>0E'PJt`OTńT̴n^"  /1~[-3<q@o)т)>zw\÷!q<q@o)т)>zwRtYe,$ZVet fi|t7viF~NIӽ#50i|r- y}t`O7~=;):hJͦeۙ9MUY&nle^fbJجStwL>0E'PJt`OVd:Z_K@C1~]"( :0E'ߏNN9,s'6iK5Kfj6iXKKi?q-ϖ}t`O7~=;՗l:Z_K@C1~]"( :0E'ߏNN9,s'6iK5EfTEWi+i|r-3;3ϮIҝ"ǧ{t9Nf xKKchf;R}tF-/SYBEXSte6Rt*%&b-/STI`jĦJ:Z[KLyn|R}tFެˬؙ?sϮIҝ"ǧ{]fyn|R}tFވf;R}tFަh (   3;㢆Kp22H } 85I)5IA Y$E@J6XlMRj;22H }ͦQoRv5IA Y$E@qA%,T CΘnk;22H }ͦ (b#Ã`)v@8,XqFgR\96Tqݦcx3q::`  k~SZjnH6nH6nH6./usr/lpp/essl.html.en_US/html/essl021.htmlen_U Guide and Reference

Guide and Reference

How to Interpret the Subroutine Descriptions

This section explains how to interpret the information in the subroutine descriptions in Part 2 of this book.

Description

Each subroutine description begins with a brief explanation of what the subroutine does. When we combine the description of multiple versions of a subroutine, we give enough information to enable you to easily tell the differences among the subroutines. Differences usually occur in either the function performed or the data types required for each subroutine.

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Syntax

This shows the syntax for the Fortran, C, C++, and PL/I calling statements:
Fortran CALL NAME-1 | NAME-2 | ... | NAME-n (arg-1, arg-2, ... ,arg-m, ... )
C and C++ name-1 | name-2 | ... | name-n (arg-1, ... ,arg-m);
PL/I CALL NAME-1 | NAME-2 | ... | NAME-n (arg-1, arg-2, ... ,arg-m, ... );

The syntax indicates:

  • The programming language (Fortran, C, C++, or PL/I)

  • Each possible subroutine name that you can code in the calling sequence. Each name is separated by the | (or) symbol. You specify only one of these names in your calling sequence. (You do not code the | in the calling sequence.)

  • The arguments, listed in the order in which you code them in the calling sequence. You must code them all in your calling sequence.

    You can distinguish between input arguments and output arguments by looking at the "On Entry" and "On Return" sections, respectively. An argument used for both input and output is described in both the "On Entry" and "On Return" sections. In this case, the input value for the argument is overlaid with the output value.

    The names of the arguments give an indication of the type of data that you should specify for the argument; for example:

    • Names beginning with the letters i through n, such as m, incx, iopt, and isign, indicate that you specify integer data.

    • Names beginning with the letters a through h and o through z, such as b, t, alpha, sigma, and omega, indicate that you specify real or complex data.

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On Entry

This lists the input arguments, which are the arguments you pass to the ESSL subroutine. Each argument description first gives the meaning of the argument, and then gives the form of data required for the argument. (To help you avoid errors, output arguments are also listed, along with a reference to the 'On Return' section.)

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On Return

This lists the output arguments, which are the arguments passed back to your program from the ESSL subroutine. Each argument description first gives the meaning of the argument, and then gives the form of data passed back to your program for the argument.

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Notes

The notes describe any programming considerations and restrictions that apply to the arguments or the data for the arguments. There may be references to other parts of the book for further information.

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Function

This is a functional, or mathematical, description of the function performed by this subroutine. It explains what computation is performed, not the implementation. It explains the variations in the computation depending on the input arguments. References are made, where appropriate, to mathematical background books listed in the bibliography. References appear as a number enclosed in square brackets, where the number refers to the item listed under that number in the bibliography. For example, reference [1] cites the first item listed.

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Special Usage

These are unique ways you can use the subroutine in your application. In most cases, this book does not address applications of the ESSL subroutines; however, in special situations where the functional capability of the subroutine can be extended by following certain rules for its use, these rules are described in this section.

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Error Conditions

These are all the ESSL run-time errors that can occur in the subroutine. They are organized under three headings; Computational Errors, Input-Argument Errors, and Resource Errors. The return code values resulting from these errors are also explained.

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Examples

The examples show how you would call the subroutine from a Fortran program. They show a variety of uses of the subroutine. Except where it is important to show differences in use between the various versions of the subroutine, the simplest version of the subroutine is used in the examples. In most cases, this is the short-precision real version of the subroutine. Each example provides a description of the important features of the example, followed by the Fortran calling sequence, the input data, and the resulting output data.

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What's New for ESSL for AIX

This section summarizes all the changes made to ESSL for AIX.

What's New for ESSL Version 3 Release 1.1

  • The ESSL POWER Library, the ESSL Thread-Safe Library, and the ESSL SMP Library are tuned for the RS/6000 POWER3.

  • The ESSL POWER Library, the ESSL Thread-Safe Library, and the ESSL Symmetric Multi-Processing (SMP) Library now support both 32-bit environment and 64-bit environment applications. For details on creating 64-bit environment applications see "Coding Your Program" and "Processing Your Program". If you are migrating to a 64-bit environment, you may need to make changes to your calls to ERRSET. For details see ERRSET--ESSL ERRSET Subroutine for ESSL.

  • ESSL for AIX provides distinct libraries for AIX 4.2.1 and AIX 4.3.2:

    • The AIX 4.2.1 ESSL Thread-Safe Library, the ESSL Thread-Safe POWER2 Library, and the ESSL SMP Library were built using the pthreads draft 7 library supplied on AIX 4.2.1. This is the same as ESSL Version 3.1.

    • The AIX 4.3.2 ESSL Thread-Safe Library, the ESSL Thread-Safe POWER2 Library, and the ESSL SMP Library were built using the pthreads library that conforms to the IEEE POSIX 1003.1-1996 specification supplied on AIX 4.3.

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Guide and Reference

Changes for ESSL Version 3

  • ESSL for AIX provides two new run-time libraries:

    • The ESSL Symmetric Multi-Processing (SMP) Library provides thread-safe versions of the ESSL subroutines for use on the RS/6000 SMP (for example 604e) processors. In addition, a subset of these subroutines are also multithreaded versions; that is, they support the shared memory parallel processing programming model. You do not have to change your existing application programs that call ESSL to take advantage of the increased performance of using the SMP processors. You can simply re-link your existing programs. For a list of the multithreaded subroutines in the ESSL SMP Library, see Table 21.

    • The ESSL Thread-Safe Library provides thread-safe versions of the ESSL subroutines for use on all RS/6000 processors. You may use this library to develop your own multithreaded applications.

      If your existing application program calls ESSL, you only need to re-link your program to take advantage of the increased performance of the ESSL SMP Library or to use the ESSL Thread-Safe Library.

  • ESSL provides new subroutines (_GETRF and _GETRS), bringing the total number of subroutines to 458.

  • For those ESSL subroutines that require extra working storage to perform computations, ESSL now provides a way to dynamically allocate storage when it does not need to persist after the subroutine call. See "Using Auxiliary Storage in ESSL".

  • The files for the Hypertext Markup Language (HTML) version of the ESSL Version 3 Guide and Reference are packaged with the ESSL product.

  • All the ESSL messages are provided in an ESSL message catalog.

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Guide and Reference

Future Migration

If you are concerned with migration to possible future releases of ESSL or possible future hardware, you should read "Planning for Future Migration". That section explains what you can do now to prevent future migration problems.

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Guide and Reference

In Brief--What's Provided in ESSL for AIX

  • ESSL provides five run-time libraries:
    • The ESSL SMP Library provides thread-safe versions of the ESSL subroutines for use on the RS/6000 SMP (for example, 604e or 630) processors. In addition, a subset of these subroutines are also multithreaded versions; that is, they support the shared memory parallel processing programming model. You do not have to change your existing application programs that call ESSL to take advantage of the increased performance of using the SMP processors. You can simply re-link your existing application programs. For a list of the multithreaded subroutines in the ESSL SMP Library, see Table 21.
    • The ESSL Thread-Safe Library provides thread-safe versions of the ESSL subroutines for use on all RS/6000 processors. You may use this library to develop your own multithreaded applications.
    • The ESSL Thread-Safe POWER2 Library provides thread-safe versions of the ESSL subroutines and is tuned for the RS/6000 POWER2 uniprocessors. You may use this library to develop your own multithreaded applications.
    • The ESSL POWER Library is tuned for the RS/6000 POWER, POWER3, and PowerPC uniprocessors.
    • The ESSL POWER2 Library is tuned for the RS/6000 POWER2 uniprocessors.

    All libraries are designed to provide high levels of performance for numerically intensive computing jobs on these respective processors. All versions provide mathematically equivalent results.

    The ESSL POWER Library, the ESSL Thread-Safe Library, and the ESSL SMP Library support both 32-bit environment and 64-bit environment applications.

  • Callable from Fortran, C, C++, and PL/I programs.

  • For a list of subroutines, refer to "List of ESSL Subroutines".

Guide Information

This part of the book is organized into seven chapters, providing guidance information on how to use ESSL. It is organized as follows:

  • Learning about ESSL
  • Designing your program
  • Setting up your data structures
  • Coding your program
  • Processing your program
  • Migrating your programs
  • Handling problems

Introduction and Requirements

This chapter introduces you to the Engineering and Scientific Subroutine Library (ESSL) for Advanced Interactive Executive (AIX*).

Overview of ESSL

This section gives an overview of the ESSL capabilities and requirements.

ESSL is a state-of-the-art collection of subroutines providing a wide range of mathematical functions for many different scientific and engineering applications. Its primary characteristics are performance, functional capability, and usability.

Performance and Functional Capability

The mathematical subroutines, in nine computational areas, are tuned for performance on the RS/6000*. The computational areas are:

  • Linear Algebra Subprograms
  • Matrix Operations
  • Linear Algebraic Equations
  • Eigensystem Analysis
  • Fourier Transforms, Convolutions and Correlations, and Related Computations
  • Sorting and Searching
  • Interpolation
  • Numerical Quadrature
  • Random Number Generation

ESSL provides five run-time libraries:

  • The ESSL Symmetric Multi-Processing (SMP) Library provides thread-safe versions of the ESSL subroutines for use on the RS/6000 SMP (for example, 604e or 630) processors. In addition, a subset of these subroutines are also multithreaded versions; that is, they support the shared memory parallel processing programming model. For a list of the multithreaded subroutines in the ESSL SMP Library, see Table 21.
  • The ESSL Thread-Safe Library provides thread-safe versions of the ESSL subroutines for use on all RS/6000 processors. You may use this library to develop your own multithreaded applications.
  • The ESSL Thread-Safe POWER2 Library provides thread-safe versions of the ESSL subroutines and is tuned for the RS/6000 POWER2 processors. You may use this library to develop your own multithreaded applications.
  • The ESSL POWER Library is tuned for the RS/6000 POWER, POWER3, and PowerPC uniprocessors.
  • The ESSL POWER2 Library is tuned for the RS/6000 POWER2 uniprocessors.

All libraries are designed to provide high levels of performance for numerically intensive computing jobs on these respective processors. All versions provide mathematically equivalent results.

The ESSL POWER Library, the ESSL Thread-Safe Library, and the ESSL SMP Library support both 32-bit environment and 64-bit environment applications.

The ESSL subroutines can be called from application programs written in Fortran, C, C++, and Programming Language/I (PL/I). ESSL runs under the AIX operating system.

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Guide and Reference

Usability

ESSL is designed for usability:

  • It has an easy-to-use call interface.

  • If your existing application programs call ESSL, you only need to re-link your program to take advantage of the increased performance of the ESSL SMP Library or to use the ESSL Thread-Safe Library.

  • It supports a 64-bit environment.

    64-bit applications can be created on any AIX 4.3.2 system, but can only run on 64-bit hardware.

    The data model used for a 64-bit environment is referred to as LP64. This data model supports 32-bit integers and 64-bit pointers. In accordance with the LP64 data model, all ESSL integer arguments remain 32-bits except for the iusadr argument for ERRSET. See ERRSET--ESSL ERRSET Subroutine for ESSL.

  • It has informative error-handling capabilities, enabling you to calculate auxiliary storage sizes and transform lengths.

  • An online book that can be displayed using an Hypertext Markup Language (HTML) document browser, is available for use with ESSL.

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Guide and Reference

The Variety of Mathematical Functions

This section describes the mathematical functions included in ESSL.

Areas of Application

ESSL provides a variety of mathematical functions for many different types of scientific and engineering applications. Some of the industries using these applications are: Aerospace, Automotive, Electronics, Petroleum, Finance, Utilities, and Research. Examples of applications in these industries are:

Structural Analysis

Time Series Analysis

Computational Chemistry

Computational Techniques

Fluid Dynamics Analysis

Mathematical Analysis

Seismic Analysis

Dynamic Systems Simulation

Reservoir Modeling

Nuclear Engineering

Quantitative Analysis

Electronic Circuit Design

What ESSL Provides

The subroutines provided in ESSL, summarized in Table 1, fall into the following groups:

  • Nine major areas of mathematical computation, providing the computations commonly used by the industry applications listed above

  • Utilities, performing general-purpose functions

To help you select the ESSL subroutines that fulfill your needs for performance, accuracy, storage, and so forth, see "Selecting an ESSL Subroutine".

Table 1. Summary of ESSL Subroutines
ESSL Area of Computation Integer Subroutines Short- Precision Subroutines Long- Precision Subroutines
Linear Algebra Subprograms:

  • Vector-scalar

  • Sparse vector-scalar

  • Matrix-vector

  • Sparse matrix-vector

 

0

0

0

0

 

41

11

32

0

 

41

11

32

3

Matrix Operations: Addition, subtraction, multiplications, rank-k updates, rank-2k updates, and matrix transposes 0 25 26
Linear Algebraic Equations:

  • Dense linear algebraic equations

  • Banded linear algebraic equations

  • Sparse linear algebraic equations

  • Linear least squares

 

0

0

0

0

 

30

18

0

3

 

32

18

11

3

Eigensystem Analysis: Solutions to the algebraic eigensystem analysis problem and the generalized eigensystem analysis problem 0 8 8
(Signal Processing) Computations:

  • Fourier transforms

  • Convolutions and correlations

  • Related computations

 

0

0

0

 

15

10

6

 

11

2

6

Sorting and Searching: Sorting, sorting with index, and binary and sequential searching 5 5 5
Interpolation: Polynomial and cubic spline interpolation 0 4 4
Numerical Quadrature: Numerical quadrature on a set of points or on a function 0 6 6
Random Number Generation: Generating vectors of uniformly distributed and normally distributed random numbers 0 3 3
Utilities: General service operations 11 0 3
Total ESSL Subroutines 16 217 225

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Guide and Reference

ESSL--Processing Capabilities

ESSL provides five run-time libraries, the ESSL SMP Library, the ESSL Thread-Safe Library, the ESSL Thread-Safe POWER2 Library, the ESSL POWER2 Library, and the ESSL POWER Library. These libraries are designed to provide high levels of performance for numerically intensive computing jobs on the RS/6000 processors. To order the IBM Engineering and Scientific Subroutine Library for AIX, Version 3 Release 1.1 , specify program number 5765-C42. Most of the subroutine calls are compatible with those in the ESSL/370 product. See the Engineering and Scientific Subroutine Library Version 2.2 Guide and Reference for details.

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Guide and Reference

Accuracy of the Computations

ESSL provides accuracy comparable to libraries using equivalent algorithms with identical precision formats. Both short- and long-precision real versions of the subroutines are provided in most areas of ESSL. In some areas, short- and long-precision complex versions are also provided, and, occasionally, an integer version is provided. The data types operated on by the short-precision, long-precision, and integer versions of the subroutines are RS/6000 architecture precisions: ANSI/IEEE 32-bit and 64-bit binary floating-point format, and 32-bit integer. See the ANSI/IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985, for more detail. (There are ESSL-specific rules that apply to the results of computations on workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".)

For more information on accuracy, see "Getting the Best Accuracy".

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Guide and Reference

The Fortran Language Interface to the Subroutines

The ESSL subroutines follow standard Fortran calling conventions and must run in the Fortran run-time environment. When ESSL subroutines are called from a program in a language other than Fortran, such as C, C++, or PL/I, the Fortran conventions must be used. This applies to all aspects of the interface, such as the linkage conventions and the data conventions. For example, array ordering must be consistent with Fortran array ordering techniques. Data and linkage conventions for each language are given in "Coding Your Program".

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Guide and Reference

Software and Hardware Products That Can Be Used with ESSL

This section describes the hardware and software products you can use with ESSL, as well as those products for installing ESSL and displaying the online documentation.

For ESSL--Hardware

ESSL runs on the IBM* RS/6000 processors supported by the AIX operating systems.

64-bit applications require 64-bit hardware.

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Guide and Reference

ESSL--Operating Systems

ESSL is supported in the following operating system environments:

  • AIX Version 4.2.1 or later modification levels (either product number 5765-655 or 5765-C34)

  • AIX Version 4.3.2 or later modification levels (product number 5765-C34)

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Guide and Reference

ESSL--Software Products

ESSL requires the software products shown in Table 2 for compiling and running.

To assist C and C++ users, an ESSL header file is provided. Use of this file is described in "C Programs" and "C++ Programs".

Table 2. Software Products Required for Use with ESSL
For Compiling For Linking, Loading, or Running
XL Fortran for AIX, Version 5.1.1 or later (program number 5808-AAR part number 04L2110) -or-

XL High Performance Fortran for AIX, Version 1.3.1 or later (program number 5765-613) -or-

IBM C, C++ compilers Version 3.6.4 or later 2 -or-

C for AIX, Version 4.3 or later (program number 5765-AAR part number 04L0675 with feature number 2163) -or-

PL/I Set for AIX, Version 1.1 or later (program number 5765-549)

XL Fortran Run-Time Environment for AIX, Version 5.1.1 or later (program number 5808-AAR part number 04L2123) -or-

XL High Performance Run-Time Environment for AIX, Version 1.3.1 or later (program number 5765-612) -and-

C libraries1

1 The AIX Version 4 product includes the C and math libraries in the Application Development Toolkit.

2 Available as a component of the VisualAge C++ Professional for AIX, Version 4, product.

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Guide and Reference

Installation and Customization Products

The ESSL licensed program is distributed on a 4-millimeter or an 8-millimeter cartridge. The ESSL Version 3 Release 1.1 Installation Memo provides the detailed information you need to install ESSL.

The ESSL product is packaged according to the AIX guidelines, as described in the IBM AIX Version 4 General Programming Concepts: Writing and Debugging Programs manual. The product can be installed using the smit command, as described in the IBM AIX Version 4 System Management Guide: Operating System and Devices manual.

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Guide and Reference

Software Products for Displaying ESSL Online Information

The ESSL Guide and Reference Version 3 is available in PostScript and HTML format on the product media.

To view the online publications shipped on the product media, you need the following:

  • Access to a common HTML document browser (such as Netscape Navigator).

  • The location of the HTML index file provided with the file sets. Contact your system administrator or installer for this location.

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ESSL Internet Resources

This section describes how you can use the ESSL resources available over the Internet.

Obtaining Documentation

The ESSL Version 3 Release 1.1 Guide and Reference is available in PDF and HTML format at the IBM RS/6000 Web site at:

http://www.rs6000.ibm.com/resource/aix_resource/sp_books

To view the ESSL PDF publication, you need access to the Adobe Acrobat Reader 3.0.1. The Acrobat Reader is shipped with the AIX Version 4.3 Bonus Pack and is also freely available for downloading from the Adobe web site at:

http://www.adobe.com.

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Guide and Reference

Accessing ESSL's Product Home Pages

The following home pages contain information on ESSL and Parallel ESSL:

  • For ESSL for AIX, use:

    http://www.rs6000.ibm.com/software/Apps/essl.html

  • For Parallel ESSL, use:

    http://www.rs6000.ibm.com/software/sp_products/esslpara.html

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Getting on the ESSL Mailing List

Information concerning ESSL's home pages and other home pages available for the RS/6000 family of products, plus late breaking information about ESSL, can be obtained by being placed on the ESSL mailing list. In addition, users on the mailing list will receive information about new ESSL function and may receive customer satisfaction surveys and requirements surveys, to provide feedback to ESSL Development on the product and user requirements.

You can be placed on the mailing list by sending a request to either of the following, asking to be placed on the ESSL mailing list:

International Business Machines Corporation
ESSL Development
Department LQJA / MS P963
522 South Rd.
Poughkeepsie, N.Y. 12601-5400
 
e-mail: essl@us.ibm.com

Note:You should send us e-mail if you would like to be withdrawn from the ESSL mailing list.

When requesting to be placed on the mailing list or asking any questions, please provide the following information:

  • Your name
  • The name of your company
  • Your mailing address
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Guide and Reference

Matrix Operations

Some of the matrix operation subroutines were designed in accordance with the Level 3 BLAS de facto standard. If these subroutines do not comply with the standard as approved, IBM will consider updating them to do so. If IBM updates these subroutines, the updates could require modifications of the calling application program. For details on the Level 3 BLAS, see reference [32]. The matrix operation subroutines also include the commonly used matrix operations: addition, subtraction, multiplication, and transposition.

Table 7. List of Matrix Operation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Matrix Addition for General Matrices or Their Transposes 
SGEADD
CGEADD


DGEADD
ZGEADD

SGEADD, DGEADD, CGEADD, and ZGEADD--Matrix Addition for General Matrices or Their Transposes
Matrix Subtraction for General Matrices or Their Transposes 
SGESUB
CGESUB


DGESUB
ZGESUB

SGESUB, DGESUB, CGESUB, and ZGESUB--Matrix Subtraction for General Matrices or Their Transposes
Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes 
SGEMUL
CGEMUL


DGEMUL
ZGEMUL
DGEMLP§

SGEMUL, DGEMUL, CGEMUL, and ZGEMUL--Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes
Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes Using Winograd's Variation of Strassen's Algorithm 
SGEMMS
CGEMMS


DGEMMS
ZGEMMS

SGEMMS, DGEMMS, CGEMMS, and ZGEMMS--Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes Using Winograd's Variation of Strassen's Algorithm
Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes 
SGEMM¢
CGEMM¢


DGEMM¢
ZGEMM¢

SGEMM, DGEMM, CGEMM, and ZGEMM--Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes
Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian 
SSYMM¢
CSYMM¢
CHEMM¢


DSYMM¢
ZSYMM¢
ZHEMM¢

SSYMM, DSYMM, CSYMM, ZSYMM, CHEMM, and ZHEMM--Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian
Triangular Matrix-Matrix Product 
STRMM¢
CTRMM¢


DTRMM¢
ZTRMM¢

STRMM, DTRMM, CTRMM, and ZTRMM--Triangular Matrix-Matrix Product
Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix 
SSYRK¢
CSYRK¢
CHERK¢


DSYRK¢
ZSYRK¢
ZHERK¢

SSYRK, DSYRK, CSYRK, ZSYRK, CHERK, and ZHERK--Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix 
SSYR2K¢
CSYR2K¢
CHER2K¢


DSYR2K¢
ZSYR2K¢
ZHER2K¢

SSYR2K, DSYR2K, CSYR2K, ZSYR2K, CHER2K, and ZHER2K--Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
General Matrix Transpose (In-Place) 
SGETMI
CGETMI


DGETMI
ZGETMI

SGETMI, DGETMI, CGETMI, and ZGETMI--General Matrix Transpose (In-Place)
General Matrix Transpose (Out-of-Place) 
SGETMO
CGETMO


DGETMO
ZGETMO

SGETMO, DGETMO, CGETMO, and ZGETMO--General Matrix Transpose (Out-of-Place)

¢ Level 3 BLAS

§ This subroutine is provided only for migration from earlier release of ESSL and is not intended for use in new programs.

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Guide and Reference

Sorting and Searching

The sorting and searching subroutines operate on three types of data: integer, short-precision real, and long-precision real (Table 16). The sorting subroutines perform sorts with or without index designations. The searching subroutines perform either a binary or sequential search.

Table 16. List of Sorting and Searching Subroutines
Descriptive Name Integer Subroutine Short- Precision Subroutine Long- Precision Subroutine Page
Sort the Elements of a Sequence ISORT SSORT DSORT ISORT, SSORT, and DSORT--Sort the Elements of a Sequence
Sort the Elements of a Sequence and Note the Original Element Positions ISORTX SSORTX DSORTX ISORTX, SSORTX, and DSORTX--Sort the Elements of a Sequence and Note the Original Element Positions
Sort the Elements of a Sequence Using a Stable Sort and Note the Original Element Positions ISORTS SSORTS DSORTS ISORTS, SSORTS, and DSORTS--Sort the Elements of a Sequence Using a Stable Sort and Note the Original Element Positions
Binary Search for Elements of a Sequence X in a Sorted Sequence Y IBSRCH SBSRCH DBSRCH IBSRCH, SBSRCH, and DBSRCH--Binary Search for Elements of a Sequence X in a Sorted Sequence Y
Sequential Search for Elements of a Sequence X in the Sequence Y ISSRCH SSSRCH DSSRCH ISSRCH, SSSRCH, and DSSRCH--Sequential Search for Elements of a Sequence X in the Sequence Y

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Guide and Reference

Interpolation

The interpolation subroutines provide the capabilities of doing polynomial interpolation, local polynomial interpolation, and both one- and two-dimensional cubic spline interpolation (Table 17).

Table 17. List of Interpolation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Polynomial Interpolation SPINT DPINT SPINT and DPINT--Polynomial Interpolation
Local Polynomial Interpolation STPINT DTPINT STPINT and DTPINT--Local Polynomial Interpolation
Cubic Spline Interpolation SCSINT DCSINT SCSINT and DCSINT--Cubic Spline Interpolation
Two-Dimensional Cubic Spline Interpolation SCSIN2 DCSIN2 SCSIN2 and DCSIN2--Two-Dimensional Cubic Spline Interpolation

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Guide and Reference

Random Number Generation

Random number generation subroutines generate uniformly distributed random numbers or normally distributed random numbers (Table 19).

Table 19. List of Random Number Generation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Generate a Vector of Uniformly Distributed Random Numbers SURAND DURAND SURAND and DURAND--Generate a Vector of Uniformly Distributed Random Numbers
Generate a Vector of Normally Distributed Random Numbers SNRAND DNRAND SNRAND and DNRAND--Generate a Vector of Normally Distributed Random Numbers
Generate a Vector of Long Period Uniformly Distributed Random Numbers SURXOR DURXOR SURXOR and DURXOR--Generate a Vector of Long Period Uniformly Distributed Random Numbers

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] =l[ l꤭`~SZj9H6H6H6 ./usr/lpp/essl.html.en_US/html/essl057.htmlen_U   "T<=>ADEILRSeort /FGHNOPailns-3BMVbcdfhmpu#.0129CUgv&,457:;X_wxy!)68Kk(QYz`C?OR9o 0G0|4=aC[sP5m,Fز>_8BD(d[K/ֵkZֵkZֵkZֵkZֵkZ.-;{Pk}3b8Bp?@` H;S.-;QACFmаB:yа M"\[Gߧ& >C[JO6!=n`D~t,44аH((-#!XEK{CC⇷@+Q аH((-#!)/=2t0V?:a8P$P#[ >/-=pr3FM4M ;k@Z(G9 Xd[K/ֵkZֵkZֵkZֵkZֵkZֵ$Kc{PbI[x[* jkXeنؓa}%}"kzˮPP`I,T# AR˰ǹ)űE( aql~oJA1-2-@(0P_9=űE( 0Ǹ?PxȷXB1-;[JET?W*;+R#| -l&l>D_rILrnGCd*;+1atqltZ4hql7k#o GYqPT;<3G ޜMѯA-FsTv.W*-bgyӉ57rIX繆}ϣg_9zq7Fr9:58XMg_9zq7Fr99[]uE5B5td{b,F&TOMb RrQk0Ǹs 9Gφr9o|r E.tkYiJnԨ\TZFֳ-|φr9o|r E.tiJil3|57k#oz)s>5eTPhXI,T PҕxtiJh1g*RrQkicZ͆=Űσg8|57k#oz)sZcJV?*;+65a3ᯜA8_9{KRgk#oN&hG R1}"kzˮY&A+F|HֳҕJEok6>aG3G ޜMѯA΍k=)_Ԩ\TZFֳ-|φr9o|r E.tiJ a3ᯜA8_9{K.E5B)_:4~DƵƔjTv.W*-m#|lkYǸs 9Gφr9o|r E.um+hu@=#|nI[cZ@O5a<a3ᯜA8_9{K!}IE? P[ 9 |r ފ\.a0ϣqlk#oN&hG RVܒ,B4mJ[ Kb O7Ƒ7$f`@Vzj'yl3|57k#oz)sVT='eCql3|57k#oz)s`9|>gųᯜA8_9{K f&ܒ6WnGcZdMUl@s>Ӊ57ҍk2gk#oN&hG R1}"kzˮ0I,T PF:Qf$L-`[rJ;]k=5U>aG3G ޜMѯA΅lQknI[k(B55JHLM5u+`M5T[jR"i+- Ԩ\TZ˖>g_9zq7Fr9:IB%σG ޜMѯAcD]QApY&AJW(DH[65䕶„-STʄTX}E,=B*XrRk.F'*;+1-|>gųᯜA8_9{Kum{R{hG'𘅄o&S]{1<@v7b,eXs>Ӊ57Э*0s>Ӊ57~H޲(4 %j  ң9Э*0H#lgSڔO F'(8=o,$pӉ57~H޲(4K$>>D98 $IƧYka5vщ umdk,XrPQ1 Q+Mb.EEl&Km@pz-mo-.B1-|>gųᯜA8_9{KmA8P~Ԣx^1<@pClme &!uTJ%s@r*+a2[jQkms|&!mr-|φr9o|r E.t*}-|φr9o|r E.s& BId‡ 7t*}$SڃɓpDby€.]ac E.P.lLB&JSXQ[ P[k1 hː {a0ϣqlk#oN&hG RFuI7sbNeCp{)T='oQ}\\TZڢSPiI!8=xUK`Y[&ۑֳ!/_:ٲ,CHU{jjL\C \'aqIl.do%ʎE/_UƔT!b%]>aG0xű'09 k&>$rr:bavcD]P8K$%rIY= []uB(I,Wԓ {k&5B>Lrۑ뭛-cݠ@0+X>5eT  %j !X,Β,avE`il~o A$1- F 07 nbsp; k\b~SZjH6H6H6./usr/lpp/essl.html.en_US/html/essl058.htmlen_U Guide and Reference

Guide and Reference

Planning Your Program

This chapter provides information about ESSL that you need when planning your program. Its purpose is to help you in performing the following tasks:

  • Selecting an ESSL subroutine
  • Avoiding Conflicts with Internal ESSL Routine Names That are Exported
  • Setting up your data
  • Setting up your ESSL calling sequences
  • Using auxiliary storage in ESSL
  • Providing a correct transform length to ESSL
  • Getting the best accuracy
  • Getting the best performance
  • Dealing with errors when using ESSL

If you are using ESSL with PL/I Set for AIX, Version 1, see the PL/I publications for details on calling subroutines and functions.

Selecting an ESSL Subroutine

Your choice of which ESSL subroutine to use is based mainly on the functional needs of your program. However, you have a choice of several variations of many of the subroutines. In addition, there are instances where certain subroutines cannot be used. This section describes these variations and limitations. See the answers to each question below that applies to you.

Which ESSL Library Do You Want to Use?

ESSL provides five run-time libraries:

  • The ESSL SMP Library provides thread-safe versions of the ESSL subroutines for use on the RS/6000 SMP processors. In addition, a subset of these subroutines are also multithreaded versions; that is, they support the shared memory parallel processing programming model. For a list of the multithreaded subroutines in the ESSL SMP Library, see Table 21.

  • The ESSL Thread-Safe Library provides thread-safe versions of the ESSL subroutines for use on all RS/6000 processors. You may choose to use this library if you decide to develop your own multithreaded programs that call the thread-safe ESSL subroutines.

    The number of threads you choose to use depends on the problem size, the specific subroutine being called, and the number of physical processors you are running on. To achieve optimal performance, experimentation is necessary; however, picking the number of threads equal to the number of online processors generally provides good performance in most cases. In a few cases, performance may increase if you choose the number of threads to be less than the number of online processors. For more information about thread concepts, see IBM AIX Version 4 General Programming Concepts: Writing and Debugging Programs.

  • The ESSL Thread-Safe POWER2 Library provides thread-safe versions of the ESSL subroutines and is tuned for the RS/6000 POWER2 uniprocessors. You may use this library to develop your own multithreaded applications.

  • The ESSL POWER2 Library is tuned for the RS/6000 POWER2 uniprocessors.

  • The ESSL POWER Library is tuned for the RS/6000 POWER, POWER3, and PowerPC uniprocessors.

The ESSL POWER Library, the ESSL Thread-Safe Library, and the ESSL SMP Library supports both 32-bit environment and 64-bit environment applications. For details see "Coding Your Program" and "Processing Your Program".


Table 21. Multithreaded ESSL SMP Subroutines
Subroutine Names 
Vector-Scalar Linear Algebra Subprograms:
SASUM, DASUM, SCASUM, DZASUM
SAXPY, DAXPY, CAXPY, ZAXPY
SCOPY, DCOPY, CCOPY, ZCOPY
SDOT, DDOT, CDOTU, ZDOTU, CDOTC, ZDOTC
SNDOT, DNDOT
SNORM2, DNORM2, CNORM2, ZNORM2
SROT, DROT, CROT, ZROT, CSROT, ZDROT
SSCAL, DSCAL, CSCAL, ZSCAL, CSSCAL, ZDSCAL
SSWAP, DSWAP, CSWAP, ZSWAP
SVEA, DVEA, CVEA, ZVEA
SVES, DVES, CVES, ZVES
SVEM, DVEM, CVEM, ZVEM
SYAX, DYAX, CYAX, ZYAX, CSYAX, ZDYAX
SZAXPY, DZAXPY, CZAXPY, ZZAXPY


Matrix-Vector Linear Algebra Subprograms:
SGEMV, DGEMV, CGEMV, ZGEMV
SGER, DGER, CGERU, ZGERU, CGERC, ZGERC
SSPMV, DSPMV, CHPMV, ZHPMV
SSYMV, DSYMV, CHEMV, ZHEMV
SSPR, DSPR, CHPR, ZHPR
SSYR, DSYR, CHER, ZHER
SSPR2, DSPR2, CHPR2, ZHPR2
SSYR2, DSYR2, CHER2, ZHER2
SGBMV¢, DGBMV¢
CGBMV¢, ZGBMV¢
SSBMV¢, DSBMV¢
CHBMV¢, ZHBMV¢
STRMV, DTRMV, CTRMV, ZTRMV
STPMV, DTPMV, CTPMV, ZTPMV
STBMV¢, DTBMV¢
CTBMV¢, ZTBMV¢


Matrix Operations:
SGEADD, DGEADD, CGEADD, ZGEADD
SGESUB, DGESUB, CGESUB, ZGESUB
DGEMUL
DGEMM, ZGEMM
DSYMM, ZSYMM, ZHEMM
DTRMM, ZTRMM
DSYRK, ZSYRK, ZHERK
DSYR2K, ZSYR2K, ZHER2K
SGETMI, DGETMI, CGETMI, ZGETMI
SGETMO, DGETMO, CGETMO, ZGETMO


Dense Linear Algebraic Equations:
SGEF, DGEF, CGEF, ZGEF
SGETRF, DGETRF, CGETRF, ZGETRF
DPOF
STRSV, DTRSV, CTRSV, ZTRSV
STPSV, DTPSV, CTPSV, ZTPSV
DTRSM, ZTRSM
STRI, DTRI


Sparse Linear Algebraic Equations:
DSRIS&


Fourier Transforms:
SCFT, DCFT
SRCFT, DRCFT
SCRFT, DCRFT
SCFT2, DCFT2
SRCFT2, DRCFT2
SCRFT2, DCRFT2
SCFT3, DCFT3
SRCFT3, DRCFT3
DCRFT3, DCRFT3


Convolution and Correlation:
SCOND, SCORD
SDCON, SDCOR, DDCON, DDCOR

Many of the dense linear algebraic equations and eigensystem analysis subroutines make one or more calls to the multithreaded versions of the matrix-vector linear algebra and matrix operation subroutines shown in this table. SCOSF, DCOSF, SSINF, and DSINF make one or more calls to the multithreaded versions of the Fourier Transform subroutines shown in this table. These subroutines benefit from the increased performance of the multithreaded versions of the ESSL SMP subroutines.

& DSRIS only uses multiple threads when IPARM(4) = 1 or 2.

¢ The Level 2 Banded BLAS use multiple threads only when the bandwidth is sufficiently large.

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What Type of Data Are You Processing in Your Program?

The version of the ESSL subroutine you select should agree with the data you are using. ESSL provides a short- and long-precision version of most of its subroutines processing short- and long-precision data, respectively. In a few cases, it also provides an integer version processing integer data or returning just integer data. The subroutine names are distinguished by a one- or two-letter prefix based on the following letters:

S for short-precision real
D for long-precision real
C for short-precision complex
Z for long-precision complex
I for integer

The precision of your data affects the accuracy of your results. This is discussed in "Getting the Best Accuracy". For a description of these data types, see "How Do You Set Up Your Scalar Data?".

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Guide and Reference

How Is Your Data Structured? And What Storage Technique Are You Using?

Some subroutines process specific data structures, such as sparse vectors and matrices or dense and banded matrices. In addition, these data structures can be stored using various storage techniques. You should select the proper subroutine on the basis of the type of data structure you have and the storage technique you want to use. If possible, you should use a storage technique that conserves storage and potentially improves performance. For more about storage techniques, see "Setting Up Your Data".

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What about Performance and Accuracy?

ESSL provides variations among some of its subroutines. You should consider performance and accuracy when deciding which subroutine is the best to use. Study the "Function" section in each subroutine description. It helps you understand exactly what each subroutine does, and helps you determine which subroutine is best for you. For example, some subroutines perform multiple computations of a certain type. This might give you better performance than a subroutine that does each computation individually. In other cases, one subroutine may do scaling while another does not. If scaling is not necessary for your data, you get better performance by using the subroutine without scaling.

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Avoiding Conflicts with Internal ESSL Routine Names That are Exported

Do not use names for your own subroutines, functions, and global variables that are the same as the ESSL exported names. All internal ESSL routine names that are exported begin with the ESV prefix, so you should avoid using this prefix for your own names.

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Setting Up Your Data

This section explains how to set up your scalar and array data and points you to where you can find more detail.

How Do You Set Up Your Scalar Data?

A scalar item is a single item of data, whether it is a constant, a variable, or an element of an array. ESSL assumes that your scalar data conforms to the appropriate standards, as described below. The scalar data types and how you should code them for each programming language are listed under "Coding Your Scalar Data" in each language section in "Coding Your Program".

Internal Representation

Scalar data passed to ESSL from all types of programs, including Fortran, C, and C++, should conform to the ANSI/IEEE 32-bit and 64-bit binary floating-point format, as described in the ANSI/IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985

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How Do You Set Up Your Arrays?

An array represents an area of storage in your program, containing data stored in a series of locations. An array has a single name. It is made up of one or more pieces of scalar data, all the same type. These are the elements of the array. It can be passed to the ESSL subroutine as input, returned to your program as output, or used for both input and output, in which case the original contents are overwritten.

Arrays can contain conceptual (mathematical) data structures, such as vectors, matrices, or sequences. There are many different types of data structures. Each type of data structure requires a unique arrangement of data in an array and does not necessarily have to include all the elements of the array. In addition, the elements of these data structures are not always contiguous in storage within an array. Stride and leading dimension arguments passed to ESSL subroutines define the separations in array storage for the elements of the vector, matrix, and sequence. All these aspects of data structures are described in "Setting Up Your Data Structures". You must first understand array storage techniques to fully understand the concepts of data structures, stride, and leading dimension, especially if you are using them in unconventional ways.

ESSL subroutines assume that all arrays passed to them are stored using the Fortran array storage techniques (in column-major order), and they process your data accordingly. For details, see "Setting Up Arrays in Fortran". On the other hand, C, C++, and PL/I programs store arrays in row-major order. For details on what you can do, see:

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How Should Your Array Data Be Aligned?

All arrays, regardless of the type of data, should be aligned on a doubleword boundary to ensure optimal performance; however, when running on a POWER2 processor, it is best to align your long-precision arrays on a quadword boundary. For information on how your programming language aligns data, see your programming language manuals.

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What Storage Mode Should You Use for Your Data?

The amount of storage used by arrays and the storage arrangement of data in the arrays can affect overall program performance. As a result, ESSL provides subroutines that operate on different types of data structures, stored using various storage modes. You should chose a storage mode that conserves storage and potentially improves performance. For definitions of the various data structures and their corresponding storage modes, see "Setting Up Your Data Structures". You can also find special storage considerations, where applicable, in the "Notes" section of each subroutine description.

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How Do You Convert from One Storage Mode to Another?

This section describes how you can convert from one storage mode to another.

Conversion Subroutines

ESSL provides several subroutines that help you convert from one storage mode to another:

  • DSRSM is used to migrate your existing program from sparse matrices stored by rows to sparse matrices stored in compressed-matrix storage mode. This converts the matrices into a storage format that is compatible with the input requirements for some ESSL sparse matrix subroutines, such as DSMMX.

  • DGKTRN and DSKTRN are used to convert your sparse matrix from one skyline storage mode to another, if necessary, before calling the subroutines DGKFS/DGKFSP or DSKFS/DSKFSP, respectively.

Sample Programs

In addition, sample programs are provided with many of the storage mode descriptions in "Setting Up Your Data Structures". You can use these sample programs to convert your data to the desired storage mode by adapting them to your application program.

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Setting Up Your ESSL Calling Sequences

This section gives the general rules for setting up the ESSL calling sequences. The information given here applies to all types of programs, running in all environments. For a description and examples of how to code the ESSL calling sequences in your particular programming language, see the following sections:

For details on the conventions used in this book to describe the calling sequence syntax, see "How to Interpret the Subroutine Descriptions". It describes how required and optional arguments are indicated in the calling sequence and the naming conventions used for different data types.

What Is an Input-Output Argument?

Some arguments are used for both input and output. The contents of the input argument are overlaid with the output value(s) on return to your program. Be careful that you save any data you need to preserve before calling the ESSL subroutine.

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What Are the General Rules to Follow when Specifying Data for the Arguments?

You should follow the syntax rules given for each argument in "On Entry" in the subroutine description. Input-argument error messages may be issued, and your program may terminate when you make an error specifying the input arguments. For example:

  • Data passed to ESSL must be of the correct type: integer, character, real, complex, short-precision, or long-precision. There is no conversion of data. Assuming you are using the ESSL header file with your C and C++ programs, you first need to define the following:

  • Character values must be one of the specified values. For example, it may have to be 'N', 'T', or 'C'.

  • Numeric values must fall within the correct range for that argument. For example, a numeric value may need to be greater than or equal to 0, or it may have to be a nonzero value.

  • Arrays must be defined correctly; that is, they must have the correct dimensions, or the dimensions must fall within the correct range. For example, input and output matrices may need to be conformable, or the number of rows in the matrix must be less than or equal to the leading dimension specified. (ESSL assumes all arrays are stored in column-major order.)

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Guide and Reference

What Happens When a Value of 0 Is Specified for N?

For most ESSL subroutines, if you specify 0 for the number of elements to be processed in a vector or the order of a matrix (usually argument n), no computation is performed. After checking for input-argument errors, the subroutine returns immediately and no result is returned. In the other subroutines, an error message may be issued.

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How Do You Specify the Beginning of the Data Structure in the ESSL Calling Sequence?

When you specify a vector, matrix, or sequence in your calling sequence, it does not necessarily have to start at the beginning of the array. It can begin at any point in the array. For example, if you want vector x to start at element 3 in array A, which is declared A(1:12), specify A(3) in your calling sequence for argument x, such as in the following SASUM calling sequence in your Fortran program: 

              N    X   INCX
              |    |     |
   X = SASUM( 4 , A(3) , 2 )

Also, for example, if you want matrix A to start at the second row and third column of array A, which is declared A(0:10,2:8), specify A(1,4) in your calling sequence for argument a, such as in the following SGEADD calling sequence in your Fortran program: 

                   A     LDA  TRANSA  B  LDB TRANSB  C  LDC  M   N
                   |      |     |     |   |    |     |   |   |   |
   CALL  SGEADD( A(1,4) , 11 , 'N'  , B , 4 , 'N'  , C , 4 , 4 , 3 )

For more examples of specifying vectors and matrices, see "Setting Up Your Data Structures".

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ified. l=~~SZjH6H6H6 ./usr/lpp/essl.html.en_US/html/essl072.htmlen_U   "<=>AERTaeinort/DFLNPSlsu2BCGHIMOUV_bcdfghm#&,-.01358;Xpxy479:vw!6KYkq)QW(pySB DxЃ* q:$8u#OAFPbڶ YP} #㎵K^v-[{{{{{AqºYp0;|,@DxQ0"Cɽ]a$Am0;BTJ& Hy7V^Rڥvr%`7  [8:L"@omo,0TT{HgPg"[|PPPW  [8҉p'XфQ#7A8yI{JL|Q8\AX/Z;΋{{{{{1DdD6:0cj/o\j-g8. p*J$$>d>$NH=YeBٵAi€́%>+][B}MU3o|y(.:0qÅAJ$$<@0P5qуHIx`(8J$$<@0P 8J$$<@0P9qуξ[E.WY_wŖmؿ;NH=".$ T8Xx{{ oU T;mѿVlE}w} ]f-b۾qbڹ[-jucMܷw|θ{ϪWVЛuS6ǒ#A?y媩eծjަ7cq+zOmŬ]D[1]Zmk:5\V3TFn{-uEؿDw|%N\J޺o:s3uS6[ZM|:22ck,knq*'fUUo=kg_f䢈e+Op1j󪙷/-ZmLn5f,]e}媪[6[v/]krxU\pgLM`|}C+צgF.WY_wŖy_wnq*N3 Z&ʯ︜g $p Zf\rX# {]ofL}ܮrs/m]^pK.pcD FB *U[qB|Qu|Yo;WĮ>3uS6qу DZ2N u':Q5'!FqDxԞdiB$+*qIԜ= 7#ƤЊm#|ZT͊Eʈ&uSkPrssqԜI(ڌy8:81?R{Qo'GI99 PFZ׊lWUo z:81?R{Qo'GI99 UZHND(99OP PM99 pP8NPGIFHQQ5'Sl(1 ^1X48=!OLbɡO "PzH1ATBPp>H.9BILb AP1&1 ~PdРg BBI#H '()7(I8$i$rFqN qPc:Rpu'::81?R{Qo'GI99 p_&K+mqŵ]8 aDcNr.pq>:81?R{Qo'GI99 %}%&Ri1'9A88IԜ= 7#ƤЊm$|\r?%7lnqg_fYr-zk pPջpN8g\'\I+61}Yr-ulZ/b٫unޫs٫r֬p^3ֶU|Qu|X%88IR"є ҉ =Ba$,ABMlڠaaMxU\|\J޺o:8 HIx`(qуHIx`F tzQ!'y1D 8N[UrZqg_f긕սt'uS6[F3v/&mk:₂YuՋdmT~q1UĜq8 g\[8$'«8q]\ݱUW \O7rު~je+/ej;p:ud٫ 0K:ppNH=ȉB& 'S6l):80(X tx,P,$ͪ'/͜taBNH=YeB&mV+ oS6qч 9 e kfF6Umc7lmgp`P@@ӒmYP XhITqH~ IF ñDyB& 'S6lt`,,pF only us k~SZj~H6H6H6~./usr/lpp/essl.html.en_US/html/essl073.htmlen_U Guide and Reference

Guide and Reference

Setting Up Auxiliary Storage When Dynamic Allocation Is Not Used

You set up the storage area in your program and pass it to ESSL through arguments, specifying the size of the aux work area in the naux argument.

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Who Do You Want to Calculate the Size? You or ESSL?

You have a choice of two methods for determining how much auxiliary storage you should specify:

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How Do You Calculate the Size Using the Formulas?

The formulas provided for calculating naux indicate a sufficient amount of auxiliary storage required, which, in most cases, is larger than the minimum amount, returned by ESSL error handling. There are two types of formulas:

  • Simple formulas

    These are given in the naux argument syntax descriptions. In general, these formulas result in the minimum required value, but, in a few cases, they provide overestimates.

  • Processor-independent formulas

    These are given in separate sections in the subroutine description. In general, these provide overestimates.

Both types of formulas provide values that are sufficient for all processors. As a result, you can migrate to any other processor and to future releases of ESSL without being concerned about having to increase the amount of storage for aux. You do, of course, need to weigh your storage requirements against the convenience of using this larger value.

To calculate the amount of storage using the formulas, you must substitute values for specific variables, such as n, m, n1, or n2. These variables are arguments specified in the ESSL calling sequence or derived from the arguments in the calling sequence.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] R{Q l ~SZj.H6H6H6./usr/lpp/essl.html.en_US/html/essl076.htmlen_U   eor<>ainstu "/=AERTcdghlmp,.01BFHILNPSVbfvy-2CDGMOUXwx#&)345;Y(*789_kqz!6:QW+?Kjp PM.=@s9^%׺t#X2Y,TF,gΙ^o5Or;~_PC8nW\+9 8^ǂkb|e3>g+/{ Rld_5R9._qB@`X 5^kH=^4fb2_o0-p 3b\x9^%!z1}-Uvmnb *o*"ʢ}ƽ7ZWW*{\O-Zg+ z^B1FD5LKڸb.gK[b[rWQfaYߣVa>ር~|ŽΑjs{K {P3mLg/ZWl6bxnW͌+ye=g~[^%mK>\W{{/K֕+sh eoYߣ2gĿ3䕽#F j2Q%1lYܱg+ 5ky$C@@Q㣳2{ y&b3v'-HF^_j2UkU/U(TEǕE7_qF/oj8/byjәoq}^FPWQpB2_sWb#6Z1 u E_YaHgmZZ\Mn/o0?Ym/`Y#F2g"IniIA]:9]/kFz1}QMD\yTS1ksj {%mnXϣl[rbpEZ +so3YJ2_tg׹^WVc+ݫy3[PeF!/F2wٌkxыZ0?,b_óxͻV%kxы3*@Gy&Lvb E7ʢ͆VT;+JeZkHInn0r^u-ZFW0@L|ʲSB 1}"dL߷Jś+V8i8w?⨦TEǕE76 ߆'/F/MրʞsMj8/bF}l`ze _3/F/o͹+/ma\ _*VR gͻV%yԹx<ʟ 6 /mB|[r;wZbu){Ifc%sxua/{In9KrTsj dP#=4 pq& o0?Ym/`Yk^y$C@D(ܠS57p(oqQ0_F ^%QEzBFm6R$IS{ M減NUT mcs&Lg )؃mܧ{g,͊5r n$ɟLgD8[)KnVS\zOq&#H\Mx)GD1$pZ09oB0io~X86|rܯPC:[EL\egF 6͊5Sy& { IE4CJ9@7"B0io~BMրjNoNy|ލDbayJf{"z/q^᩼()q$r(?DY$"Q"@, Q@Þ$p+kn0a\c*ތ\Poλ~ XXR~!`YX2ɥG(+ыF/{|K1nuqkpO$1B&E7qQMZR|B Z|$(0f \)!>|֭n_7 \M#" P$r 9㢽v$WMJܭ#+U0j#c 3S6 ).{ MDFK L)$ܢorOpkS P$@N ۏE w#Ǧ͑`( )4%b4ླྀ_agr Guide and Reference

Guide and Reference

Providing a Correct Transform Length to ESSL

This section describes how to calculate the length of your transform by use of formulas or error-handling capabilities provided in ESSL.

What ESSL Subroutines Require Transform Lengths?

For the ESSL subroutines listed in Table 24, you need to provide one or more transform lengths for the computation of a Fourier transform. These transform lengths are supplied through one or more arguments, such as n, n1, n2, and n3, in the calling sequence for the ESSL subroutine. Only certain lengths of transforms are permitted in the computation.

Table 24. ESSL Subroutines Requiring Transform Lengths
Subroutine Names 
Fourier Transforms:
_CFT  _RCFT  _CRFT  _COSF  _SINF
SCOSFT  _CFT2  _RCFT2  _CRFT2  _CFT3
_RCFT3  _CRFT3  SCFTP  SCFT2P  SCFT3P

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Guide and Reference

Who Do You Want to Calculate the Length? You or ESSL?

You have a choice of two methods for determining an acceptable length for your transform to be processed by ESSL:

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Guide and Reference

How Do You Calculate the Length Using the Table or Formula?

The lengths ESSL accepts for transforms in the Fourier transform subroutines are listed in "Acceptable Lengths for the Transforms". You should use the table in that section to find the two values your length falls between. You then specify the larger length for your transform. If you find a perfect match, you can use that value without having to change it. The formula provided in that section expresses how to calculate the acceptable values listed in the table. If necessary, you can use the formula to dynamically check lengths in your program.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] TO kꠇ~SZjE2 K62 K62 K6E./usr/lpp/essl.html.en_US/html/essl08.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui l~SZjV)H6H6H6./usr/lpp/essl.html.en_US/html/essl080.htmlen_U  er<>Aanost ",/=EILNRTcghilmpu.0123BCDFHPSVbdfy#)-4GMOUXYvw&(*578;x69:_k!QWqz+?Kj =pirKɆQ1M%ȠrKƠJ`Ip#IHM鮢Q }Bl1c 鮢Q6 #[%/+((((((((((((QNPz]C6_3 % /F` <,?,g  &ZcDPh(_8]Zq`  q@(H`\B@52|.w*'/ŀ 'AD%& L'n7'qኮqI0o bXX(:&-%.0jeQ;o7cWIU8.?(P,` `$B0@? bȜ D$H`XW(N-:S4%t`9hҡ~K&_7WKQEQEQEQEQEQEQEQEQEQEQEa.D `'6p D/t/F2`N&]YĈ,H'eCw\^acN*eE )^>n0{L7kٺ3(QNw `!6Mu/c[-TG޽HE=؛>gK3v$$K 8Lh ,`-v#^f>!^ 5m-6^=b9Qs5c7޾6 W]ۭ9" `M8Dgnݮ񂆊qf̳|f'ORכtaz,q sZ4dygn?p/w-i5{Vvn'|`N0Q8tGnݮ񂆊TC6e5Ե<Y4Bn?w Y-cjYۣ"z/vZl !a^wdi h/ܻL/^f{j&"l v17g#W},b嶯ݺer>pc6ṇ7jtclHE;Qn7k Nn&6w3_wl1^e\3v^f{j=;˿XngΖ1]wfy/Y&(QOwݬ`.9l1^eQG},biw2ݮf=6`ngY-AZ>z.$$K 8Lh ,`tWk9f` tiv'l?6O6ǯ2g穏Z4dya]6pY5{Vvn'|`N*d3w4Wb,[U~MR!t/Yb.' 3Wgfny綠ݝۭ=L{:XuQ#:_ަl1vM粖gn{vF*/*f 3WzOz=6`ϝn{kM~Ռ=uٸ H'L()i$4m8L{n8[;Kq7k`vqf̳$F-$N"PR&Hi"/pz.8WYCw&[;w3{vmYٺ'PrޕfC=ۭ8YOv ?y|r޴TG޼}nǛl rK$EpX@Ĉ,<0GD.y=;7NL3WzOz=6`ϝn{kM~Ռ=uٸ H'L()i$tH%ێ?k-NRE(hݾ͙g0I'%ՐZHE0LI"/p8J>ᅯ6N+7cj:[:Ӈ݂Z&yQ.81b -֌!Z&t`9hzJтaa\;,e fQtN"/t,[&g :0@zJрFAr  (M8A1 IdN,Ν \y1S%7[6>g]uk`` q]*nלێ?kqv͋Z[`>S3tPݮ5:q3Mnyɰze!7Sqf̳ۣ/di h뵜Bn?1VKF@~_Ov.nyyqʈ \zJ(ND&R' :&uY(Ra3FVAr#q1uyYA6.B0AudaJrΌ! `H6.B0\`N2 cיg3gK3v gc-cث?vϩ3Ho7gKl\`F D•D .B0'@БB6.B0\`N+ rzo cיg3gK3v gc-cku1o7gKl\`F/!%`N0$DQ8q,£,qx!(f5ov͋Z[`>S3(QN>fVlߝۭ9Xh-lC)ۧM.n 6'%ՐZHHxaQEys88.nz-ernzD3L޾Ζ1]wfZJ0@t`9h9Dň$MD.KD•D"ܬL)TNgF+7 :&F Q0Q9Dw+% U.`YрɋZpS4WNtb,tKk@tM %aJr\VJ&' ]pѓn (hD3fY-ern`% тD.CD•D"ܬL)TN\0сY-AZ J@-g@ (D"D hP"k<  (h`#y(RDW;*ˆ:!|y؝ۭ;Vvn~}20l\`% UHrQ0Q9pgD>8u{j߯ϰZq"M ]hul!aJr\VJ&'.L % YP˂-gP+%H VKD Bxr% YP hHCF!.CD•D"ܬL)TN\0ѕ;tyOY}ܥn{kM۷k?!7=ϛZg n4$P7q'vN՝׫l\`% UHrQ0Q9pgE"iovMǁY-AZ>v" _vmݺ׶?΅ۣ"z/vZlnݻY.'q v|ݮZg;v" 'PrޕUןh +K\]Xly؝ۭ;Vvn6.B0Au'`H&0VAi"q,|XΝ*ԾtۧYٻ% H&&S1LC>nxCEwD3fY?^bnәڳtO~jݬ]6roW]eV)zݬ\َfϛ0PF1Q ٖs=ZngΖ1]wf[3Iq=s5{Vvni_޾2v*]3y^@t}mYٺ'yhk.k-םܥۧɰ{j[=їsqʈFM\y̰ 8rMčwgC;g<%,NYcF j H tNH7^sQ8fwxKY"(Ej Hל,!(ޛ9.IdvZ w\$AdƝלàl R&Cvfxz ?lz{;嶯ݺg{qܝݬ+QZ`o3W3 '\Db)Et-Dmڨ3W/~wF}=-~btgEs[3Wgf_M=nϖڿvȯZmqܝgbs(krݻVKF@i.f{jXov˿T ߷oJv Y-=er3}?ƼL.'v|gZzg}:M9AW)zݬZ 1}˿Tv.mO]NnҷB>?ݬ6di h/+~|. 7v`wn=gni\Pyěq]YS (< :$[ekӔ|L7}::XuަQ% :0M\pb[ .B0M5P4r\+ ?(RG ? g48a48`9I@0O 09Q@@`8$<(I@0'48X|CEP5WH)%=5S\,808,<0J)Yb(h 4P9*h_]5CR7QN]`(CEP`d Y^,T@FAr  (M8l~#:_޽YgY#k7mϵg=o7+FMns5~-~bﳥW]eV.Idp1Š"CNs Dmڨ3W/~wF}=-~;ttnջFGװtt{b֖fc>g嶯ݺet$DQ;-g1-]PytHl|_\Db)Et-$F-$N"QAWHi"/p@F}9Gm_tzqܝ[n7=3^6sl_W]ePBQD/!%,A%`т#DY(G,j"SY2*(>J$hp |CC( 0 A@`s|CA48`P UtX U5MQÏMQMQMQT(R g .NoJ)Yb(h 4P9*h_]5CR7QN]`(B(XE:K,Qu yA7\5exIRk%Z&F.`Q6KF%,N^bsg{'6rJ< ,`H+\D최w\:$A G xKY"("ĵw\:*.!( 0 rK$EpPZ/QEys=LqxXW%"k8,%1" *@888B AP@tyVAi"p 08t`l@>0@jD@ kZ~SZj] H6H6H6] ./usr/lpp/essl.html.en_US/html/essl081.htmlen_U Guide and Reference

Guide and Reference

Getting the Best Accuracy

This section explains how accuracy of your results can be affected in various situations and what you can do to achieve the best possible accuracy.

What Precisions Do ESSL Subroutines Operate On?

Both short- and long-precision real versions of the subroutines are provided in most areas of ESSL. In some areas, short- and long-precision complex versions are also provided, and, occasionally, an integer version is provided. The subroutine names are distinguished by a one- or two-letter prefix based on the following letters:

S for short-precision real
D for long-precision real
C for short-precision complex
Z for long-precision complex
I for integer

For a description of these data types, see "How Do You Set Up Your Scalar Data?". The scalar data types and how you should code them for each programming language are listed under "Coding Your Scalar Data" in each language section in "Coding Your Program".

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Guide and Reference

How does the Nature of the ESSL Computation Affect Accuracy?

In subroutines performing operations such as copy and swap, the accuracy of data is not affected. In subroutines performing computations involving mathematical operations on array data, the accuracy of the result may be affected by the following:

  • The algorithm, which can vary depending on values or array sizes within the computation or the number of threads used.

  • The matrix and vector sizes

For this reason, the ESSL subroutines do not have a closed formula for the error of computation. In other words, there is no formula with which you can calculate the error of computation in each subroutine.

Short-precision subroutines sometimes provide increased accuracy of results by accumulating intermediate results in long precision. This is also noted in the functional description for each subroutine. See the RS/6000 POWERstation and POWERserver Hardware Technical Reference Information--General Architectures manual for details.

For the RS/6000 POWER and POWER2, the short-precision, floating-point operands are stored by the hardware in the floating-point registers as long-precision values, and, as a result, all arithmetic operations are performed in long-precision. Where applicable, the ESSL subroutines use the Multiply-Add instructions, which combine a Multiply and Add operation without an intermediate rounding operation.

For the ESSL POWER Library, ESSL Thread-Safe Library, and ESSL SMP Library, results obtained by 32-bit environment and 64-bit environment applications using the same ESSL library are mathematically equivalent but may not be bit identical.

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Guide and Reference

What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?

The data types operated on by the short-precision, long-precision, and integer versions of the subroutines are ANSI/IEEE 32-bit and 64-bit binary floating-point format, and 32-bit integer. See the ANSI/IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985 for more detail.

There are ESSL-specific rules that apply to the results of computations using the ANSI/IEEE standards. When running your program, the result of a multiplication of NaN ("Not-a-Number") by a scalar zero, under certain circumstances, may differ in the ESSL subroutines from the result you expect.

Usually, when NaN is multiplied by a scalar zero, the result is NaN; however, in some ESSL subroutines where scaling is performed, the result may be zero. For example, in computing alphaA, where alpha is a scalar and A is a matrix, if alpha is zero and one (or more) of the elements of A is NaN, the scaled result, using that element, may be a zero, rather than NaN. To avoid problems, you should consider this when designing your program.

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How is Underflow Handled?

ESSL does not mask underflow. If your program incurs a number of unmasked underflows, its overall performance decreases. For the RS/6000, floating-point exception trapping is disabled by default. Therefore, you do not have to mask underflow unless you have changed the default.

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Guide and Reference

Where Can You Find More Information on Accuracy?

Information about accuracy can be found in the following places:

  • Migration considerations concerning accuracy of results between releases, platforms, and so forth are described in "Migrating Your Programs".

  • Specific information on accuracy for each area of ESSL is given in "Performance and Accuracy Considerations" in each chapter introduction in Part 2.

  • The functional description under "Function" for each subroutine explains what you need to know about the accuracy of the computation. Varying implementation techniques are sometimes used to improve performance. To let you know how accuracy is affected, the functional description may explain in general terms the different techniques used in the computation.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] . To a l~SZjKH6H6H6 ./usr/lpp/essl.html.en_US/html/essl086.htmlen_U    eor<=>ainstu "AEcdfhlmp,-./123DFHILMNPRSTbgvy#0BCOUXw&45689:;VY_x!()7GWQqz?Kh<0,4\`ТdE$ŏ2?2P,ZEPy"d7Hy2H4V*_[ےvfI&h$O q#pRTs9s9s9s9s9w%'`>`,G9{KS;†p pp ,;&P_?"^{@!"yaOy\9A qpP$Q#(&L $ reHRC2S.@G;'N0 qA@`P҉G2di c9C2S.@(# x0H(p iDș2XJ@41X=TEL!h8А0h(  ,H{BxP L0 @y$P PAT:$̹Т==))<'9s9s9s9s9sq"^{QQ(3Ai/Jr;h#{Hy"5GDN@.@0D9r94Rap88k##~_ ӖW5I;h^dFʢQ' HxpEU9CÇ9Z/=I8p*h^{QQ($<8pdhʢQ' Hxp=yGDN@Æ=y,+F~v#\ؚ?/֍EӳM/}sXck0OΦ+H(y"9GDN@.=$9r94Rap88A:ЧW_⑑fZQUKr6w霓 Z\!$Z-Y=I8pBh^{QQ($<8pနhʢQ' HxpBP{Eڏ*D!Ç -yTJ$ 8hhʢQ' HxpG E紳̭{~}Hˇr8Kņw_ٽW5I;;7`W#{~~xWk`hi7i2O -|'|7fu4ὃ𒝈Sb)e赟~3HV;7 '`@0 ؅6 ,BOf#)9i4kHR9)L3N zx5)pkR;sRB71aE}Nixlv*"R9)#_f\-Nhdm{0OΦ+Hi2OֽNq4i\:ت+Of#:0Txa )D84GQkE/)R_3ZBLGS[|>SD84Q,$߇6*_xήoԅ!.w+3MO,Ӯig'b 412*z27׆HR,\ DR  DCr؅6 GLK^q ܹܮnK4ed671l5Es}B>^\R,\ogq;Dx|ΞM]䍉gm89/ xpX(5Ӣ +)u?0do  ߂7)!Hr7a^7X%X/1GL%~u5ZFW؁ަy#|=h~-{˟PW$p^27?,s]By5tf\!$;8ߢS[egƋei (7JqC ^_:ܭ#+ЧSix؎ru0TAS*\=hS<ӂGLg~_0d!yi_tu^E4_Λ/syܮ3Z|ߕ5RZ8߄Wk2Mku5ZFW1=;f\, )egZ~tة|ϫw Cu>~Y2Mku5ZFW?LMh@*f\!$WzB2>_i 6*_9s1.HzB:p]xߍ>I;kyb^z8+r\7a#ww CqEwMAztUԅ |gtdG;uC+JjS[eb ޥ^%赞`rC2 )Lӫ ev7 M\ Guide and Reference

Guide and Reference

Where Can You Find More Information on Performance?

Information about performance can be found in the following places:

  • Many of the techniques ESSL uses to achieve the best possible performance are described in the "High Performance of ESSL".

  • Migration considerations concerning performance are described in "Migrating ESSL Version 2 Programs to Version 3".

  • Specific information on performance for each area of ESSL is given in "Performance and Accuracy Considerations" in each chapter introduction in Part 2.

  • Detailed performance information for selected subroutines can be found in reference [30], [41], [42] and on the IBM RS/6000 web site at:

    http://www.rs6000.ibm.com/software/Apps/essl.html

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Guide and Reference

Dealing with Errors when Using ESSL

At run time, you can encounter different types of errors or messages that are related to the use of the ESSL subroutines:

  • Program exceptions
  • ESSL input-argument errors
  • ESSL computational errors
  • ESSL resource errors
  • ESSL attention messages

This section explains how to handle all these situations.

What Can You Do about Program Exceptions?

The program exceptions you can encounter in ESSL are described in the RS/6000 architecture manuals. For details, see:

  • The ANSI/IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985

  • The RS/6000 POWERstation and POWERserver Hardware Technical Reference Information--General Architectures manual

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Guide and Reference

What Can You Do about ESSL Resource Errors?

This section gives an overview on how you can handle resource errors.

All Resource Errors

ESSL returns a resource error and terminates your program when an attempt to allocate work area fails. Some ESSL subroutines attempt to allocate work area for their internal use. Other ESSL subroutines attempt to dynamically allocate auxiliary storage when a user requests it through calling sequence arguments, such as aux and naux. For information on how you could reduce memory constraints on the system or increase the amount of memory available before rerunning the application program, see "ESSL Resource Error Messages".

You can determine the resource errors that can occur in a subroutine by looking under "Error Conditions" in the subroutine description in Part 2 of this book. Error messages for all resource errors are listed in "Resource Error Messages".

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Guide and Reference

What Can You Do about ESSL Attention Messages?

This section gives an overview on how you can handle attention messages.

All Attention Messages

ESSL returns an attention message to describe a condition that occurred, however, ESSL is able to continue processing. For information on how you could reduce memory constraints on the system or increase the amount of memory available, see "ESSL Resource Error Messages".

For example, an attention message may be issued when enough work area was available to continue processing, but was not the amount initially requested. An attention message would be issued to indicate that performance may be degraded.

For a list of subroutines that may generate an attention message, see Table 31. For a list of attention messages, see "Informational and Attention Error Messages".

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ml#HD lH~SZj +H6H6H6./usr/lpp/essl.html.en_US/html/essl093.htmlen_U   T"<=>AELaenort /DGINOPRhilsu-1235FHSVbcdfgmpy.08CMYv#,4679BUXw&:_x!();?WQzKkqjp9 N\nϦ M暄aq>(d`c\Ᲊ1چP=Wĺ'\Vju cpR[ WJ 1{{{{{ÇA>8)OF 3R\~ 70p8r@dWcppH@{qHOx8 , Ç ǍL<ĩo I:^$WHxH$'E-y-n*AI#G^#  @Qa(C`cwS =YlU#h]EtQ@6Q:$pgQ 0`@,9@-j(6^ cpV%քH{{{{{{p1.Onܰ 0:`c4DebU.IѢ c*F!r{FC䢍TǶ0@@urUp&(,C`6ueWqu㍥.r+XE٫k|_!ç_"ZB 8ƍBp0P`:=@`@p:=@`@q tz1 ځB[b'6601.Oj b'Ŷ601.Oj b' rpVcR9Z/EǢ0Zrǹ> Xʵ:~[+^4 Arp1@tz1 4$'V5c㫞U/l{}l`Fc_!ç_岵B בEZ׷ʴW]$8t5ZgCk+[>#vmKy0+j♩/\b=MNJ(tl6qa[XVIѕh5cՌЂ0Iu~hAzj1ֳԐ9) ш\NJ(tl6:~X-o y'FWE٫60q8 v9pRJ% cH4E휆kx{AIn8)b'_.1Bc1PpAP1;F{g!>3@^R[TqOu _ p  q6Ѿ3@^f=AmPE+r|.1Bc 5G@#@`4E휆kx{AIn8X ~Ђp_0:h*8Aڤ v4E휆kx{u$"cӠ;h9 I.URF8:Ӽf轳oz/n`$Xni3@^f=RME#nAڤ v4E휆kx{u$"#\]"cӠ;h9 I.URF8:Ӽf轳oz/n`$Xni3@^f=RM nAڤ v4E휆kx{u$"t;NC5|f轺m.ˬUlԑ7Nz/l3[h۩& 1v4E휆kx{u$1q@i3@^f=RM,PHq!*W($8HI7Nz/l3[h۩&VIt;NC5|f轺mbqtx{g!>3@^I62Cqtnj1w=r4EԓlH@$cӠ;h9 I qtx{g!>3@^I6Hf8:Ӽf轳oz/n`!8:7TH;h9 IE $@@Gʸ%E $@@ni3@^f=RMcӠ;h9 IE1v4E휆kx{u$1q@i3@^f=RM,PHq"t;NC5|f轺m.ˬUlԑ7Nz/l3[h۩& ,Aw=r4Eԓl$8Avx{g!>3@^I6HA!Ĉ.ExbC Q7Nz/l3[h۩&1v4E휆kx{u$"Řt;NC5|f轺mdt8R cz/l3[h۩& ($88:Ӽf轳oz/n` [5$cӠ;h9 IE1v4E휆kx{u$1q@i3@^f=RM,PHq" WdHA!Ĉ$Aw=r4Eԓlt;NC5|f轺mbqtx{g!>3@^I62Cqtnj1w=r4EԓlH"U(,PHq"qtx{g!>3@^I6e*jHAw=r4EԓlcӠ;h9 IcӠuH;TӼf轳oz/n`$XD .ExbC$cӠ;h9 I qtx{g!>3@^I6Hf8:Ӽf轳oz/n`!8:7TH;h9 IE $@A7Nz/l3[h۩&VIt;NC5|f轺mbqtx{g!>3@^I62Cqtnj1w=r4EԓlH!?*W($88:Ӽf轳oz/n` Aw=r4EԓlcӠ;h9 IcӠuH;TӼf轳oz/n`$XD WdHA!Ĉ 1v4E휆kx{u$]X٩#ni3@^f=RM,Y7Nz/l3[h۩&Hq7N R;NC5|f轺mbC|Q^$XDt;NC5|f轺mni3@^f=RM,Y7Nz/l3[h۩&Hq7N R;NC5|f轺mbCt;NC5|f轺m.ˬUlԑ7Nz/l3[h۩& ,Aw=r4Eԓl$8Avx{g!>3@^I6HA!Ĉ>U(,PHq"!&8:Ӽf轳oz/n` Aw=r4EԓlcӠ;h9 IcӠuH;TӼf轳oz/n`$XDA&8:Ӽf轳oz/n` Aw=r4EԓlcӠ;h9 IcӠuH;TӼf轳oz/n`$X@*W($80$cӠ;h9 I qtx{g!>3@^I6Hf8:Ӽf轳oz/n`F8:7TH;h9 IE $.ExbCA&8:Ӽf轳oz/n` [5$cӠ;h9 IE1v4E휆kx{u$R1q@i3@^f=RM,PHq qvJ+ċH  1v4E휆kx{u$8:Ӽf轳oz/n`$Xni3@^f=RME#nAAS rp1@tz1 4AMۃxU-. V)_kvjZ*:EqNgibr*׆vjZ*pǹl{^~F|_%-2Cg+Zkrg.S*UڅV#E.aibcvjǦ~j.#Kz. |UpX[ZЂ0~ XʵQAb^J(;T2Cg+Z~*j\?|zrU`-orfeZ(hEDxļPvd.VXVTǶ0`A( Qh/+.V q9N8hF!r{F%l::2ibV}fu|[+^hAzg8Ѡh 2y{܊J^inU1|{ױ#Kr7qu5ZgBūf85Z_!ç_ {"\Tb̕!\{Yj[rg~ ~J*Lrp1E.OhЕp-XʬugXm,IuSb+[e4hvyZ,T6 V̕!\{XX1|u]{jH܉qT^^b~kZ24eX?kе,Z@,/^Pt'4\B W0`vo!-o@uZ/fJX/'|eVl`FNg~ ~VX9Ldm\bB ( Tk^^r,d*f]j`c7 WՏ||`TbU/X-R9X6ߗ8rP~2oU^%" -{{tx$1 . W%W 9DN2QFZgR2Bp,Y 2Vϖ1|0c~2~QAߛWUfZ*G+^ܝݜrعLPUPbХz=a RGIhU-b2㧋 98b'mhJ 6~}s,N+L{x9W*Y+^_l`FNg~ 8to\uN+L{z9nT9V1||g.S*_^Jԗ$Ya`/b=a!A(ZrQFZgR2Bp,ij/_ f:ߒe(o+^k{#nN9l\(T(bFhRkwGRCڰUNP)#4*䖱NvqbZcs pptz1 6%\׸7Ler|2ıe#L{|ߍmx. cq%ֱO+حml`FN9l\(W-Lgܟ^)_BŗX2+BKg/jX\b/⾶k|_\b-o^*=ZאleqR\%-zنqV>UZ٩L-U1|{׷/^ V1lR\~Ueb1BpjUkz[бe#L{ǧ+|Wh$a%" -{{tx$1 9A;Tl6Kp˳¯lcK]k[Θ)_ŗX2 _j)b]JkpX[ZcHVЂq|un e(/%{ʵTV;㗻8rRXJ5I {j W%W 9@`ЫZ8eO)krp1@tz1 4DxQpo eWqu㍥.uLbˬFZc/P5k.Xy[8Zu|[+^hAzg8Ѡhb'[b'- tz1 ځB tz1 ځBb'601.Oj  ll`cш\ 6601[rg.S+c3OӯL= ]b4װ=nمW*&^'y;X#K{"@ csN@/v{,WEKz9l\(Wbr*׆vjZ* l{|ЂK'|Zřz`%klnT9V/U9l\(WokXVZ+H6b&NsR@ V G+X[,{bZ.X1B\}qb1Bk|_BŗX2by(o*sbU]ַʵ/rZ/OZ>3FJ/=a,ij/_ y{[)h_ ኮ_Й:I˳пjK5hmfk|_\b-o^_KV.ekە~kV+23h51Ƃ5Z^fZc bU%'\r_icEK@k|S Wбe#L{}b*ʵQAb^J(;TEe`= -Leb,Lzdz%HW_%ՏKz8/~U1|{׷/^1Zz#VX/vqb1BJ(;TVf*~B@ֶoL1|{׷/^O+{(RenJQAb^J(;T+B*Y 2VϮ1|{2Tqb\ʖg4׵/fJ9;UX0-䢃7ʵ]ַʵ/rZ¨J(:#3L!RZX>^b} #EA},KZۑ(S?*ؗ>^b} ]!çܩfsL{17*Yz椁ՏjeEE5X'}qܶVhAzj/^^bz|3k-bU0xmf},KZfl߭=EE5Xkn^عLPYJ?|rU]{un #rUVWfklַ/vqb1BS{Ʃ!mXaqAA(lrKX 81Bq9N TǴ ,$LN*m{Х\la@ ` 5I {j D ⡫]:t)Wp HGRCڰ0:*`@|R60 @0=a6MЂ'¯laTǶ0@urUp ӔD`0!bUla@ $PqsB"`2t)V8RGAƍй@v kr~SZjH6H6H6./usr/lpp/essl.html.en_US/html/essl094.htmlen_U Guide and Reference

Guide and Reference

How does Error Handling Work in a Threaded Environment?

When your application program or Fortran first creates a thread, ESSL initializes the error option table information to the default settings shown in Table 26. You can change the default settings for each thread you created by calling the appropriate error handling subroutines (ERRSET, ERRSAV, or ERRSTR) from each thread. An example of how to initialize the error option table and change the default settings on multiple threads is shown in "Example of Handling Errors in a Multithreaded Application Program".

ESSL issues error messages as they occur in a threaded environment. Error messages issued from any of the existing threads are written to standard output in the order in which they occur.

When a terminating condition occurs on any of the existing threads (for example, the number of allowable errors was exceeded), ESSL terminates your application program. One set of summary information corresponding to the terminating thread is always printed. Summary information corresponding to other threads may also be printed.

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Guide and Reference

Where Can You Find More Information on Errors?

Information about errors and how to handle them can be found in the following places:

  • How to code your program to use the ESSL error-handling facilities is described in "Coding Your Program".

  • All ESSL error messages are listed under "Messages".

  • The errors and return codes associated with each ESSL subroutine are listed under "Error Conditions" in each subroutine description in Part 2.

  • Complete diagnostic procedures for all types of ESSL programming and documentation problems, along with how to collect information and report a problem, are provided in "Handling Problems".

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Guide and Reference

Setting Up Your Data Structures

This chapter provides you with information that you need to set up your data structures, consisting of vectors, matrices, and sequences. These techniques apply to programs in all programming languages.

Concepts

Vectors, matrices, and sequences are conceptual data structures contained in arrays. In many cases, ESSL uses stride or leading dimension to select the elements of the vector, matrix, or sequence from an array. In other cases, ESSL uses a specific mapping, or storage layout, that identifies the elements of the vector, matrix, or sequence in an array, sometimes requiring several arrays to help define the mapping. These elements selected from the array(s) make up the conceptual data structure.

When you call an ESSL subroutine, it assumes that the data structure is set up properly in the array(s) you pass to it. If it is not, your results are unpredictable. ESSL also uses these same storage layouts for data structures passed back to your program.

The use of the terms vector, matrix, and sequence in this book is consistent with standard mathematical definitions, and their representations are consistent with conventions used in mathematical texts. Special notations and conventions used in this book for describing vectors, matrices, and sequences are explained in "Special Notations and Conventions".

Overlapping Data Structures: Most of the subroutines do not allow vectors, matrices, or sequences to overlap. If this occurs, results are unpredictable. Where this applies, it is explained in "Notes" in each subroutine description. This means the elements of the data structure cannot reside in the same storage locations as any of the other data structures. It is possible, however, to have elements of different data structures in the same array, as long as the elements are interleaved through storage using strides greater than 1. For example, using vectors x and y with strides of 2, where x starts at A(1) and y starts at A(2), the elements reside in array A in the order x1, y1, x2, y2, x3, y3, ... and so forth.

When you use this technique, you should be careful that you specify different starting locations for each data structure contained in the array.

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Vectors

A vector is a one-dimensional, ordered collection of numbers. It can be a column vector, which represents an n by 1 ordered collection, or a row vector, which represents a 1 by n ordered collection.

The column vector appears symbolically as follows:



Figure ESYGR21 not displayed.


A row vector appears symbolically as follows:



Figure ESYGR22 not displayed.


Vectors can contain either real or complex numbers. When they contain real numbers, they are sometimes called real vectors. When they contain complex numbers, they are called complex vectors.

Transpose of a Vector

The transpose of a vector changes a column vector to a row vector, or vice versa:



Figure ESYGR23 not displayed.


The ESSL subroutines use the vector as it is intended in the computation, as either a column vector or a row vector; therefore, no movement of data is necessary.

In the examples provided with the subroutine descriptions in "Reference Information" of this book, both types of vectors are represented in the same way, showing the elements of the array that make up the vector x, as follows: 

                        (1.0, 2.0, 3.0, 4.0, 5.0, 6.0)

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Conjugate Transpose of a Vector

The conjugate transpose of a vector x, containing complex numbers, is denoted by xH and is expressed as follows:



Figure ESYGR24 not displayed.


Just as for the transpose of a vector, no movement of data is necessary for the conjugate transpose of a vector.

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Guide and Reference

Matrices

A matrix, also referred to as a general matrix, is an m by n ordered collection of numbers. It is represented symbolically as:



Figure ESYGR25 not displayed.


where the matrix is named A and has m rows and n columns. The elements of the matrix are aij, where i = 1, m and j = 1, n.

Matrices can contain either real or complex numbers. Those containing real numbers are called real matrices; those containing complex numbers are called complex matrices.

Transpose of a Matrix

The transpose of a matrix A is a matrix formed from A by interchanging the rows and columns such that row i of matrix A becomes column i of the transposed matrix. The transpose of A is denoted by AT. Each element aij in A becomes element aji in AT. If A is an m by n matrix, then AT is an n by m matrix. The following represents a matrix and its transpose:



Figure ESYGR26 not displayed.


ESSL assumes that all matrices are stored in untransformed format, such as matrix A shown above. No movement of data is necessary in your application program when you are processing transposed matrices. The ESSL subroutines adjust their selection of elements from the matrix when an argument in the calling sequence indicates that the transposed matrix is to be used in the computation. Examples of this are the transa and transb arguments specified for SGEADD, matrix addition.

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Guide and Reference

Conjugate Transpose of a Matrix

The conjugate transpose of a matrix A, containing complex numbers, is denoted by AH and is expressed as follows:



Figure ESYGR27 not displayed.


Just as for the transpose of a matrix, the conjugate transpose of a matrix is stored in untransformed format. No movement of data is necessary in your program.

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In Storage

A matrix is usually stored in a two-dimensional array. Its elements are stored successively within the array. Each column of the matrix is stored successively in the array. The leading dimension argument is used to select the matrix elements from each successive column of the array. The starting point of the matrix in the array is specified as the argument for the matrix in the ESSL calling sequence. For example, if matrix A is contained in array A and starts at the first element in the first row and first column of A, you should specify A as the argument for matrix A, such as in: 

   CALL  SGEMX (5,2,1.0,A,6,X,1,Y,1)

where, in the matrix-vector product, the number of rows in matrix A is 5, the number of columns in matrix A is 2, the scaling constant is 1.0, the location of the matrix is A, the leading dimension is 6, the vectors used in the matrix-vector product are X and Y, and their strides are 1.

If matrix A is contained in the array BIG, declared as BIG(1:20,1:30), and starts at the second row and third column of BIG, you should specify BIG(2,3) as the argument for matrix A, such as in: 

   CALL  SGEMX (5,2,1.0,BIG(2,3),6,X,1,Y,1)

See "How Leading Dimension Is Used for Matrices" for a complete description of how matrices are stored within arrays.

For a complex matrix, a special storage arrangement is used to accommodate the two parts, a and b, of each complex number (a+bi) in the array. For each complex number, two sequential storage locations are required in the array. Therefore, exactly twice as much storage is required for complex matrices as for real matrices of the same precision. See "How Do You Set Up Your Scalar Data?" for a description of real and complex numbers, and "How Do You Set Up Your Arrays?" for a description of how real and complex data is stored in arrays.

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Guide and Reference

How Leading Dimension Is Used for Matrices

The leading dimension for a two-dimensional array is an increment that is used to find the starting point for the matrix elements in each successive column of the array. To define exactly which elements become the conceptual matrix in the array, the following items are used together:

  • The location of the matrix within the array
  • The leading dimension
  • The number of rows, m, to be processed in the array
  • The number of columns, n, to be processed in the array

The leading dimension must always be positive. It must always be greater than or equal to m, the number of rows in the matrix to be processed. For an array, A, declared as A(E1:E2,F1:F2), the leading dimension is equal to:

(E2-E1+1)

The starting point for selecting the matrix elements from the array is at the location specified by the argument for the matrix in the ESSL calling sequence. For example, if you specify A(3,0) for a 4 by 4 matrix A, where A is declared as A(1:7,0:4): 

                 *                             *
                 | 1.0   8.0  15.0  22.0  29.0 |
                 | 2.0   9.0  16.0  23.0  30.0 |
                 | 3.0  10.0  17.0  24.0  31.0 |
                 | 4.0  11.0  18.0  25.0  32.0 |
                 | 5.0  12.0  19.0  26.0  33.0 |
                 | 6.0  13.0  20.0  27.0  34.0 |
                 | 7.0  14.0  21.0  28.0  35.0 |
                 *                             *

then processing begins at the element at row 3 and column 0 in array A, which is 3.0.

The leading dimension is used to find the starting point for the matrix elements in each of the n successive columns in the array. ESSL subroutines assume that the arrays are stored in column-major order, as described under "How Do You Set Up Your Arrays?", and they add the leading dimension (times the size of the element in bytes) to the starting point. They do this n-1 times. This finds the starting point in each of the n columns of the array.

In the above example, the leading dimension is:

E2-E1+1 = 7-1+1 = 7

If the number of columns, n, to be processed is 4, the starting points are: A(3,0), A(3,1), A(3,2), and A(3,3). These are elements 3.0, 10.0, 17.0, and 24.0 for a11, a12, a13, and a14, respectively.

In general terms, this results in the following starting positions of each column in the matrix being calculated as follows:

A(BEGINI, BEGINJ)
A(BEGINI, BEGINJ+1)
A(BEGINI, BEGINJ+2)
.
.
.
A(BEGINI, BEGINJ+n-1)

To find the elements in each column of the array, 1 is added successively to the starting point in the column until m elements are selected. This is why the leading dimension must be greater than or equal to m; otherwise, you go past the end of each dimension of the array. In the above example, if the number of elements, m, to be processed in each column is 4, the following elements are selected from array A for the first column of the matrix: A(3,0), A(4,0), A(5,0), and A(6,0). These are elements 3.0, 4.0, 5.0, and 6.0, corresponding to the matrix elements a11, a21, a31, and a41, respectively.

Column element selection can also be expressed in general terms. Using A(BEGINI,BEGINJ) as the starting point in the array, this results in the following elements being selected from each column in the array:

A(BEGINI, BEGINJ)
A(BEGINI+1, BEGINJ)
A(BEGINI+2, BEGINJ)
.
.
.
A(BEGINI+m-1, BEGINJ)

Combining this with the technique already described for finding the starting point in each column of the array, the resulting matrix in the example is:



Figure ESYGR28 not displayed.


As shown in this example, a matrix does not have to include all columns and rows of an array. The elements of matrix A are selected from rows 3 through 6 and columns 0 through 3 of the array. Rows 1, 2, and 7 and column 4 of the array are not used.

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Guide and Reference

Positive Definite or Negative Definite Symmetric Matrix

A real symmetric matrix A is positive definite if and only if xTAx is positive for all nonzero vectors x.

A real symmetric matrix A is negative definite if and only if xTAx is negative for all nonzero vectors x.

In Storage

The positive definite or negative definite symmetric matrix is stored in the same way the symmetric matrix is stored. For a description of this storage technique, see "Symmetric Matrix".

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Guide and Reference

Positive Definite or Negative Definite Complex Hermitian Matrix

A complex Hermitian matrix A is positive definite if and only if xHAx is positive for all nonzero vectors x.

A complex Hermitian matrix A is negative definite if and only if xHAx is negative for all nonzero vectors x.

In Storage

The positive definite or negative definite complex Hermitian matrix is stored in the same way the complex Hermitian matrix is stored. For a description of this storage technique, see "Complex Hermitian Matrix".

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Guide and Reference

Positive Definite or Negative Definite Symmetric Toeplitz Matrix

A positive definite or negative definite symmetric matrix A of order n is also a Toeplitz matrix if and only if:

aij = ai-1,j-1    for i = 2, n and j = 2, n

The elements on each diagonal of the Toeplitz matrix have a constant value. For the definition of a positive definite or negative definite symmetric matrix, see "Positive Definite or Negative Definite Symmetric Matrix".

The following matrix illustrates a symmetric Toeplitz matrix of order n; that is, it has n rows and n columns:



Figure ESYGR30 not displayed.


A symmetric Toeplitz matrix of order n is represented by a vector x of length n containing the elements of the first column of the matrix (or the elements of the first row), such that xi = ai1 for i = 1, n.

The following vector represents the matrix A shown above:



Figure ESYGR31 not displayed.


In Storage

The elements of the vector x, which represent a positive definite symmetric Toeplitz matrix, are stored sequentially in an array. This is called packed-symmetric-Toeplitz storage mode. Following is an example of a positive definite symmetric Toeplitz matrix A and how it is stored in an array X.

Given the following matrix A: 

                    *                        *
                    | 99  12  13  14  15  16 |
                    | 12  99  12  13  14  15 |
                    | 13  12  99  12  13  14 |
                    | 14  13  12  99  12  13 |
                    | 15  14  13  12  99  12 |
                    | 16  15  14  13  12  99 |
                    *                        *

the array is: 

                    X = (99, 12, 13, 14, 15, 16)

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Guide and Reference

Positive Definite or Negative Definite Complex Hermitian Toeplitz Matrix

A positive definite or negative definite complex Hermitian matrix A of order n is also a Toeplitz matrix if and only if:

aij = ai-1,j-1    for i = 2, n and j = 2, n

The real part of the diagonal elements of the Toeplitz matrix must have a constant value. The imaginary part of the diagonal elements must be zero.

For the definition of a positive definite of negative definite complex Hermitian matrix, see "Positive Definite or Negative Definite Complex Hermitian Matrix".

The following matrix illustrates a complex Hermitian Toeplitz matrix of order n; that is, it has n rows and n columns:



Figure ESYGR32 not displayed.


A complex Hermitian Toeplitz matrix of order n is represented by a vector x of length n containing the elements of the first row of the matrix.

The following vector represents the matrix A shown above.



Figure ESYGR33 not displayed.


In Storage

The elements of the vector x, which represent a positive definite complex Hermitian Toeplitz matrix, are stored sequentially in an array. This is called packed-Hermitian-Toeplitz storage mode. Following is an example of a positive definite complex Hermitian Toeplitz matrix A and how it is stored in an array X.

Given the following matrix A: 

     *                                                      *
     |  (10.0, 0.0)   (2.0, -3.0)  (-3.0, 1.0)   (1.0, 1.0) |
     |   (2.0, 3.0)   (10.0, 0.0)  (2.0, -3.0)  (-3.0, 1.0) |
     | (-3.0, -1.0)    (2.0, 3.0)  (10.0, 0.0)  (2.0, -3.0) |
     |  (1.0, -1.0)  (-3.0, -1.0)   (2.0, 3.0)  (10.0, 0.0) |
     *                                                      *

the array is: 

        X = ((10.0, 0.0), (2.0, -3.0), (-3.0, 1.0), (1.0, 1.0))

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Guide and Reference

Symmetric Band Matrix

A symmetric band matrix is a symmetric matrix whose nonzero elements are arranged uniformly near the diagonal, such that:

aij = 0    if |i-j| > k

where k is the half band width.

The following matrix illustrates a symmetric band matrix of order n, where the half band width k = q-1:



Figure ESYGR41 not displayed.


In Storage

The two storage modes used for storing symmetric band matrices are described in the following sections:

Upper-Band-Packed Storage Mode

Only the band elements of the upper triangular part of a symmetric band matrix, including the main diagonal, are stored for upper-band-packed storage mode.

For a matrix A of order n and a half band width of k, the array must have a leading dimension, lda, greater than or equal to k+1, and the size of the second dimension must be (at least) n.

Using array ASB, which is declared as ASB(lda,n), where p = lda = k+1, the elements of a symmetric band matrix are stored as follows:



Figure ESYGR42 not displayed.


where "*" means you do not store an element in that position in the array.

Following is an example of a symmetric band matrix A of order 6 and a half band width of 3.

Given the following matrix A: 

                    *                        *
                    | 11  12  13  14   0   0 |
                    | 12  22  23  24  25   0 |
                    | 13  23  33  34  35  36 |
                    | 14  24  34  44  45  46 |
                    |  0  25  35  45  55  56 |
                    |  0   0  36  46  56  66 |
                    *                        *

you store it in upper-band-packed storage mode in array ASB, declared as ASB(4,6), as follows. 

                    *                        *
                    |  *   *   *  14  25  36 |
              ASB = |  *   *  13  24  35  46 |
                    |  *  12  23  34  45  56 |
                    | 11  22  33  44  55  66 |
                    *                        *

Following is an example of how to transform your symmetric band matrix to upper-band-packed storage mode: 

      DO 20 J=1,N
         M=K+1-J
         DO 10 I=MAX(1,J-K),J
            ASB(M+I,J)=A(I,J)
      10 CONTINUE
   20 CONTINUE

Lower-Band-Packed Storage Mode

Only the band elements of the lower triangular part of a symmetric band matrix, including the main diagonal, are stored for lower-band-packed storage mode.

For a matrix A of order n and a half band width of k, the array must have a leading dimension, lda, greater than or equal to k+1, and the size of the second dimension must be (at least) n.

Using array ASB, which is declared as ASB(lda,n), where q = lda = k+1, the elements of a symmetric band matrix are stored as follows:



Figure ESYGR43 not displayed.


where "*" means you do not store an element in that position in the array.

Following is an example of a symmetric band matrix A of order 6 and a half band width of 2.

Given the following matrix A: 

                    *                        *
                    | 11  21  31   0   0   0 |
                    | 21  22  32  42   0   0 |
                    | 31  32  33  43  53   0 |
                    |  0  42  43  44  54  64 |
                    |  0   0  53  54  55  65 |
                    |  0   0   0  64  65  66 |
                    *                        *

you store it in lower-band-packed storage mode in array ASB, declared as ASB(3,6), as follows: 

                    *                        *
                    | 11  22  33  44  55  66 |
              ASB = | 21  32  43  54  65   * |
                    | 31  42  53  64   *   * |
                    *                        *

Following is an example of how to transform your symmetric band matrix to lower-band-packed storage mode: 

      DO 20 J=1,N
         DO 10 I=J,MIN(J+K,N)
            ASB(I-J+1,J)=A(I,J)
      10 CONTINUE
   20 CONTINUE

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Guide and Reference

Positive Definite Symmetric Band Matrix

A real symmetric band matrix A is positive definite if and only if xTAx is positive for all nonzero vectors x.

In Storage

The positive definite symmetric band matrix is stored in the same way a symmetric band matrix is stored. For a description of this storage technique, see "Symmetric Band Matrix".

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Guide and Reference

Complex Hermitian Band Matrix

A complex band matrix is Hermitian if it is equal to its conjugate transpose:

H = HH

In Storage

The complex Hermitian band matrix is stored using the same two techniques used for symmetric band matrices:

Following is an example of a complex Hermitian band matrix H of order 5, having a half band width of 2.

Given the following matrix H: 

           *                                              *
           | (11,  0) (21, -1) (31,  1)  (0,  0)  (0,  0) |
           | (21,  1) (22,  0) (32, -1) (42, -1)  (0,  0) |
           | (31, -1) (32,  1) (33,  0) (43, -1) (53, -1) |
           |  (0,  0) (42,  1) (43,  1) (44,  0) (54, -1) |
           |  (0,  0)  (0,  0) (53,  1) (54,  1) (55,  0) |
           *                                              *

you store it in a two-dimensional array HP, as follows:

  • In lower-band-packed storage mode: 
              *                                           *
          | (11,  *) (22, *) (33, *) (44,  *) (55, *) |
     HP = | (21,  1) (32, 1) (43, 1) (54,  1)    *    |
          | (31, -1) (42, 1) (53, 1)    *        *    |
          *                                           *

  • In upper-band-packed storage mode: 
              *                                              *
          |    *        *     (31,  1) (42, -1) (53, -1) |
    HP =  |    *     (21, -1) (32, -1) (43, -1) (54, -1) |
          | (11, *)  (22,  *) (33,  *) (44,  *) (55,  *) |
          *                                              *

where "*" means you do not have to store a value in that position in the array. The imaginary parts of the diagonal elements of a complex Hermitian band matrix are always 0, so you do not need to set these values. The ESSL subroutines always assume that the values in these positions are 0.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] |  l~SZjP#÷H6÷H6÷H6./usr/lpp/essl.html.en_US/html/essl117.htmlen_U   <>aeinort "/=ARTVdglpsu&-012345;BEILMNPbfhmw#*,.6CDFHOSUcxy|()8GXkv+9:JY_j!7KqzQx  X4WX|ЉBF$bbr H`Ö0^*dzŖ.VfĸэYA)=%,\x3)&gL:QbVՐlÞiiiiiiiiiixA`̨|@-]LJדLR!#)!@l `@ QE 2<c1s5@WA Ɔ˦Ʉp+2hax I H$`YL9e *%MH9G 2S+&n!_&8x Hh @#`a,haP,hMN#XDh!_4hpF= A| ,c~nw;!LAr޺Iu8э먬(QXbUip'X3ˍ7[nVmqsN V%ьoi)-RMەq.fF3ѽ';+@ +WR=*]6rюhS;m8іsFG: 4(\ѝ6'kHOߛɠXyJu.xH=uqi&7Cn6L] >gz+ %Vmɋfۍz+%.A5s71ɩ.K3&7i3:J6v[5ZNerFZS=bav7 HMP-h =uIuJA먬&%VH@DۍL] 1qbm8QX/Iub$uޒ+t6dųrbmɽu).A59Vќoi)-Ri/hNqm8їZq1>:ى[z+ =%Vnܭ͋A먬OIuqi&V. D1ɠPyE`AʙXycAPѓ؉fqB&lvʈH[Mڈ FH,VHQX PS711= .A78TӍu711>g9ΕRqvuB춪~Viƌ.;z79IXvH8ZE} HMP-h =uOIuJA먬&%VH@DۍL] 1qbm8QXM.Cn6L[W&.lQX/IuH0yюMHw8\mhљ1I4)Qs؜Y":w;-2ԺvYtHsb@rh<0`ҁkAA먬zK "R]Ea0%.zB% .lbmɍۍCn6OoIub$uQX &-Cn6M먬OIuH0y,vYʴOΌ{LOyH< "zE!ͦrnܭӍu70s]Eaک[71:QX+t6dCn6M먬#]E`,"cA+2U`i'A,:Mٲ %ҷ؛I0YS+V(HvVZIezQ]دJrӓR=*_;nN9ӹq(@65SD Po|/$+7‚T?zF+jwrߝ7[nV{N4e֣aF7 TܽmeN6Ez@EbIe!&i1-H@#ʴO,4X+^@+CX?8# yU_y GȒAX<!H"^4X P MT78ӐO6Ժzh)Wmr{N4e֣lh{LOw[nV{N4e֣aF7 Tܽme HMT?0?7e` 0C  &a@Hp@ 0d?V(0ɈKd?p,<ŌCp+2"UiRA栃ÉAfT0U4C.z-iƌl0hQIBYbI)J=Lʆ `ICD`S2X0@)eCSD`$"R<ЧsƌrjGKsک[2Qm~8TӍuqA&btgg:lN,֑h#zLg+dXe8їZDRcrYzH%B` .Xe4K Yu M#@+)dfr)\j\ hi(qj#y޽fY RQrLRۍtw9JmMڮt:qnBSYnW* _rMHJw;,oijGKvrBh)Wq(jQFi1ΙZhaܽme @6ؕ6ܔ0Jq恃2PA"f%!Νqv%'6:ӝ3QNw;73u(%d,B`@6㞍1= .ASlĭu+4cOٸьعާz+S]E`:ˎԢJN޺"QXlQƝm;z+S]E`F;,v[ؙQ֙\nHc}Eˁ]E`@Eў6E7 Q,3f$wj6\A-zK-+QX4^+]E`ϝQXG] 4ۍz+]E`.lbmɽu).w;.MHw:U?+wF]j6ƌg{LOzLqs9VA+2U`i'#]d .YQҷ؛I0YS+V(H*@wxQPATyզ~tgiN'j]Ngj{4pl{ک[2Qn4aѽ'޻U?+ 2Qq*Ln^2U $&*VɎ2a1d?pܲ`0A@B` .@) @ L@ C ̬XȉT?7 2c"%PzF+jV)` {Õ* U <j,-VE=%!<=aʕ M8C摁H:j,) X0ISTjh8vFd'|YRaȓަPѓ44h<2 F DRs /*]Ǥ1`Ç'L9R'hA'ny`LA `ZHM) X0@) PT)[QfH)@H9cC lp~SZjw÷H6÷H6÷H6m ./usr/lpp/essl.html.en_US/html/essl118.htmlen_U    <>/=ATaenort ",BERVdgils&.012345;CDHILMNPSUfhmp#-8FObcuwxy|()*79:GX_!6YvQjKkzQ(T"\2m2ItAD%'|K #aH-QH2%bYMVg'i2x蕉e5 sarVa{{{{{p}C p~EJp5XLiNC8 p dp%%PZq`~0]HÔLY膟^WAF"2;_8]HÔLY`+3G 1 E蚕Gpp8 ,8 Ԍ9H:Uϳi$b*/Z%ƎAE GPR &t$*ABBBBBQ0pO4'*Rv'-j()9/;uf[.{{{{{{ 1=,(,̘;h$5E9;NaC ܇1=ՊI<BōvIƌ#v^6evdi&y^̇㡸⃵ 1=C㡨d9Q& @pw C!@D0Pjga؝'i5fWfLVke.zW!8[hS|ڙZ|ChB;>2bIvBfLVi/*/ ?;.̘$שs{2G쮃- E &[ .fa2{A]9h+h{A]d㖂4m=@ϑx|&m"&m#8-urWD GA]@ϑȉ[gȌCLv'gU{2ii^mv'IM{ٕٓ{2sbWiZ bP;b{ICQ]¥<Ҥb0| E脲m4U[lRq% EEwWq&u (?r JNKҴ[I㡸 1=C 㡨d9Q& @p㡨d9Q& @pCPs.*L8w C U;>2bIvBfLVi/*/ ?;.̘$כS=:]Sg؝'i4ɊMsҽwoI]1ZIi1>zW١6zt~<&ɊMsҽK{v⻎OR9eoA]9h+J޶{IѶkI5kHAF:K_#'u/*.F::1Լwjtb~-v堮Î!1Xxeiy3FBfLVkͩ.]_ɾr5Pă"-3Q0=Ic@ϑx|#KWeZ 64m=I Z A]Fqƍ|Z A]FƍqB$-vX堮i-v堮A]9h+A"3hzCLvT_֐Iᕥ;>zLӮ̘$כS=:]whMT9 rAEu &Or1xD-^&m"Š4m=3hzŠ4m=7hzccOR KWeZ 6WiLrWqƍD1jgeEi|uLtc2~g ?;.̘$כS=:]whMT9 rAEu &Or11ю[gȼM>DA]9h+hz0n8Ѵ/A]9h+hz4v8Ѵ$1h+-w OG6 cc$y#Šqi_#!&[P:vvG#Go^nN쮁٭;4;?+ZB:Vs4cߕEavm٤POSf+n2pA1~-v堮%f֒kP]ѿ+9 þ=֐#%u$21xR#Sn~K٭$j5~V nkհF;wX - Cn&+&Fۆ[f{eSfBVy&fg Cn'כad٭:$w&YMC Wq= G9ET z ѩXQY14v5?sCxaGQU^mŽHLf2bxo 0h?!*Kћm0`Q۞i102bxo 0h?!*Kћm,XGnyhԒ;?Siy㡼H0(/Fm恊T0P(Cx`88 ]Gn 8 3&&㡨c &n8K u k~SZjķH6ķH6ķH6./usr/lpp/essl.html.en_US/html/essl119.htmlen_U Guide and Reference

Guide and Reference

Symmetric Tridiagonal Matrix

A tridiagonal matrix A is also symmetric if and only if its nonzero elements are found only on the diagonal, subdiagonal, and superdiagonal of the matrix, and its subdiagonal elements and superdiagonal elements are equal; that is:

(aij = 0   if |i-j| > 1)    and    (aij = aji   if |i-j| = 1)

The following matrix illustrates a symmetric tridiagonal matrix of order n:



Figure ESYGR49 not displayed.


In Storage

Only the diagonal and subdiagonal elements of the positive definite symmetric tridiagonal matrix are stored. This is called symmetric-tridiagonal storage mode. The elements of a symmetric tridiagonal matrix A of order n are stored in two one-dimensional arrays C and D, each of length n, where array C contains the subdiagonal elements, stored as follows:

C = (*, a21, a32, a43, ..., an,n-1)

where "*" means you do not store an element in that position in the array. Then array D contains the main diagonal elements,stored as follows:

D = (a11, a22, a33, ..., ann)

Following is an example of a symmetric tridiagonal matrix A of order 5: 

                      *                    *
                      | 10   1   0   0   0 |
                      |  1  20   2   0   0 |
                      |  0   2  30   3   0 |
                      |  0   0   3  40   4 |
                      |  0   0   0   4  50 |
                      *                    *

which you store in symmetric-tridiagonal storage mode in arrays C and D, each of length 5, as follows: 

                       C = (*, 1, 2, 3, 4)
 
                       D = (10, 20, 30, 40, 50)

Note:Some ESSL subroutines provide an option for specifying at least n additional locations at the end of each of the arrays C and D. These additional locations are used for working storage by the ESSL subroutine. The reasons for choosing this option are explained in the subroutine descriptions.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] " kꂈ~SZjE2 K62 K62 K6E./usr/lpp/essl.html.en_US/html/essl12.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui k]~SZjŷH6ŷH6ŷH6./usr/lpp/essl.html.en_US/html/essl120.htmlen_U Guide and Reference

Guide and Reference

Positive Definite Symmetric Tridiagonal Matrix

A real symmetric tridiagonal matrix A is positive definite if and only if xTAx is positive for all nonzero vectors x.

In Storage

The positive definite symmetric tridiagonal matrix is stored in the same way the symmetric tridiagonal matrix is stored. For a description of this storage technique, see "Symmetric Tridiagonal Matrix".

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ZIR[.$&Fh='ۃژFiKyy QZg:mOn[-Ґ[/rn0[bS\W-EƧgGRk˦#tNJ't6{St拲4&|.4Ú]7JGr1\_pc1WKIфP|)Ga&ЍЁ;ϲd)Ggm~ţ `fM͔nyy e`JAjO4]ɶiݙPۚQM/=ɸß> R;](҈٩?lbю>!J| tyL(J[Shı9Maϙ2Bϰ5 [H-l.h3 %4Em&|kcty Qز)GJ] [ӥ X<$PB6 WG.A4,i)mv(1eE2ن) DnDF.S\WIcKϵҏ>!:Q)Ggmy^>ɒ}B6vZ d_0!~҉( 7gx~;7g A_~;7g  xXv2o?0/<07ɿ>;ߎ d wpv2o#m(E6nhlbщmC@BЋwɒ0󰳰;'J<}$'J<b[bqW 6l"4]iY'-.Swy 0"2vaBvQ;Dmy] iOJ\-Gk}ySh!:QI#` {epI R;](҈(h@`.|w w㱓q^ w㱓q wc& c&~;|47C|w,;ɿAF%. B, 8xQPQaagaAaaaaaaacCÃ҄X hF!d]to ]} }B6Mw=j|;)Kw7EQWeQۅғZ i  6` "Qm%`,#sc'f8Q tNF[;0"(h@$!`GٴHGm)HL e ](HL[,JAl^)#$m [H-zAlH-52@ɒ}xI!H-byhGsJ"m [iH-BBdn[, ](HLD]}$'J<R e`8׊Ho XF[;0"(h@$!`GٴHGm)HL e ](HL[-*R exI ͢D>!J<R e X(`i AlH- eiH-ͤm)HL e ](HLD]}$!J<R eJAl8׊Hf)H-w4& et$&FJt!26m%tyL( H-R ec^)#$ƼRFHR e:R ecͥiD,Q! 6lYE@!hLCh@ et$&FҐ[.$&FvaϜe R6҉(tzAlҐ[,$'kZ1Gz-H-ZR eM$'kZ1)Gz-H-H-7s h@1hĺ<&HNyz<&HByh= [H-ܛ9I1 6nf]PC3`(x@ ,K"ρu)w !=lϜKhxPa &0$ o*zR07Kuaa0.6D$Ҁch k~SZjoǷH6ǷH6ǷH6o./usr/lpp/essl.html.en_US/html/essl122.htmlen_U Guide and Reference

Guide and Reference

Sequences

A sequence is an ordered collection of numbers. It can be a one-, two-, or three-dimensional sequence. Sequences are used in the areas of sorting, searching, Fourier transforms, convolutions, and correlations.

Real and Complex Elements in Storage

Sequences can contain either real or complex data. For sequences containing complex data, a special storage arrangement is used to accommodate the two parts, a and b, of each complex number, a+bi, in the array. For each complex number, two sequential storage locations are required in the array. Therefore, exactly twice as much storage is required for complex sequences as for real sequences of the same precision. See "How Do You Set Up Your Scalar Data?" for a description of real and complex numbers, and "How Do You Set Up Your Arrays?" for a description of how real and complex data is stored in arrays.

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One-Dimensional Sequences

A one-dimensional sequence appears symbolically as follows, where the subscripts indicate the element positions within the sequence:

(x1, x2, x3, ... xn)

In Storage

A one-dimensional sequence is stored in an array using stride in the same way a vector uses stride. For details, see "How Stride Is Used for Vectors".

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Two-Dimensional Sequences

A two-dimensional sequence appears symbolically as a series of columns of elements. (They are represented in the same way as a matrix without the square brackets.) The two subscripts indicate the element positions in the first and second dimensions, respectively:



Figure ESYGR51 not displayed.


In Storage

A two-dimensional sequence is stored in an array using the stride for the second dimension in the same way that a matrix uses leading dimension. It uses a stride of 1 for the first dimension. For details, see "How Leading Dimension Is Used for Matrices". (In the area of Fourier transforms, a two-dimensional sequence may be stored in transposed form in an array. In this case, the stride for the second dimension is 1, and the stride for the first dimension is the leading dimension of the array.)

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Guide and Reference

Three-Dimensional Sequences

A three-dimensional sequence is represented as a series of blocks of elements. Each block is equivalent to a two-dimensional sequence. The number of blocks indicates the length of the third dimension. The three subscripts indicate the element positions in the first, second, and third dimensions, respectively:



Figure ESYGR52 not displayed.


In Storage

Each block of elements in a three-dimensional sequence is stored successively in an array. The stride for the third dimension is used to select the elements for each successive block of elements in the array. The starting point of the three-dimensional sequence is specified as the argument for the sequence in the ESSL calling statement. For example, if the three-dimensional sequence is contained in array BIG, declared as BIG(1:20,1:30,1:10), and starts at the second element in the first dimension, the third element in the second dimension, and the first element in the third dimension of array BIG, you should specify BIG(2,3,1) as the argument for the sequence, such as in: 

   CALL  SCFT3 (BIG(2,3,1),20,600,Y,32,2056,16,20,10,1,1.0,AUX,30000)

See "How Stride Is Used for Three-Dimensional Sequences" for a detailed description of how three-dimensional sequences are stored within arrays using strides.

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How Stride Is Used for Three-Dimensional Sequences

The elements of the three-dimensional sequence can be defined as aijk for i = 1, m, j = 1, n, and k = 1, p. The first two subscripts, i and j, define the elements in the first two dimensions of the sequence, and the third subscript, k, defines the elements in the third dimension. Using this definition of three-dimensional sequences, this section explains how these elements are mapped into an array using the concepts of stride. (Remember that the elements aijk are the elements of the conceptual data structure, the three-dimensional sequence to be processed by ESSL. The sequence does not have to include all the elements in the array. Strides are used by the ESSL subroutines to select the desired elements to be processed in the array.)

The sequence elements in the first two dimensions are mapped into an array in the same way a matrix or two-dimensional sequence is mapped into an array. It uses all the items listed in "How Leading Dimension Is Used for Matrices", such as the starting point, the number of rows and columns, and the leading dimension. The stride for the first dimension, inc1, of a three-dimensional sequence is assumed to be 1, as for matrices. The stride for the second dimension, inc2, of a three-dimensional sequence is equivalent to the leading dimension for a matrix.

The stride for the third dimension, inc3, is used to define the array elements that make up the third dimension of the three-dimensional sequence. The stride for the third dimension is used as an increment to step through the array to find the starting point for each of the p successive blocks of elements in the array. The stride, inc3, must always be positive. It must always be greater than or equal to the number of elements to be processed in the first two dimensions; that is, inc3 >= (inc2)(n).

A three-dimensional sequence is usually stored in a one-, two-, or three-dimensional array; however, for the sake of this discussion, a three-dimensional array is used here. For an array, A, declared as A(E1:E2,F1:F2,G1:G2), the strides in the first, second, and third dimensions are:

inc1 = 1
inc2 = (E2-E1+1)
inc3 = (E2-E1+1)(F2-F1+1)

Given an array A, declared as A(1:7,1:3,0:3), where the lengths of the first, second, and third dimensions are 7, 3, and 4, respectively, the resulting strides are inc1 = 1, inc2 = 7, and inc3 = 21.

The starting point for a three-dimensional sequence in an array is at the location specified by the argument for the sequence in the ESSL calling statement. Using the array A, described above, if you specify A(2,2,1) for a three-dimensional sequence, where A is defined as follows, in four blocks, for planes 0 - 3, respectively: 

1.0  8.0 15.0    22.0 29.0 36.0    43.0 50.0 57.0    64.0 71.0 78.0
2.0  9.0 16.0    23.0 30.0 37.0    44.0 51.0 58.0    65.0 72.0 79.0
3.0 10.0 17.0    24.0 31.0 38.0    45.0 52.0 59.0    66.0 73.0 80.0
4.0 11.0 18.0    25.0 32.0 39.0    46.0 53.0 60.0    67.0 74.0 81.0
5.0 12.0 19.0    26.0 33.0 40.0    47.0 54.0 61.0    68.0 75.0 82.0
6.0 13.0 20.0    27.0 34.0 41.0    48.0 55.0 62.0    69.0 76.0 83.0
7.0 14.0 21.0    28.0 35.0 42.0    49.0 56.0 63.0    70.0 77.0 84.0

then processing begins in the second block of elements at row 2 and column 2 in array A, which is 30.0. The stride in the third dimension is then used to find the starting point for each of the next p-1 successive blocks of elements in the array. The stride, inc3, is added to the starting point p-1 times. In this example, the stride for the third dimension is 21, and the number of blocks of elements, p, to be processed is 3, so the starting points in array A are A(2,2,1), A(2,2,2), and A(2,2,3). These are elements 30.0, 51.0, and 72.0. These array elements then correspond to the sequence elements a111, a112, and a113, respectively.

In general terms, this results in the following starting positions for the blocks of elements in the array:

A(BEGINI, BEGINJ, BEGINK)
A(BEGINI, BEGINJ, BEGINK+1)
A(BEGINI, BEGINJ, BEGINK+2)
.
.
A(BEGINI, BEGINJ, BEGINK+p-1)

Using m = 4, n = 2, and p = 3 to define the elements of the three-dimensional data structure in this example, the resulting three-dimensional sequence is defined as follows, in three blocks, for planes 0 - 2, respectively:



Figure ESYGR53 not displayed.


As shown in this example, the three-dimensional sequence does not have to include all the blocks of elements in the array. In this case, the three-dimensional sequence includes only the second through the fourth block of elements in the array. The first block is not used. Elements of an array are selected as they are arranged in storage, regardless of the number of dimensions defined in the array. Therefore, when using a one- or two-dimensional array to store your three-dimensional sequence, you should understand how your array elements are stored to ensure that elements are selected properly. See "Setting Up Arrays in Fortran" for a description of array storage.
Note:Three-dimensional sequences are used by the three-dimensional Fourier transform subroutines. By specifying certain stride values for inc2 and inc3 and declaring your arrays to have certain number of dimensions, you achieve optimal performance in these subroutines. For details, see "Setting Up Your Data" and the "Notes" section for each subroutine.

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Guide and Reference

Coding Your Program

This chapter provides you with information you need to code your Fortran, C, C++, and PL/I programs using ESSL.

Fortran Programs

This section describes how to code your Fortran program using any of the ESSL run-time libraries.

Calling ESSL Subroutines and Functions in Fortran

In Fortran programs, most ESSL subroutines are invoked with the CALL statement: 



   CALL  subroutine-name  (argument-1, . . . , argument-n)

An example of a calling sequence for the SAXPY subroutine might be: 

   CALL SAXPY (5,A,X,J+INC,Y,1)

The remaining ESSL subroutines are invoked as functions by coding a function reference. You first declare the type of value returned by the function: short- or long-precision real, short- or long-precision complex, or integer. Then you code the function reference as part of an expression in a statement. An example of declaring and invoking the DASUM function might be: 

   DOUBLE PRECISION DASUM,SUM,X
       .
       .
       .
   SUM = DASUM (N,X,INCX)

Values are returned differently for ESSL subroutines and functions. For subroutines, the results of the computation are returned in an argument specified in the calling sequence. In the CALL statement above, the result is returned in argument Y. For functions, the result is returned as the value of the function. In the assignment statement above, the result is assigned to SUM.

See the Fortran publications for details on how to code the CALL statement and a function reference.

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Setting Up a User-Supplied Subroutine for ESSL in Fortran

Some ESSL numerical quadrature subroutines call a user-supplied subroutine, subf, identified in the ESSL calling sequence. If your program that calls the numerical quadrature subroutines is coded in Fortran, there are some coding rules you must follow:

  • You must declare subf as EXTERNAL in your program.

  • You should code the subf subroutine to the specifications given in "Programming Considerations for the SUBF Subroutine". For examples of coding a subf subroutine in Fortran, see the subroutine descriptions in that chapter.

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Setting Up Scalar Data in Fortran

Table 27 lists the scalar data types in Fortran that are used for ESSL. Only those types and lengths used by ESSL are listed.

Table 27. Scalar Data Types in Fortran Programs
Terminology Used by ESSL Fortran Equivalent
Character item1

'N', 'T', 'C' or 'n', 't', 'c'

CHARACTER*1

'N', 'T', 'C'

Logical item

.TRUE., .FALSE.

LOGICAL

.TRUE., .FALSE.

32-bit environment integer

12345, -12345

INTEGER or INTEGER*4

12345, -12345

64-bit environment integer2

12345, -12345

INTEGER*83

12345_8, -12345_8

Short-precision real number4

12.345

REAL or REAL*4

0.12345E2

Long-precision real number4

12.345

DOUBLE PRECISION or REAL*8

0.12345D2

Short-precision complex number4

(123.45, -54321.0)

COMPLEX or COMPLEX*8

(123.45E0, -543.21E2)

Long-precision complex number4

(123.45, -54321.0)

COMPLEX*16

(123.45D0, -543.21D2)

1 ESSL accepts character data in either upper- or lowercase in its calling sequences.

2 In accordance with the LP64 data model, all ESSL integer arguments remain 32-bits except for the iusadr argument for ERRSET.

3 INTEGER may be used if you specify the compiler option -qintsize=8.

4 Short- and long-precision numbers look the same in this book.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] HREF= k~SZjE3 K63 K63 K6E./usr/lpp/essl.html.en_US/html/essl13.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui lj~SZj @ͷH6ͷH6ͷH6+./usr/lpp/essl.html.en_US/html/essl130.htmlen_U   <>T /Daenrs"12=APRVilot#&(),-34;EFGILNObcdfhmpu+568BCHMgwy.Svx|*079UX:J_q!KYj{}Qk?WZzpEH +J {O|6/C yc/C 兑RҚT>xZFIḼSJX d\jTHT5_M4M4M4M4M4M4M4M4M4M4O70RNeFk)QǪT0WѸ!xpڟD@p#ž8@ qGT{" |#Hp崉mMT`gHw.etR(!: TiM HăEiښh3;2D~@d1GT{" A)*‹%4ΐ\} z)@`[ȩ NL(R&D$I u '&r :u4 [`4o#3\.Q5*ue馚iiiiiiiiii|hu(E45 Cd[XZ8L3]i%32q;-NY@)YbECDhJdRl"5 t0 &ba=ԥ"^&#;zo 9eH#p 9eH#p2ަ: " t2ަ:)HƒeF䚣tS2&9/R!q;!>tĩ6LS휗f.ѿ(90vS O?K-[Mtĩ7̎m[Or^4Ck͵OIEj- RS3.^3tMMO DNfC=P4k)e&\e#q^p3.e#ȷب|Sn,NMuz-35u[p㒤g%̴t2y:bTWzdkƮ6,Q":W U@\!B YlѢfb [#o]E@NZ-[Ql]EŽu9k1o]Ep_6<{SLS>4Cp^ (0YlѢfb [#o]E@NZ--ux [ [Ql0c o]Ep_6<{a 4N58JD=eP(o31SDOkF޺e[PQl]EŽu9k0o8/j=`VY#o\ujrMDԨXjqgJD>ަ[\$<{SLS>4C䚣t  c>eR  U.[DH`RzZrTR߻K'e79/R!=mدl-J03^pVWr\ҧ0휗 f\e:휓JD< 휗eShv+~elHSNxױ?{) 8O˱_Kr\ҧ0W@+\S9 4N58JD?_Vڤv'S2&Tl-l5:T1m<^1նx5IV휗6,{T5*  Aת#ˆ*B@rպ(LhiuPeI\Fb N2j┤CdvSn2߫eu &Y44e2]oe2D]woeVպ(LhiuPe:߰e "*"~-^PBeCK^-\~/a^(( z]B M .{ pHoe2"ua[/VT2ɡC/a < Z$@r2߫eu &Y44e2"uaWA[l[R&U ZE!QE^-\PQq[l%H\$<:y 휗){w;F2kg8S2$k>x "X PgU#~5 D 8]wd{ "H @B3r\O˱_sms~>x4~fIxHb5͵O(90vS L 5O DNfUsF&#>b!8ZI&VRD q `bdCCȆ.3pdC&D1q L`3$g"H3vcNZjBl[Tgli/Vڤv'9.mYh9̦i".Fb NI3˶׫GeFmD6^&RUg;g%dFg%ͱJQn3F7=IV5>Q Z~vbO0y AJa&fXofQ~[oղnJ,]T2jN2j₈ H z]B M .{ pHoe2"ua[/VT2ɡC/a < Z$@r2߫eu &Y44e2"uaWA[l[R&U ZE!QE^-\PQD]woeVպ(LhiuPe4 E^-\PQFDvh Zd߰~u*PdꡗWHiAQ Z{(0 z]B M .{ pHoe2"ua[/VT2ɡC/a < Z$@r2߫eu &Y44e2"uaWA[l[R&U Z(CH I uaW/wE!D]woeVպ(LhiuPe4 T]oe2Da$ARo*ƍ0eO-[l[R&U Z(CH I I^-\PQAQITq[l[R&U Z$@r2j:߰~u*PdꡗWA^-\ x9[oղnJ,]T2j:߰eD ~-^PBeCK^-\E!QDXI $[{ rqADRI! z]B M .{ pQDb$4"QuaW/wEo*ƍ0eO$Dn2߫eu &Y44e2DRETE$HI^-\PQAQI!vua[/VT2ɡC/a < Z$@r2߫eu &Y44e2"uaWA[l[R&U Z$@r2j:߰~u*PdꡗWHiAQ(CI" [{ rqADRI!vua[-IR8W 5O DNfC;g%s){w=OHؑg%P0`1BV܎m()!"s-M6&9.mYh9LD Ub9'QDw?!9/p w#Gjװ|Mґlv'Tlkʹ-TH`P$DUT &^PBeCK^-\ZLŶZ3jvuaW,'"ȳ#}x~Vua[/VT2ɡC/a"~/a"VFؘo2'Ϣ-[l[R&U Z Z.2߫eu &Y44e2η2j]oeVպ(LhiuPeoe2Rh Zd߰~u*PdꡗWoe2Ēua[/VT2ɡC/a߰eua[/VT2ɡC/aη2j#6y>v|p[oղnJ,]T2jC Z[oղnJ,]T2j~/a~-^PBeCK^-\DuaWUѶ&̴ź߰~u*PdꡗWE^-\H{ z]B M .{ qoe2oeV~T:12sə (0hliH-3cIu-uQl  v޺`Sn,MJ,-<́[ѓ'1*KնvZ3j|8UTkد `)qe&͵Cوp+‹*VȚ5IVw–{m,~&RUk- xUK~ <e#]o\ujrM-k#3NRRuaŞ)zo pT휤4e:ύ)&#>8{GZ5!":XP0`1BPQD E!  $DH`zZrTR߻K'e79/R!_{E~ Ff4mIvKt4~f79.mU2 (iƘ;)ՆQm<3^pVw %o35w oF휗>x4~f79.mU2 (k`VFzJed< f79.Aҧ0%~elHSNxױ7^(("D"H @C .߻;g%dO\ĺ _$JGj4!9)#\T򉓜e:TDe^ѾI7H0 !){SDhJdp`)β . h0 KGx׀Q-4hSGO3 G$1*}gSmPbNRR3qѿɾѷ6LS>4C3ն#^6؋,'S2&Tl-l5:&KƔxy3#uO&NvÚiXjqgƔ%TR6k76r\$aj`^R}휤4e:ύ)񝳒x~]}إ[p4/Vbfi(UZ~Qٕkdbh AJaU}~Wf>vuؙJ5{2[ 2eo$KјShl伙M'9/R!Du%H%J P4:nJ,]T2jf-*јS Za8)E۷ z]B M .{ q[{ rqADQIo*ƍ0slRɱn2߫eu &Y44e2ĝoe2DMvua[/VT2ɡC/a < Z$@r2߫eu &Y44e2"uaWA[l[R&U Z$@r2j:߰~u*PdꡗWHiAQ(CI" [{ rqADRI!o[oղnJ,]T2j) "*"Hi$AQ Z{($h.߻U[bakMua[/VT2ɡC/a "(,Q$D`:߰e "*# "0 z]B M .{ pHoe2"ua[/VT2ɡC/a < Z$@r2߫eu &Y44e2"uaWA[l[R&U Z(CH DITEH[ Z{) $h.߻2߫eu &Y44e2DRETE$H,RBQuaW/wEMTsqlL7r]P׬~-^PBeCK^-\E!QDXI $*"$,'[{ rqADQDa$ARo[oղnJ,]T2j:߰eD ~-^PBeCK^-\ x9[{ pHoeVպ(LhiuPeD ~/a < z]B M .{ pQDb$4 M!~/a^((C $0CQo[oղnJ,]T2j) "*"Hi$AQ(CI,[{ rqADQIo*ƍ0vcVd-[l[R&U Z(CH DITE&hI^-\PQAQIo[oղnJ,]T2j:߰eD ~-^PBeCK^-\ x9[{ pHoeVպ(LhiuPeD ~/a < z]B M .{ pQDb$4 M!D[ Z{) $i oJZ}9.j߬Wr\CI MSqKiH-zFX6čv'9/K,vSoƾqAH @pH`HH`3\Sn,MIeohߣAeO{FM A֪sOO~OEz)#?OE)%?z(;QHtG<K\'C'OI$K"<ȥ$ $;ȏ.#Gjװ|Mґlv'Tlkʹ-TH`P$DUT &^PBeCK^-\ZLŶZ3jvuaW,'"ȳ#}x~Vua[/VT2ɡC/a"~/a"tĩW\!az-fb&&f*xG6=N5FƞV2|RC-uv[OLLTAjaORFX Ohd>tĩW[Wj六D X$̀eb$3Z=7Ոlg! X$Еj)Se Vb+zĈE!="I4%Z@:Z@ LD"k$Y@T:ȍh) ؘf]|))[zĈE!="I4%Z=`)β . (aI"ަq"q-9jrM̀eb$2ަ:^ +JGj kH~SZj ηH6ηH6ηH6 ./usr/lpp/essl.html.en_US/html/essl131.htmlen_U Guide and Reference

Guide and Reference

Creating Multiple Threads and Calling ESSL from Your Fortran Program

The example shown below shows how to create up to a maximum of eight threads, where each thread calls the DURAND and DGEICD subroutines.
Be sure to compile this program with the xlf_r command and the -qnosave option. 



       program matinv_example
       implicit none
!
!  program to invert m nxn random matrices
!


       real(8), allocatable :: A(:,:,:), det(:,:), rcond(:)
       real(8)              :: dummy_aux, seed=1998, sd
       integer              :: rc, i, m=8, n=500, iopt=3, naux=0
!
! allocate storage
!


       allocate(A(n,n,m),stat=rc)
       call error_exit(rc,"Allocation of matrix A")
       allocate(det(2,m),stat=rc)
       call error_exit(rc,"Allocation of det")
       allocate(rcond(m),stat=rc)
       call error_exit(rc,"Allocation of rcond")


!
! Calculate inverses in parallel
!
!SMP$  parallel do private(i,sd), schedule(static),
!SMP$&    share(n,a,iopt,rcond,det,dummy_aux,naux)
       do i=1,m


         sd = seed + 100*i
         call durand(sd,n*n,A(1,1,i))
         call dgeicd(A(1,1,i),n,n,iopt,rcond(i),det(1,i),
     &               dummy_aux,naux)
       enddo
 


       write(*,*)'Reciprocal condition numbers of the matrices are:'
       write(*,'(4E12.4)') rcond
!
!
!
       deallocate(A,stat=rc)


       call error_exit(rc,"Deallocation of matrix A")
       deallocate(det,stat=rc)
       call error_exit(rc,"Deallocation of det")
       deallocate(rcond,stat=rc)
       call error_exit(rc,"Deallocation of rcond")
       stop


       contains
         subroutine error_exit(error_code,string)
         character(*)    :: string
         integer         :: error_code
         if(error_code .eq. 0 ) return
         write(0,*)string,": failing return code was ",error_code
         stop 1
         end subroutine error_exit
       end

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Guide and Reference

C Programs

This section describes how to code your C program.

Calling ESSL Subroutines and Functions in C

This section shows how to call ESSL subroutines and functions from your C program.

Before You Call ESSL

Before you can call the ESSL subroutines from your C program, you must have the appropriate ESSL header file installed on your system. The ESSL header file allows you to code your function calls as described in this section. It contains entries for all the ESSL subroutines. The ESSL header file is distributed with the ESSL package. The ESSL header file to be used with the C compiler is named essl.h. You should check with your system support group to verify that the appropriate ESSL header file is installed.

In the beginning of your program, before you call any of the ESSL subroutines, you must code the following statement for the ESSL header file: 

   #include <essl.h>

If you are planning to create your own threads for the ESSL Thread-Safety or SMP Library, you must include the pthread.h header file as the first include file in your C program. For an example, see "Creating Multiple Threads and Calling ESSL from Your C Program".

Coding the Calling Sequences

In C programs, the ESSL subroutines, not returning a function value, are invoked with the following type of statement: 



    subroutine-name (argument-1, . . . , argument-n);

An example of a calling sequence for SAXPY might be: 

   saxpy (5,a,x,incx,y,1);

The ESSL subroutines returning a function value are invoked with the following type of statement: 



function-value-name=subroutine-name (argument-1, . . . , argument-n);

An example of invoking DASUM might be: 

   sum = dasum (n,x,incx);

See the C publications for details about how to code the function calls.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] (= lꮥ~SZj ҷH6ҷH6ҷH6C ./usr/lpp/essl.html.en_US/html/essl135.htmlen_U   e<=>Aailnorstu "/EHRTcdghm,-.124CFLMNPSbfpy#&03589;BDIOV_qwx67:UYv!()GWXkQKh E@AdzJ'IAT$t(O %OyIBԑsjE&dnm֠zyF Ẽح굲s9s9s9s9s9<#tK'ˑOëZvչ|4z]2Gp >`!0ch0g."S֜p?+PO8 Q!QB pKGU*%i 8¸`p0@҈h&Z(" UDJ 8ర8x$ =4*C"@.?EMJs"UGRDŽqqǃ `)9$G Mǂ$H"G懔9tȸ6£W {CSyF nmOUs9s9s9s9s9s y͇+*hH@ȑhA5!3%Ƌthg9Q}]3[r"oҾ*3gi:7kozժ2qq\s|uj o;L_WƭC"*4yHVqlV  {e1r8nѽn`Bj y,N~q;L_\vjV7s&3P1c9ʋv(įv g+IV3lYbF0M򢢭@F _h{-@F _oj0M@3Y5"l9XEǥ[l$<(y;L_qfFչ}=3+Bj y,U)*/mc||Fsu5ߛme m3+yʘ~Tcz%˓s<`PÚY5  "ح{kDڄ y,Y|g?|G:&qcyʘmz[~ 7Ƴv*)O*{MqTI4I88a!L {d,YckV"{Ph=*vä$~@p0RLh{#VMgīHϣkV"{P߁J53.\;L_ s@EGah}#V#ekorⶊsI0Mj0M6_,1~fis5>{Ejro9ԓ{C^ 83$$T!"E*k-Ap&'-mʹ-Ap&'f{9hB I4I8 Guide and Reference

Guide and Reference

Setting Up a User-Supplied Subroutine for ESSL in C

Some ESSL numerical quadrature subroutines call a user-supplied subroutine, subf, identified in the ESSL calling sequence. If your program that calls the numerical quadrature subroutines is coded in C, there are some coding rules you must follow for the subf subroutine:

  • You can code the subf subroutine using only C or Fortran.

  • You must declare subf as an external subroutine in your application program.

  • You should code the subf subroutine to the specifications given in "Programming Considerations for the SUBF Subroutine". For an example of coding a subf subroutine in C, see "Example 1".

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Guide and Reference

Setting Up Scalar Data in C

Table 28 lists the scalar data types in C that are used for ESSL. Only those types and lengths used by ESSL are listed.

Table 28. Scalar Data Types in C Programs
Terminology Used by ESSL C Equivalent
Character item1

'N', 'T', 'C' or 'n', 't', 'c'

char *

"n", "t", "c"

Logical item

.TRUE., .FALSE.

Specify it as described in "Setting Up Complex and Logical Data Types in C".2
32-bit environment integer

12345, -12345

signed int
64-bit environment integer3

12345l, -12345l

long
Short-precision real number4

12.345

float
Long-precision real number4

12.345

double
Short-precision complex number4

(123.45, -54321.0)

Specify it as described in "Setting Up Complex and Logical Data Types in C".2
Long-precision complex number4

(123.45, -54321.0)

Specify it as described in "Setting Up Complex and Logical Data Types in C".2
1 ESSL accepts character data in either upper- or lowercase in its calling sequences.

2 There are no equivalent data types for logical and complex data in C. These require special procedures. For details, see the referenced section.

3 In accordance with the LP64 data model, all ESSL integer arguments remain 32-bits except for the iusadr argument for ERRSET.

4 Short- and long-precision numbers look the same in this book.

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Guide and Reference

Setting Up Complex and Logical Data Types in C

Complex and logical data types are not part of the C language; however, some ESSL subroutines require arguments of these data types.

Complex Data

ESSL provides identifiers, cmplx and dcmplx, for complex data types, defined in the ESSL header file, as well as two macro definitions, RE and IM, for handling the real and imaginary parts of complex numbers: 

   #ifndef _CMPLX
   #ifndef _REIM
   #define _REIM  1
   #endif
   typedef union  { struct { float __re, __im;}
                    __data;  double __align;} cmplx;
   #endif
   #ifndef _DCMPLX
   #ifndef _REIM
   #define _REIM  1
   #endif
   typedef union  { struct { double __re, __im;}
                    __data;  double __align;} dcmplx;
   #endif
   #ifdef _REIM
   #define RE(x)  ((x).__data.__re)
   #define IM(x)  ((x).__data.__im)
   #endif

You must, therefore, code an include statement for the ESSL header file in the beginning of your program to use these definitions. For details, see "Calling ESSL Subroutines and Functions in C".

Assuming you are using the ESSL header file, if you declare data items to be of type cmplx or dcmplx, you can pass them as short- and long-precision complex data to ESSL, respectively. You may want to write a CSET macro to initialize complex variables, using the RE and IM macros provided in the ESSL header file. Following is an example of how to use the CSET macro to initialize the complex variable alpha: 

   #include <essl.h>
   #define CSET(x,a,b)  (RE(x)=a, IM(x)=b)
   main()
   {
   cmplx alpha,t[3],s[5];
   .
   .
   .
   CSET (alpha,2.0,3.0);
   caxpy (3,alpha,s,1,t,2);
   .
   .
   .
   }

If you choose to use your own definitions for complex data, instead of those provided in the ESSL header file, you can define _CMPLX and _DCMPLX in your program for short- and long-precision complex data, respectively, using the following #define statements. These statements are coded with your global declares in the front of your program and must be coded before the #include statement for the ESSL header file. 

   #define _CMPLX
   #define _DCMPLX

If you prefer to define your complex data at compile time, you can use the job processing procedures described in "Compiling".

Logical Data

By coding the following simple macro definitions in your program, you can then use TRUE or FALSE in assigning values to or specifying any logical arguments passed to ESSL: 

   #define FALSE 0
   #define TRUE 1

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Guide and Reference

Setting Up Arrays in C

C arrays are arranged in storage in row-major order. This means that the last subscript expression increases most rapidly, the next-to-the-last subscript expression increases less rapidly, and so forth, with the first subscript expression increasing least rapidly. ESSL subroutines require that arrays passed as arguments be in column-major order. This is the array storage convention used by Fortran, described in "Setting Up Arrays in Fortran". To pass an array from your C program to ESSL, to have ESSL process the data correctly, and to get a result that is in the proper form for your C program, you can do any of the following:

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Guide and Reference

Creating Multiple Threads and Calling ESSL from Your C Program

The example shown below shows how to create two threads, where each thread calls the ISAMAX subroutine. To use the AIX pthreads library, you must specify the pthread.h header file as the first include file in your program.
Be sure to compile this program with the cc_r command. 



#include <pthread.h>
#include <stdio.h>
#include <unistd.h>
#include <essl.h>
 


/* Create structure for argument list  */
typedef struct {
  int        n;
  float     *x;
  int     incx;
} arg_list;
 


/* Define prototype for thread routine */
void *Thread(void *v);
 
int main()
{
 float  sx1[9] = { 1., 2., 7., -8., -5., -10., -9., 10., 6. };
 float  sx2[8] = { 1.,12., 7., -8., -5., -10., -9., 19.};
 pthread_t first_th;
 pthread_t second_th;
 int rc;
 arg_list a_l,b_l;
 


 /* Creating argument list for the first thread  */
 a_l.n = 9;
 a_l.incx = 1;
 a_l.x = sx1;
 
 /* Creating argument list for the second thread  */
 b_l.n = 8;
 b_l.incx = 1;
 b_l.x = sx2;
 


 /*  Creating first thread which calls the ESSL subroutine ISAMAX  */
 rc = pthread_create(&first_th, NULL, Thread, (void *) &a_l);
 if (rc) exit(-1);
 
 /*  Creating second thread which calls the ESSL subroutine ISAMAX */
 rc = pthread_create(&second_th, NULL, Thread, (void *) &b_l);
 if (rc) exit(-1);
 
 sleep(1);
 exit(0);
}
 


/* Thread routine which call ESSL routine ISAMAX */
void *Thread(void *v)
{
 arg_list *al;
 float *x;
 int n,incx;
 int i;
 
 al = (arg_list *)(v);
 x = al->x;
 n = al->n;
 incx = al->incx;


 /* Calling the ESSL subroutine ISAMAX  */
 
 i = isamax(n,x,incx);
 if ( i == 8)
   printf("max for sx2 should be 8 = %d\n",i);
 else
   printf("max for sx1 should be 6 = %d\n",i);
}

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ile. l~SZj\׷H6׷H6׷H6 =./usr/lpp/essl.html.en_US/html/essl141.htmlen_U   er <>AERainost/STcdhlmpu",.;=BDHILNOPVbfgy#&01235CFMUvw()*-49GX_x678Yk:z!KWZq{}?Qh* Zaq@s%89 ÎANcH M#s$ yQ/3Gr'1jtrIf5ԭÇ8pÇ8pÇ8pÇ8pÇ8ps p?GZ f d|A0x8@/0o,L@w8hHIqZpR\( |€04B?Ax"HK J!9To @А s!Ƞ`<$\p)T(GXg*P$ A`4$ƈA( XI+Or¸qR@VXg*XHHO 040y4yR7 *LG 4+ŒJ53"ɠ8P4|\ƀDY8u+~8pÇ8pÇ8pÇ8pÇ8pÇ8psQ) *dZFSsƑ p)yTK ƜD)ig1CDHDF c²P$$,bwU@t* qQ%'C^yӀ_xw-5U=8cD0Ssƿx $@Ӟ5 8Q" 0)9c_nxXHPoFY0 NBYzy֞zF%f$FWizLIpkN.M.R,=YsU:gK,kδ9im_tnb1 A9 4LZIO <}my}˯,^I%;δ UZ{ק-*IC`=ζ˜9mTKI_e cN&4F0(" pTY{ߦW) Å4ʉv "B!cb(:y֒Svo:pM9 YIN:ׯlO-̑Y׺ySI)y_^7UtJcK8>]KI,Ke2N[ 3~'*_޾t%&s<]eVOySIܒ^(<4 5 8Q+DTY{ߦW) Úsӈƿx $@E9=kG@pDa 1!‰`Ssƿx $@QNisQ(X`sN{B/)ӣ]F^yi_IEtmTK޾BYbS[%}U%y֜wn4LIN"-Oty[jkd΂yڂK ΘӐAU~c:5K0w=p<芋9^Wrkf ' 69y__-2] {ΙP>oI) :su_zS[%F17O:՝/nj{kR7U5T_7UZV0o#J)y2QI׶*DsM.RNWʝj)yE )|K7ynzrJN4';y,{E9O%e+KgA<ަ|/o-pk"O,Ɠ$|]I. iyi^wplԜߵ5i2Niwiyi^weKvcN&4!kG@p9$A`S-~CR[/-)isN"4>.cPxpĄ#"A1:_.kIkN. QtiwͥEΕJ;sP -%TN.k?E9oiwֽv[iBXӋFh9iE9'1sKR[Vthטν.fxy?`6wU:gBD@gzQt/-Ζ"-}%-) \>}>n@ʉ):\{ΙaoZnI|]^a/E9lmWQ.c:5KyaғS3Uk+1Ƒ s_.cKFiw^KZqwwVdf-IU%Vm|\ƜZ0sGisNB.id`l1x@HFX, PU)=uvO=y[jWi%>nl1,4kQ(I,``HP|[:J)sӈ,揋<8!5sGih7mF|\%V.cN@rsNAig ђ[n}QE}]Z$K6F&e]\nl/{ף%OPߙn9ReU%y7U5Wʉsu_ʗO>nxO6i}wk=Lݮہ]tT58(neFYlFfMXncK98Dy$S]ʞfi+iwӥdN. ƖsIqeC~W-HvZF*s4ҪqwV4LO<n|ѕ&]t״֜]p8D΄b|kђb\wOwTNT|ѕNDdaLL_iwͥZqwuK<_=L𰐠]a!#9#,7UzRjΒ|ݮr֑n'UKΉM_<0ԯ1,]S3].tf]p[';`n'UKlFfMf妗W)q(˧O71>yi^u3lFfgy[cK8>ԝg~%tj0JJ/NWʜukb\đ\{Ι^3q)([*0p|(_ DnNR%%yntb]z%zl7kZX WO.cjLe٭y֞t奷~}ӝƜ"Mig1C\DrIfC8l5}C-=ĻѼF4!cK9{^Wr3Uk<_))(zOub^t.zN.ΞŸ3[=+KۭfӕK7δK-=ĸyOt#Q,S>n}QE}΂yl/f+#}<>%%{eye妗W){{ץVt&]ʖ3]W[<%+Ξ[jkd΂yښ+Ց%-_ks9ۣ]F^Ziur`_92sr4!kG@p9$A N>F~%OB-9iC\ƖsGjNyX{\94`viwUaͥEnI|\ƜZ0sGisNB.i.cKFc?1sZ0wpAĬxqRa!A0*^xpJǼ"X`yI@@Wy2~X{P?$%DP$R'?(_$,ρ<g`t~H?FA|B(rG <1lbV<,$(7?@8"D9w%g#rE:aZiur@8h 5'ÊxrcIsK./nf6jNZo4'44/nfﲥUe ];1sYi5 8QYS-~CR[/-)isN"4>.cPxpĄ#"A1:_ҢJAsU H๯i%粡=E|\5Η.t2.t2-%TN.k?E9oiwֽv[iBXӋFh9iE9'1yl/o:"gKR[kFus'D]ʟ7ktD {.cjLRt~z$Ӌ.kV"H6J΂ylN1y<0Iשn[jkd1kPߩt󙪊TNt5y_^cH!ˆE80g+RwZs5&ӞD5ig4|\ƠǕGE@>.cKFx$[K. IuU6,"J5%TN. 58`揋Ӑ 朄\ӐxpsY';{^WrE-x(WsU:g|;JLyynYU:F|M<.%TN.],EIn%K*%o.Gf`}w˥GfZz$PS(OUV^I)va|WIFJ)=h3ojInz2(n"Z{:Jf]tY(Ӌ.dF}YP߼|)ݮr֑ʜut%.tf]p8D΄b|kђb\wOwT4|WrNT52]K)(Xgy1}O6i{'UKl}i}ϊL)库/:v Y@HH'Eʂ9 S8P4gBJ)Q>n[jkdz5E|vOtlFfg} c*+|]:VJ4{eĤnJʜ(gy1liw^KZqwu3lFfgy[o O.1;W)vT<PJ{Z+_| [ cK8>ԝgnl/3`8VZUȐt8PuUY{$we#YaM9],;5:τb|F0+-ePщUȐE8P< cK9"濔x #K4)äM=8khAÏ+sP4|\ƖޒW8Zh(XMP 5|\ə.zN.]vӋ.g3c?ə.zN.]vӋ.g3c?v4у>.cN@rsNAig ђ[<<~˘گy<،-l~zVrR%% i@8h 5&4L]u$4ffyot.cjLƖsIsK. i 0`|]-5F]wyE^/-4Kg}Քy 䢞ZiurrFTVKJ(o%/snX1sܢ81,$|]=E|]otΗYJ[9S:5:NϹur˩y UO:5Ue+δfs'칍0cY&I.Q i .Q i ^kOsI<6tΖYםiNZ[~g9ۘ@qi$Ɩr |W){s5QJΆI3}˯,ʝl5|me7h3DT],FJ):$P<3]sΆk.cN@rsNAig ђ[<$P<6t|M9ܺKyuY^u~m:1ZIO i@8h 5&4L]tiV]wϛލRS>tnZi2Niwӡ]w.t2Ӌ.O<|5ĥ2{b^ıO.cKF7m}F|\%V*pU<7k:rrY޲M^u| NZׅ/y<ݮ^uBFip]lYLF0( b**…4cYiCQ( blYLM9iAcK9n1;W)*DOލR/4S_ioZc#|f: 4y,O:)/.m8/KXsj/%:dOz29+Oy*%}K.0ɜF3|ݮD s5ߛNvmMlƖsGjNyX{\94`g3pL !F%cȼd3a801gDT], \nZ L Ƴiţ4|\Ɩ8 Bt×|I Guide and Reference

Guide and Reference

C++ Programs

This section describes how to code your C++ program.

Calling ESSL Subroutines and Functions in C++

This section shows how to call ESSL subroutines and functions from your C++ program.

Before You Call ESSL

Before you can call the ESSL subroutines from your C++ program, you must have the appropriate ESSL header file installed on your system. The ESSL header file allows you to code your function calls as described in this section. It contains entries for all the ESSL subroutines. The ESSL header file is distributed with the ESSL package. The ESSL header file to be used with the C++ compiler is named essl.h.

In the beginning of your program, before you call any of the ESSL subroutines, you must code the following statement for the ESSL header file: 

   #include <essl.h>

If you are creating your own threads for the ESSL Thread-Safe or SMP Library, you must include the pthread.h header file in your C++ program. For an example, see "Creating Multiple Threads and Calling ESSL from Your C++ Program".

Coding the Calling Sequences

In C++ programs, the ESSL subroutines, not returning a function value, are invoked with the following type of statement: 



    subroutine-name (argument-1, . . . , argument-n);

An example of a calling sequence for SAXPY might be: 

   saxpy (5,a,x,incx,y,1);

The ESSL subroutines returning a function value are invoked with the following type of statement: 



function-value-name=subroutine-name (argument-1, . . . , argument-n);

An example of invoking DASUM might be: 

   sum = dasum (n,x,incx);

See the C++ publications for details about how to code the function calls.

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Guide and Reference

Passing Arguments in C++

This section describes how to pass arguments in your C++ program.

About the Syntax Shown in This Book

The argument syntax shown in this book assumes that you have installed and are using the ESSL header file. For further details, see "Calling ESSL Subroutines and Functions in C++".

No Optional Arguments

In the ESSL calling sequences for C++, there are no optional arguments, as for some programming languages. You must code all the arguments listed in the syntax.

Arguments That Must Be Passed by Value

All scalar arguments that are not modified must be passed by value in the ESSL calling sequence. (This refers to input-only scalar arguments, such as incx, m, and lda.)

Arguments That Must Be Passed by Reference

Following are the instances in which you pass your arguments by reference (as a pointer) in the ESSL calling sequence:

Arrays

Arguments that are arrays are passed by reference, as usual.

Subroutine Names

Some ESSL subroutines call a user-supplied subroutine. The name is part of the ESSL calling sequence. It must be passed by reference.

Character Arguments

Character arguments must be passed as strings, by reference. You specify the character, in upper- or lowercase, in the ESSL calling sequence with double quotation marks around it, as in "t". Following is an example of how you can call SGEADD, specifying the transa and transb arguments as strings n and t, respectively: 

   sgeadd (a,5,"n",b,3,"t",c,4,4,3);

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Guide and Reference

Setting Up a User-Supplied Subroutine for ESSL in C++

Some ESSL numerical quadrature subroutines call a user-supplied subroutine, subf, identified in the ESSL calling sequence. If your program that calls the numerical quadrature subroutines is coded in C++, there are some coding rules you must follow for the subf subroutine:

  • You can code the subf subroutine using only C, C++, or Fortran.

  • You must declare subf as an external subroutine in your application program.

  • You should code the subf subroutine to the specifications given in "Programming Considerations for the SUBF Subroutine". For an example of coding a subf subroutine in C++, see "Example 1".

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Guide and Reference

Setting Up Scalar Data in C++

Table 29 lists the scalar data types in C++ that are used for ESSL. Only those types and lengths used by ESSL are listed.

Table 29. Scalar Data Types in C++ Programs
Terminology Used by ESSL C++ Equivalent
Character item1

'N', 'T', 'C' or 'n', 't', 'c'

char *

"n", "t", "c"

Logical item

.TRUE., .FALSE.

Specify it as described in "Setting Up Short-Precision Complex Data Types and Logical Data Types in C++".2
32-bit environment integer

12345, -12345

signed int
64-bit environment integer3

12345l, -12345l

long
Short-precision real number4

12.345

float
Long-precision real number4

12.345

double
Short-precision complex number4

(123.45, -54321.0)

Specify it as described in "Setting Up Short-Precision Complex Data Types and Logical Data Types in C++".2
Long-precision complex number4

(123.45, -54321.0)

complex5
1 ESSL accepts character data in either upper- or lowercase in its calling sequences.

2 There are no equivalent data types for logical and short-precision complex data in C++. These require special procedures. For details, see the referenced section.

3 In accordance with the LP64 data model, all ESSL integer arguments remain 32-bits except for the iusadr argument for ERRSET.

4 Short- and long-precision numbers look the same in this book.

5 This data type is defined in file <complex.h> for C++.

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Guide and Reference

Setting Up Short-Precision Complex Data Types and Logical Data Types in C++

Short-precision complex data types and logical data types are not part of the C++ language; however, some ESSL subroutines require arguments of these data types.

Short-Precision Complex Data

ESSL provides an identifier, cmplx, for the short-precision complex data type, defined in the ESSL header file, as well as two member functions, sreal and simag, for handling the real and imaginary parts of short-precision complex numbers: 

   #ifndef _CMPLX
   class cmplx
     {
     private:
       union { struct { float _re,_im; } _data; double _esvalign;};
     public:
       cmplx() { _data._re = 0.0; _data._im = 0.0; }
       cmplx(float r, float i = 0.0) { _data._re = r; _data._im = i; }
       cmplx(cmplx &c) { _data._re = c._data._re;
                              _data._im = c._data._im; }
       friend inline float sreal(const cmplx& a) { return a._data._re; }
       friend inline float simag(const cmplx& a) { return a._data._im; }
     };
   #endif

You must, therefore, code an include statement for the ESSL header file in the beginning of your program to use these definitions. For details, see "Calling ESSL Subroutines and Functions in C++".

Assuming you are using the ESSL header file, if you declare data items to be of type cmplx or complex, you can pass them as short- or long-precision complex data to ESSL, respectively. Following is an example of how you might code your program: 

   #include <complex.h>
   #include <essl.h>
   main()
   {
   cmplx alpha,t[3],s[5];
   complex beta,td[3],sd[5];
   .
   .
   .
   alpha = cmplx(2.0,3.0);
   caxpy (3,alpha,s,1,t,2);
   .
   .
   .
   beta = complex(2.0,3.0);
   zaxpy (3,beta,sd,1,td,2);
   .
   .
   .
   }

If you choose to use your own definition for short-precision complex data, instead of that provided in the ESSL header file, you can define _CMPLX in your program, using the following #define statement. This statement is coded with your global declares in the front of your program and must be coded before the #include statement for the ESSL header file. 

   #define _CMPLX

If you prefer to define your short-precision complex data at compile time, you can use the job processing procedures described in "Compiling".

Logical Data

By coding the following simple macro definitions in your program, you can then use TRUE or FALSE in assigning values to or specifying any logical arguments passed to ESSL: 

   #define FALSE 0
   #define TRUE 1

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Guide and Reference

Setting Up Arrays in C++

C++ arrays are arranged in storage in row-major order. This means that the last subscript expression increases most rapidly, the next-to-the-last subscript expression increases less rapidly, and so forth, with the first subscript expression increasing least rapidly. ESSL subroutines require that arrays passed as arguments be in column-major order. This is the array storage convention used by Fortran, described in "Setting Up Arrays in Fortran". To pass an array from your C++ program to ESSL, to have ESSL process the data correctly, and to get a result that is in the proper form for your C++ program, you can do any of the following:

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Guide and Reference

Creating Multiple Threads and Calling ESSL from Your C++ Program

The example shown below shows how to create two threads, where each thread calls the ISAMAX subroutine. To use the AIX pthreads library, you must remember to code the pthread.h header file in your C++ program.
Be sure to compile this program with the xlC_r command. 



#include "essl.h"
#include <iostream.h>
 
/* Define prototype for thread routine */
void *Thread(void *v);
 


/* Define prototype for thread library routine, which is in C */
extern "C" {
#include <pthread.h>
#include <stdlib.h>
int pthread_create(pthread_t *tid, const pthread_attr_t *attr,
                   void *(*start_routine)(void *), void *arg);
}
 


extern "Fortran" int isamax(const int &,  float *, const int &);
 
/* Create structure for argument list  */
struct arg_list {
  int      n;
  float   *x;
  int   incx;
};
 


void main()
{
 float  sx1[9] = { 1., 2., 7., -8., -5., -10., -9., 10., 6. };
 float  sx2[8] = { 1.,12., 7., -8., -5., -10., -9., 19.};
 pthread_t first_th;
 pthread_t second_th;
 int rc;
 struct arg_list a_l,b_l;
 


 a_l.n = 9;
 a_l.incx = 1;
 a_l.x = sx1;
 
 b_l.n = 8;
 b_l.incx = 1;
 b_l.x = sx2;
 


 /* Creating argument list for first thread  */
 rc = pthread_create(&first_th, NULL, Thread, (void *) &a_l);
 if (rc) exit(-1);
 
 /* Creating argument list for second thread  */
 rc = pthread_create(&second_th, NULL, Thread, (void *) &b_l);
 if (rc) exit(-1);
 


 sleep(20);
 exit(0);
}
 


/* Thread routine which calls the ESSL subroutine ISAMAX */
void* Thread(void *v)
{
 struct arg_list *al;
 float *t;
 int n,incx;
 int i;
 


 al = (struct arg_list *)(v);
 t = al->x;
 n = al->n;
 incx = al->incx;
 


 /* Calling the ESSL subroutine ISAMAX  */
 
 i = isamax(n,t,incx);
 if ( i == 8)
   cout << "max for sx2 should be 8 = " << i << "\n";
 else
   cout << "max for sx1 should be 6 = " << i << "\n";
 return NULL;
}

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  li~SZjX޷H6޷H6޷H6:./usr/lpp/essl.html.en_US/html/essl149.htmlen_U    er <>AERainost/STcdhlmu",.;=BDFHILNOPVbfgpy#&-01235CMUvw()*489GXx+67Y_k:qz{}!?KWZQh,^"9q A'N<)O>E9$G4Aj$)ךyDpnQ,iDƀHONK$o%]Nߎ8pÇ8pÇ8pÇ8pÇ8pÇB\ƀHOǿZ fNᄼA4^I+]8x iwie:u*'^LsRJjs'zydfgC~gi$|[){0W/kyӝmMƗqYuo;$?#RNsjodj/_:N|ݮ/:FU*'RT7gO Iuo헼nw·Le/DؕV8y֒Wvگ队|8-|9oe*'O:vHHE8Q:^/kvt )TO`DprD"DaNU/y^uݮΜ<4֟$nI?u%~V Pc]i CU(-q:^/kvt H5 8Q2)=c_ʼB A!5 8Q2)=c_ʼB A NisU(`h 1܍}LSJ?:^2k@ڛ-D%vڛ-+δ櫾gδ JY2y ]lnmÙ*'B&y_^tS9DC~Ö;Uڛ,3|9ndU~_ܱJN9y֒TgziΆg嘴ѬJΔW޾G9]ƗsH9*K$h /8X{L^|[){sU)]D՝ s]1yHqBj1sQ 揋<z:O6Wsg5]u3m1ǁg8 B1gJpn6u$ k?Ut+qs k?Ut+qk*1@w|pn4v*EqtVt/$ai4|\ƞ4b枋ǔEL{^WŲJ|NwSͫS2W+-yL涴vmMS;QEz(J|纙ڊ.Ut񧢋kR櫾gd榮 詝6=]y__+-2g] {ΙT1>oI+ f:s_yͩcI9^!‰dCAB  IYn{Ķr2Y.cODz1sOEAiw"ĕ弹Ηat״֞apz9׶'W})vjL*Y VJ'ĕ幪ﺙĥr{b|u1ATOeOЕ%D՝ .gjLVu%~z$>./kpOIIlm2WU>g:UO󧺙ғS\Wsjod yP﩯u3Nt7y_^cI9^!‰dAB  t%fey֜w 4s]1yHqBj1sQ 揋i^OZywV~]e5j -&,u|\ƞ^0sGisOF.iyH9.Tj{k^WŲL既WiVthdb -S<@lt3+_-JJou3R%3^{Փ9+0 g]9y֞2 jfiɮZyvJq6NyTO\Kk]<[3ikR%fv]eA(*D@R%B`<7k WVZW71:*e-J~T:{):5}6K0JJozF(Aiw1DINBs~%UPy4AcK5).m*.׮rM<04>.cODz1sOEAiwQRv=N^u;tS<#cs;U:`&Sܳ.|ݮOyW>Rwݮ.JN 2v^#y!}ZnI|\/4Ie3*yگa0ŗIUq>jI)ӼWr!,Rv^ؘ%kC|NwQ|w1Akt5]ZWiN&]Lv)UOlPy*&F(#~ٝ#eP(4bYPrD" E8Q*S_֗ګ]D՝ ӔRKs'z Q?볡 _g󧺙]^uFFUᜍwʟX/3`DprD#`0)  M=ou3l󷺙%wU/y^u]4AkW@pY$@@`!|9Æ~%UP-=i䃜ƗsGj/Ryh{\H94`v~iͥEΕ. cO/94H94^<Ɨr*N)*#칝0_Ri)䪵;U;I-4zNfgWi ')ySw$ k/:Jt'q*fʹ:^2k_1?zG֊5~'*_޾BUljӤ:jS{%;Uu3·]lkU)Rvfn}畭3tUįĤo:s_yͩMLvKr5Vv0yO0 _eCykv i_*21): ƈs](MĪ'֕l$JTI%U ėO|էI_,+L>.+%Zyf4PTO*oIO޶Z7)SsgK| JiiBnELAگy< F"|wL]sJWi^OZyf4cKGJJUߒIOޕSv9RsJW+s9:fU͑pg#]AL>.m*>.RRBYĤkz ɚaگKZtw_J~tNiJygaiӥV4L9_J~1);¹ﺓ]lw.zL#[){{)UOcO ƗqVvu3RU|_Ξg]9y֞2 %+-eBAʃ<#"##* Z{3tl^uu3·Le/va~ٝ#eX T|8Q:^/kwe#yaM=|ydg/:ׄbJ0"8VZʆC)u$B"1B7š{1.4sU(I< 4)_pgU_(H9iw4|\Ƣ!ǖGD>.cKOR$p,X kO0iԚyp as? 0zO0av>.g11v4>.cODz1sOEAiw"ҒLA˙گy.O0sLDW[){ cK ORiyLoFUΕiyNu3]}Jܲt6[UΕV8y֜i؟2wLA˙گy1҄ ap ƞap*'^ju<=r̫r:UO%%W{+Ng9Z؟3$kj=3^{ׇ ΊĤn9…|0_5]I%<@vtΞY-δ/- 5֠H4cK$濕x %O4 9  ig!}^uO{y 1܊lS=+KۚNt7{W~I%8_=M{ܺNSr1?:v|]=I|]Ok詖kR3MWV` f%y֞v姸S)}Ī%#*I/S-וlŤw$'Ĥol=:^Zmuzy|էLjedokdo<%)#**'SM. cO/94H94^<Ɨp'o;ySZI_knJJou31A.gjLԚJ{^dӾtS>n}N^uwwu|].ԚJy$֙s޼WsjodB1AIUq>jI)ӼWrz ;{_yۯlL[$ho{_p@j9 WVZzӉS7UuS~_gk(Yk* T#NJ|>T=Wny)Q5gC~t%f:ބC~TO-zo—xnyciёp4g#]l8 %( pdpSO{[L<{Il5]UcWi%~nl1.4!*H` `N6KW{Ɨr*NXDOޕS/4Les׺OL>.֞apcOiԒ֞ap3^g,s~v}䭫D支RW[-i{)UON_ƗsGj/Ryh{\H94`3c?ಅAED|@bZ<ʐ(L,{?O N b?Ut+qk*1@wD]< lkIa~:$VmtQӄEqkP /[RՀD84\~%  cO/94`3cyx1@<(s7):p(f:_@^O@ :_IcEΗX0:_6:VJ*X\KR,tk?/%Xh/;c?ƞ^0sGixreIM ,:$˖&UG[JK#(+9kI򕾬5<`揋3i4|\Ɨ&T$PâJ S1-X$yHg鮲x5j -5XYcai4|\Ɨ~1cO/94`8h!û`X"!yR @Dx9x*2<#A"Z< si^; gd†ҫe5$CYT7w v~m*.t:Uld z.{*04>.cKc?1s^0wx= x NDXt AB8FC3dz$ʼDɕ."&DnF%ù`3IC*L@DCyAx$ &AсLD9W92Q,DBQTD"#<rx_&Bey8qjEΉE.tIqgEΉAr\.\kRp4>.cK3x<J0@D$ʂ2Q,DBQPF gL*gyR:$AqI*=g<`揋3sz$ыz/RcK$s_ʼB!%p}V8y֒Wvo:pisO$s]:yui^u*x NV8y֒WwΌNsjod$S=+KΞg5UUorjS{%1 pGjTO1A9^bX~[/:JM9ߺle-֐S>5_׺Ғ1B+ӣ9Vvu+lbUXZI_{+˯ҼO*'^&SL{_[Ne;_FG YPPP.D "XSL!0ANJIJ\{Θiw1DIH!|9~TLjfJ/+Kۚ O %*&m.cODz1sOEAiw"[=+KsT>';}թ+ĕs[Z_;U6K 詝t=]^sT>g WsLEΉ*xE̵|[)~jx NI.K|]: : ,u|\5^-4AcA!ѣXh=i=)1܊le/o:$OV[櫾gFUu'})DF}>nBTVt7칝29Z=ԕL>.ܓO0%%JֆtTg]TU.Mu:>.P(P|\,tZk֞apkyx1A= 4S:+KS< "[s;U:g|;J纙>nYSi#*&nv|\,u|]o.$o8*yF%9y֞ "X_gk(Yk* T2K t "X$& Svui^uI)Q5sz گy{3[=+KJ|t'EN,󧺙ғS\Wsjod tbƗsH9*K$h)_p@䪾0H9iw4|\Ƣ!ǖGD>.cK7m!.iͥEnI|\ƞ^0sGisOF.iyH9.**NvδÃ'nVgb{.gjLԚJ{^eϛSi#*_NUߛE3atiVawRwNT}%:*NÞvho2Ü/F(#wΆJ=i˩e*1Db/3`Dp R*HPy %Q>*pU|7k:rypJWindBBS*'㖽vt7KtS<7kδʸB{3S6KWfvH` CE8TX)cI9^!‰dC9 3*bisO$4>.cQxD#bA1+L>.m*.ti iӥV4d4i'nxO6iwo=M|ݮـV|]= O0 /)S%3GBÁ#XjI)vҪyWxINv9R)*u1._,3 A}O6i{)UOl}iϊ̫庳?:v]eARʑ, de\`( *DگozK΋瑫SwΞgJf *K|]:VJ{eĤJJ,g#]AJf[o O.fwA)hR+-_:ypfwA)hRa~76wo '(1.5.S^gO IuoĤol=)vr=e\J0"8VZʅF)uDyHF$ ENFFUg#]ٜOS^gJ^I+va~ٝ#eX T@ScWi%|7kC-i ipF%9y֞# lQ²T41K9" ՝iw1DIF~%UA=i䃜ƗsGj/Ryh{\H94`at.;KgM0`iv|]=I|]0R i3;?4ݦjA<1cO/94H94^<Ɨr*N)*)칝3ʉĦjyo.geD3z_hĪq)*X_S7Á#XcK RKcO0%3S{ Ss;U:`4Pat[O0Xi4=:^TO{^Zmu4Pat&aw:_ )le\V:S>t}.Cl5[LUcWif'})칝0c](M0R i 0R i ~櫷SsL+kLʹ.ӥTRU|JTs2H 涫{ Ss;U:gxx 詜JJJ3(_ Q?jUۨ3.ӥTIU[<[38^n6։,S {.gjLٜO:rS^NjcO(4S=+KۚNt7I)u3]}JJT3MWV'~I^uΊUt)*oR%Zt5]ZWifLfnE[Kfg1ƒs_ʼB'9  >JLwҼF4AcK5).cODz1sOEAiw"ҒL&ToeWIOuҼO#5^,tbUXZI_ jf@8p$k"Qiw474ҭ<;{7k*|ܵ1҄CAm<*'ath1|]yo7Kg#*ɉc.yui^u&nv|]: iy֙b~\JJwU~nl/ʉNyvn5]vaecku-6K_-JI+[){sU3tl^u@R%fv]eA(*D@R%B`<7k WVZW71J9.4sU(I< 488S>g1%WyLj:ɉf٧.ԚJtKtS:+δ3Ut+δ7kcI9^!‰dCQNBs~%UPF-=i䃜ƗsGj/Ryh{\H94`viͥEnI|\ƞ^0sGisOF.iyH9.**NvδÃ'nVgb1\{Θ/4w,S/||ݮOyW90yO0ڭFBL{^#y!}ZnI|\/4Ie3*yگab1/k9Ғ|wS/yگKRv^ؘIZ|w1Ar5:+δS.o;U׶21D#lQ²T(1K9"AF"(D|zKUݮRjΆKMyÊ)^u NZ߅/yLݮ/:ӣ#* ipFO,q\J0"8P9"0 Pcঞ cK$s_ʼB!N6KW{Ɨr*N40$SldsJʟ7k*0,(8_Ri).޴%j: 87Lx 5s+e/0m<7KXsj/%:椕OzR9;Py*'}K.c0ѬJ5|ݮ%D՝ Uߛ̝Rv6K cK5).cODz1sOEAiw4\%(  r59kiw  %+-eB]eE8꽹yƞpp@Q²T,8(bYQN+7=a#ePP`.ESSL V3 R1.1 Guide and Reference ɉ k4k~SZj?޷H6޷H6޷H6?./usr/lpp/essl.html.en_US/html/essl150.htmlen_U Guide and Reference

Guide and Reference

PL/I Programs

If you are using ESSL with PL/I Set for AIX, Version 1, see the PL/I publications for details on calling subroutines and functions.

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Guide and Reference

Processing Your Program

This chapter provides information on how to process your program. It contains processing procedures that apply to Fortran, C, and C++ programs using ESSL. It describes only the ESSL-specific changes you need to make to your job setup procedures. For complete examples of job setup procedures, see your programming language's programming guide.

Notes:

  1. For the ESSL SMP Library, you can use the XL Fortran XLSMPOPTS environment variable to specify options which affect SMP execution. For details, see the Fortran publications.

  2. If you are using either the ESSL POWER2 Library or the ESSL Thread-Safe POWER2 Library, you must be running on a RS/6000 POWER2 processor.

Compiling

This section describes how to compile your program.

General Procedures

You can use any procedures you are currently using for compiling your program. ESSL requires no changes to the compile-time setup procedures for C or C++ programs, because the ESSL header file, essl.h, which is used for C and C++ programs, is installed in the /usr/include directory.

ESSL supports the XL Fortran compile-time option -qextname. For details, see the Fortran manuals.

You must use only the allowable compilers or assemblers listed in Table 2.

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Using Your Own Complex Data Definitions in C Programs

If you want to specify your own definitions for short- and long-precision complex data, add -D_CMPLX and -D_DCMPLX, respectively, to your compile command, as shown here:
ESSL Library Name Command
SMP -or-

Thread-Safe -or-

Thread-Safe POWER2 

   cc_r  -cO  -D_CMPLX  -D_DCMPLX  xyz.c

POWER2 -or-

POWER 

   cc  -cO  -D_CMPLX  -D_DCMPLX  xyz.c

where xyz.c is the name of your C program. Otherwise, you automatically use the definitions of short- and long-precision complex data provided in the ESSL header file.

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Using Your Own Short Complex Data Definitions in C++ Programs

If you want to specify your own definition for short-precision complex data, add -D_CMPLX to your command, as shown here:
ESSL Library Name Command
SMP -or-

Thread-Safe -or-

Thread-Safe POWER2 

   xlC_r  -cO  -D_CMPLX  xyz.C -qnocinc=/usr/include/essl

POWER2 -or-

POWER 

   xlC  -cO  -D_CMPLX  xyz.C -qnocinc=/usr/include/essl

where xyz.C is the name of your C++ program. Otherwise, you automatically use the definition of short-precision complex data provided in the ESSL header file.

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Guide and Reference

Compiling and Linking

You can use any procedures you are currently using to link or run your program, as long as you make the necessary modifications for ESSL. This section describes these modifications. For details on the complete procedures, see your operating system and programming language manuals.

64-bit environment

The ESSL libraries for AIX 4.3.2 which support a 64-bit environment are:

  • The ESSL POWER Library

  • The ESSL Thread-Safe Library

  • The ESSL SMP Library

64-bit environment applications can be created on any AIX 4.3.2 system, but can run only on 64-bit hardware.

If you are accessing ESSL from a 64-bit environment program, you must add the -q64 compiler option. This compiler option is language independent.
ESSL Library Name Command
SMP 
   xlf_r  -O  -qnosave -q64 xyz.f  -lesslsmp

Thread-Safe 
   cc_r  -O -q64 xyz.c  -lessl_r

POWER 
   xlC  -O  -q64 xyz.C  -lessl

where xyz.f is the name of your Fortran program, xyz.c is the name of your C program, and xyz.C is the name of your C++ program.

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Guide and Reference

Fortran Programs

If you are accessing ESSL from a Fortran program and want to compile and link in one step, you can use the following command:
ESSL Library Name Command
SMP 
   xlf_r  -O  -qnosave xyz.f  -lesslsmp

Thread-Safe 
   xlf_r  -O -qnosave xyz.f  -lessl_r

Thread-Safe POWER2 
   xlf_r  -O -qnosave xyz.f  -lesslp2_r

POWER2 
   xlf  -O  xyz.f  -lesslp2

POWER 
   xlf  -O  xyz.f  -lessl

where xyz.f is the name of your Fortran program.

If you want to compile and link your Fortran program in separate steps, you can use the following commands:
ESSL Library Name Command
SMP 
   xlf_r  -O  -c  -qnosave  xyz.f
   xlf_r  xyz.o  -lesslsmp

Thread-Safe 
   xlf_r  -O  -c  -qnosave  xyz.f
   xlf_r  xyz.o  -lessl_r

Thread-Safe POWER2 
   xlf_r  -O  -c  -qnosave  xyz.f
   xlf_r  xyz.o  -lesslp2_r

POWER2 
   xlf  -O  -c  xyz.f
   xlf  xyz.o  -lesslp2

POWER 
   xlf  -O  -c  xyz.f
   xlf  xyz.o  -lessl

where xyz.f is the name of your Fortran program, and xyz.O is the name of your object file.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  l&~SZjkH6H6H6 ./usr/lpp/essl.html.en_US/html/essl156.htmlen_U    <>T "/=AELRe-DGINOPVacors.BCFHM_dfilmnptxy0123SXghuz#&,4579:;Wbw!68Uv()QYkjK N\nL4-q:|-.<6r TJ*mX7q"mU!V=*gB!B!B!B!B!Cn63M O@z^887$'880pq /_4mfA !~ R$`pq:[7u3dd TH;I; `/A@K:[7u3ddJqi#G`KP_Ib$ pX  v6Kcl uL$Yw;$vP7 ,_pX  v66Q;,RGN(iM*AA 1@q2rJՃCwvXcl4"8ڪBzUe !B!B!B!B!B!7qt\MP-М<@ JRV4'ilh^\z1(8mU!Vq:L (Hyr` %/*Q|r4/qt |qGJ@аH@)o6+uXUe$yOar^/ap)*C*__BĉתT&H䊵r엩`72YRAW&Mpy"eT(bˊJJcoFу!fڒnO̐9X3T[oѹz7h9 u#4Fԓv \EJmFiޝ4FkjI&~xу!fڒntX{i ԍ0JߚNMoNتeN~/i;h9 u#4Fԓvh{Tmv^4FkjIiYYa~j_ڑ2#oR4+~i:16ޝ%T~^zw=:rFh&*C#^@P);/NNC5Hу$ݍ,?5/H `4Yd`6ޝ%T~^zw=:rFh&M l~xу!fڒnt[Rԍ0z [IՖI m:Kb;-L6z0t3]Ԍ=;RMؚ@Q;/NNC5Hу$ݍ,a"6#L擫,m:Kb;-LENXObԍ0z Z߻<}yUJWG\A/ap(% G\C*__BĉתT&HWc\9JrE{5, dK2W*U ח"2k.PH2PclU19qIPS{Lmz0t3]Ԍ=;RM 0z*m7/Fу!fڒnk(R ܷ2ӼfӐwR3@`I7``oӲz0t3]Ԍ=;RMNcVKR4DmF%fX{ڑ2#oR4+i:6Jm:Kb;-L6z0t3]Ԍ=;RM١UR*Fzw=:rFh&mIge%a"6#L,=H `4YdoNتeN~/i;h9 u#4Fԓvh{TM l~xу!fڒntX{_ڑ2#oR4+7ԍ0z _IՖI mXLoe1xу!fڒnʁMzw=:rFh&mIge5d#L0dFޤiVoe#L0dFޤiWueeoNتeN~/i;h9 u#4Fԓv&6TmNӼfӐwR3@`I7cm:K?-%a"6#L-a"6#L擫,m:Kb;-LENXObԍ0z Z߻<}yUJWG\A/ap)*Ai ԍ0MʪRr< uU$Km<@.^9peY nequ*\ʓ&g3^\>H*H%ʳ#]Ji5ʇYIK.P:.,9"!>UHUD:W嘩ˮKUbDL+1*DRE.UEv;!\ީRC,V$%(XLr T A "/=AELRTe-CDGINOPVacilnors.BFH_dfmptuxy123MSghz#&+,04589:;WXbqw!67Uv()QYkKj LJ|eiP;˄2ᐄ~X6'#ucI~n>]o_p.uYhs1c1c1c1c1c<_pu8qv\wzDp &.+@/]e`!0(d@`/KP  ?/]Y"cӋɑ+1-4-i`? K +;K.` =8\&b(ZhZ/PtU%g  Z0_rVjY`:*`@#A@h` v@dnMQ#Dk,XӸuL*0($=a 2`wu][C1c1c1c1c1c1ye] /$F]oĵ03r] /u] -ef./m$Ap%0h` ~td=7k] /<Ѐ` ~%t$8B(PS!-v%Y'jEUmpu8!܋LMF rLY Ee`w]f+cdDʲOw/Dz`? 08 Ue*⑩-]vXg9gCB/)?fOE-(^.셆nXg9gCB/)?Ld셆nBܵuc|yE?*r}I7rP1vNT<""ؕg5GNU҅Dr ʺ$^dL2+m,«x]vA*]  Wre!byEE'P}ݐ7ʇ_Ir!ī=\~Z?r*#NU&~ZmƵ 0ʫ0`M Luv]=v ,n-]W]7ʇ_Ir!j,9A"Bݐ7ʇ_Ir!ī=\~Z?r*#NU&~ZmȅkaVaXmAf\ Vz]XZ,o9gCB/)?RD:!cw,o9gCB/)?]V{( tImr!| j 0^mݐJ`BBܵuc|yE?*r}I c|yE?*r}IJE](TG LkL2+m,«x]vA*]  WrO]hLB_r*#NU9ki [zB(B$܋L.Ĵ$MF rLY Ee`w]f/*6`7"!(CnTiBW풹$0mߒY$(@P,]vZB>):eyP1vNT<""aX"ҍ(BXfyP1vNT<""HQXf-]W]7ʇ_Ir!;!cw,o9gCB/)?]V{(ZQtQr3aVaXmAf\0k_r*#NU,Zm E=v ,n-]W]7ʇ_Ir!j,9AvBX*r}&.ʇ_dR~Qsj'*B9]g,$*±6ڂ*1ׅmw.`ֿ](TG Yki=v ,n-]W]7ʇ_Ir!j,9A"Bݐ7ʇ_Ir!ī=\~ZʺPA9WD 0ʫ0`M Luv]59WJ'*-{BƴUWr7r嫮CB/9POԑΡnXCB/9PObU.g媏 tI j 0^m(r*#NU,Zmȅ gU܅܅jP1vNT<""$A7rP1vNT<""ؕgj'*B9]g,$*±6ڂ*1ׅmw.g tK?mwd*w!i ؖ.'*B9]$!~„/ʲM` ȴ{JNU҅Dr ʺ'-mg!koYHB_?d]..=e%ؖ$!~EU`~U}Bd %Qumߒq0%c? D"Yaʲ@)(INE@u\wYHBaPu^yVI-\}푺+ʲM2cX̱DzlAEk2UWN9BJ|F [y&\UmuY AF67d&W@sUui jlW]CB/9POهS`J4W !aCB/9PO AF;!aw-]vX*r}&.ʇ_dR~셍ܱT<"L]ӕ9Ȥv%Y kFr*#NU&~Zm GaVaXmAf\ Vz]XZ,o9gCB/)?l Xr셍ܱT<"L]ӕ9Ȥv%Y kFr*#NU&~ZmƵ 0ʫ0`M Luv]=v ,n-]W]7ʇ_Ir!j,9A"Bݐ7ʇ_Ir!ī=\~Z?hRCNU҅Dr ʺ$^rL2+m,«x]vA*]  Wre!byEE'HPd,n!byEE'*3Fr*#NU&~ZmȅL2+m,«x]vA*]  Wre!byEE'HWd,n!byEE'*3Fr*#NU&~Zma&Uf aUɎ.k gU܅܅ HU'ķWpV@wq fHDxԔ̑(|@A`wqP( HԚ$(|@A`wqP($@ߨ\.Kx |YkbD \ fL~@2+I@A`wq$($@ L@v@Vd ^)CQ5%K]r,Mp@s_ʼB k*~SZj, H6H6H6, ./usr/lpp/essl.html.en_US/html/essl158.htmlen_U Guide and Reference

Guide and Reference

Migrating Your Programs

This chapter explains many aspects of migrating your application programs to use the ESSL subroutines. It covers:

  • Migrating ESSL Version 3 programs to Version 3 Release 1.1
  • Migrating ESSL Version 2 programs to Version 3
  • Planning for future migration
  • Migrating between RS/6000 processors
  • Migrating from other libraries to ESSL

Migrating ESSL Version 3 Programs to Version 3 Release 1.1

This section describes all the aspects of migrating your ESSL Version 3 application programs to Version 3 Release 1.1.
Note:For a list of the new features added in ESSL Version 3 Release 1.1, see "What's New for ESSL for AIX".

ESSL Subroutines

The calling sequences for the subroutines in ESSL Version 3 and ESSL Version 3 Release 1.1 are identical.

Distinct libraries are provided for AIX 4.2.1 and AIX 4.3.2

  • For AIX 4.2.1, the ESSL Thread-Safe Library, the ESSL Thread-Safe POWER2 Library, and the ESSL SMP Library were built using the pthreads draft 7 supplied on AIX 4.2.1. (This is the same as ESSL 3.1)

  • For AIX 4.3.2, the ESSL Thread-Safe Library, the ESSL Thread-Safe POWER2 Library, and the ESSL SMP Library were built using the pthreads library that conforms to the IEEE POSIX 1003.1-1996 specification supplied on AIX 4.3.

Threaded applications built using ESSL 3.1 will continue to run with ESSL 3.1.1.

If you are migrating to a 64-bit environment you may need to make changes to your call to ERRSET. See ERRSET--ESSL ERRSET Subroutine for ESSL.

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Guide and Reference

Migrating ESSL Version 2 Programs to Version 3

This section describes all the aspects of migrating your ESSL Version 2 application programs to Version 3.
Note:For a list of the new features and subroutines added in ESSL Version 3, see "Changes for ESSL Version 3".

ESSL Subroutines

The calling sequences for the subroutines in ESSL Version 2 and ESSL Version 3 are identical. This includes the new ESSL SMP and Thread-Safe Libraries that are included in the ESSL Version 3 product. You do not have to change your existing application programs that call ESSL subroutines when migrating to the ESSL Version 3 product. You must, however, re-link your application program. Therefore, you can simply re-link your existing programs to take advantage of the the increased performance of using the ESSL SMP Library on the SMP processors.

For the _GEF and _GEFCD subroutines, the first column of the matrix L with the corresponding Uii = 0 diagonal element is identified in a computational error message. Previously, the last column was identified. You do not have to make any modifications to your existing application programs that call these subroutines.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] si kʗ~SZjE3 K63 K63 K6E./usr/lpp/essl.html.en_US/html/essl16.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui k8~SZjc)555c)./usr/lpp/essl.html.en_US/html/essl160.gifen_UGIF87aRRR))Z11ZZ1))!)!9)s!R9)c)!B)B)J1B1R)!sB9J)B)Z9R91kBc1sBc9cBk1{JZ9cBJB9{Z)cBkJ{1scޜR9!{sRޥ1kR!ZJ)91!k{!k!BJ9J{c1)JRRRJ1skRƭJJRcֽ9νB9)ZZJRRJsRZBRBB9JBBBRRRR!1!9B919BJ)s119R{ZR!!9!c1k1c11{JBkR9kR9ZJBZRZZ1RR1BBBRRsJZZ)111999BBk{{sBJJJRRs{ck!RRZ!1B9{BJ19B9cks{k{ZckBJRcks19BRZc!99B91BB)Z9RBZc1{c){k1ZBc{JcRcs9k9{RZR9ZsZ{RccBkck{cZZRBk!Z)!{9kcBZB1cR9csRZB{ZZkJcZkc!s!kck9kZ{ck9)sck)sRscRJsR{R9BBs99R!Z)1{)Z!1!B)B)Js!19R9RJcJRcJZZkZcBR19Zcckƽ{{{ssskkkcccZZZRRR111)))!!!, dTFs!!&(b1RxQ"NJ?r,h'C>(F 3He:.on &C+r!Lv4 %dDy&uҡO,Vʒeģ{tM5}ރrşh7Ƶ`s=J,̮ ̔nvg72|lJKGif܁n-۝M۸s߆B3*t Nȓ3ݛwoУ3uAfN];#+vNxӧ]|{÷?y~G ́FEy 蠁`f7Uxjסxn!~!2z{"'b(ߌvga8}wbC9oTa #⸣G"i#\eE\:嗌%aY)cUȜaf`yer %rzg|&^֥v)2RŠ=梏爧X(gg}喏Y]nij>ꧬ٪ʦeB H2)Az}JکV cvϖ :n{06+oªkK./:/+o:0>Oܯ֩rr 4r BQg 7㝇 L YDsTD3Nxw77ʶ^'/m}b.T6{.7o?b 0pYg){/~O_~.|,ಶ#,Ĭ 3ըRB!3Ho6@ۙO $? rPd/pA~SǼC >POox=CpTA:Q+݈AGa)9E u>oaL q$ l46'OH@|c$Gy%W I5C^TeHP=po('t*oB]:st+6^o#)H[RR$Ih{hLI!QhMoZ9FW͞g39t|8#jV dBnHF[IPsz8c:ѸN* D&<^O'P034QjR~ mhCm h4Õ4(gύ{Q}@:<t*5JTjPQ:PڜNSƔEJ1AT0 ٺ ?JUvj`jճʂX nB}FZΚzulRZWj~lZO[U b+HbTۼMGY4p.|Ow)e-\vX[[[4,YhӺХ&[I*]Tz^4-^jZ7H,bg]*ŭDm J'A6翘AP{]GmK݁0宁{[[m_3 +$'0Wzg$>U=Cŋ=Z{|Q5V(蘓`c>z^9pN1Cm+)7({ilq9Y .˻k/a™(2g;~F#n 8_B  #LB;K%8H׾5;V;^\<prS=4Pk@'Nxz;(pM\׻FwսHr/>6ˀs:c~B#.&ҥ:32`β^xDęl:9x4/{;@Md4Np @`̽q™7 WlXCGu_x{s]: 2/͋_8ȅ2?D&}0vg< R3 Y+uDDv܅dw=e(p3>u"σs:k\Os*a0u=s9G/}ט`1ӗ|DWO{wB0סnL{M fFsb%3·}ү;蠿>'idꅧz ?YH//'/Qe`|L <_=/zun7zt7~Uq'^ }|:78D5c.$h}%sD}rT~'^llpԀ~+vTO>1p:YC6xIhv@7{op p Kh/c!8hkmI5`aWuW}e}W~ 6 K(b]@Iݳ"//HCT8\7hLjq& 1o 0 v@e@h:@ Ic+Luh>=hkkHr7 ȂFwlk#؀HHt(k(p yЊy{Yh`&)B@*0RjPcLvu9byK hٝn閰9 m Ai{ٝȝXi銾31kD ٞ YI o9mI\?)LoQhU ѹ٢+: by.\]9yy(-vH J:1I0WhdIY/ڟWJVZht0Eʥpp:<ږ@)DZ5-0LCR-@7TW: )cP[Penjz.-pN3H wLՃP UCZ`jvŸvЪjʨt{I*az(-=m؇>=h7 )jzD5ڢZi@\ɫʉʑj.&q\<3 ʪ(kvP JZmyc@hd Jv0 e)mԪ K0ۭv`5K*/; 8+ˉj\`۪BBk7v0y7@ʩ kSJc\; 5;X;l˪Xk2pˉrk*sKUK~h`J۫dp[Zp wxt{j#S7;s8۶uk6[t;˷ytDKiKFꐸRaДnz;W;:˺[?k[jkJZv [>{[${TP>7{`< ;m;*ۿ쿰4;+nIKLPˉ} ,k0L|&IvG;=̪ ,|Dܭĉ*tKFl`'LFv@{[ >, \L_\v c ŀ,9L٠[|V<+wܿBpJ 7`I< L|94{EW|ʫ_ǁ tܠpµ=XK;ȱZ-`(zvp 4̠ Al ʢJ li+ ސ'  Elj.`-7P}pЫz mMl~DZm NjP 2{ ]}}`P}@HG}N~ GK}-^\R%}ް,l`2л|v>vԴL  O S-] V]KlkٛkPJ}ޠ ۠ m-ppYUmZc@ـ]ְ8M` pؐ=QVەۻD`pBݜ}h=ڦ J`o-ӭP=H-}ݾm mS ںѭG`@ ݛ۽}ަ}ғ Ӭ- .߲..0`Rm-}Ԑ ِIJvKs@^㱽!Nޔ HzH*X. LnLnܫp0p 0|~ ǭJ>n`Iy q~Q`.p_f>.o>p.MꪞJ~p 08n~ PS>q^n~ꊞ[n7 ,֦^夾Qn|.mp@~ྮC]ӈՎpn ^rPnmwB~}۞POߠ 1nڐ:>?鐐A0`a`WS]&n腎Z^q~پ/ڞ|0M>t$m ܐn`xZmLywo>wv~~}w~~aZY\_KdJ>[䣟яԟ_/Yompщ܊Vُ?| !X` !%ɢ|XP"B| ޥ%'{1ݣgo彔)R`Ď{SƆvLjt`HM)*H5C 9\Jc$mϞEbc*0Kc/ )*L FlC9,,iK-l⃁y.эt KfQ7( <̣JxCK| /33bǏ L2CT%ݴ K u|R QCYKzJd44yBB泊(GSʇ3@;I* ;M389Ɩԙ* εF 1GTDvAu1&@F*SO^>Ԥ<&=À6$*XÊLgMkAi -#P%cΑ7IbSN[NjTǙS-U+FVJ6f~`gzZq&bĒ_mlGލ-=meͺ7esnֹ֨4ap" =h\qVo/s{@`0w·便|ru=+f V 1xkSs_ի=X̤O}D;ٮﶵj:Z;0y"KXlZלǶ1mx[/=1 PP[>|VQn|h2xwgVգ!z}x&\Dା@<:GM˜x9?ddn^ܶ4לՈļ'D=>zׇ͏{/'; w{C*i3f{nOs>`f<٫mv">&2 \9O|i~q(%HգMnx>&<?~0E{ЕrgnٓwP=?ƲX{TN쏏?գX9˾ OHཨ-Roo;Ap@礷!}5>':ۿ3Qhc@AJSߛ@Dl  ;B+Bƣ#;O $+@ ?)hB'\'1̹9~ÜC$=. Q QC1Ú2LC=C=xI~P FÎ"CpA>DBMLLE=Js9R°ۄTPDSSRE>,BGU4E$\@EbDLCA?i'eҒD|9<545J(PGPmdKR&5sN(MR%8(eR%5J*RQhDPZC0QX5S="S 3̟E ~ =yTN]T!%G5S KTmT(}Q-~HQB\R0SRX21 TdMV5 R$DJX1"PVNV;}^S_eU)VlUEuQԄXNT-5MheTFzWGQ EK mTj=O RO-PVQom&fSWX]S{%UFvUQxIB-5{XXf5SYKԟ%XҖ֐וU{Y-PX=XX]ZeTZT,|eZe"YـZ [ڠ]TXp5ێ[SNׅMȄUXۻW}3Zmeۥ=ؼ=\X5\%[e\e\R[$\V ܾ 5[Mݔ-=|\S˅P\]]۴e]8 ]^uܱZ][]%]譄>Wẍ́m\ݍŝ ߘM[-^\5=߻E55P,m]]\^_ (_m}_~]E] JW` v`EV`666{ V\a^a W#V&vbvmb=O).b&>X'sZ&Va`,vbFc/5[V!$ba#a `>b/vaa7d@@NbB.dG\Gb (X?c,&a26'.1&!]3.bPdAF5&>3bSfo!b%DbYnaJ. PdRVfU.6e_e`~bPfif'Qc~nfTfgv>rf5NPRkvg-~e*VcZfr6J pfWxgyMN]fTfnh~5vgpfHU^goYfhhii.ibJVvni~i^f&iivin霾;d k.~SZjH6H6H6./usr/lpp/essl.html.en_US/html/essl160.htmlen_U Guide and Reference

Guide and Reference

ESSL Messages

The text of message format has changed. ESV has been removed from the message text. For details on the new format, see "Message Format".

Some input-argument and computational error message numbers have been changed. The old message numbers can still be used when calling ERRSET, however, you should migrate to the new message numbers. The following describes which error messages have been modified:
Old Error Message Number New Error Message Number
2074 2608
2123 2609
2128 2700

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Guide and Reference

Planning for Future Migration

With respect to planning for the future, if working storage does not need to persist after the subroutine call, you should use dynamic allocation. Otherwise, you should use the processor-independent formulas or simple formulas for calculating the values for the naux arguments in the ESSL calling sequences. Two things may occur that could cause the minimum values of naux, returned by ESSL error handling, to increase in the future:

  • If changes are made to the ESSL subroutines to improve performance
  • If changes are necessary to support future processors

The formulas allow you to specify your auxiliary storage large enough to accommodate any future improvements to ESSL and any future processors. If you do not provide, at least, these amounts of storage, your program may not run in the future.

You should use the following rule of thumb: To protect your application from having to be recoded in the future because of possible increased requirements for auxiliary storage, use dynamic allocation if possible. If the working storage must persists after the subroutine call, then you should provide as much storage as possible in your current application. In determining the right amount to specify, you should weigh your storage constraints against the inconvenience of making future changes, then specify what you think is best. If possible, you should provide this larger amount of storage to prevent future migration problems.

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Guide and Reference

Migrating between RS/6000 Processors

This section describes all the aspects of migrating your ESSL application programs (back and forth) between the RS/6000 PowerPC, POWER, POWER2, and POWER3 processors.

Auxiliary Storage

The minimum amount of auxiliary storage returned by ESSL error handling may vary among the RS/6000 processors for the following subroutines: all the Fourier transform subroutines, SCONF, SCORF, and SACORF. Therefore, to guarantee that your application programs always migrate from any platform to any other platform, you should use the processor independent formulas to determine the amount of auxiliary storage to use.

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Guide and Reference

Bitwise-Identical Results

Because of hardware and ESSL design differences, the results you obtain when migrating from one ESSL Library to another may not be bitwise identical. The results, however, are mathematically equivalent.

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Guide and Reference

Migrating from Other Libraries to ESSL

This section describes some general aspects of moving from an IBM or non-IBM engineering and scientific library to ESSL.

Migrating from ESSL/370

There is a high degree of compatibility between ESSL/370 and ESSL for AIX. However you may need to make some coding changes for certain subroutines. See the Engineering and Scientific Subroutine Library Version 2.2 Guide and Reference for details.

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Guide and Reference

Migrating from Another IBM Subroutine Library

If you are migrating from other IBM library products--such as Subroutine Library--Mathematics (SL MATH) or Scientific Subroutine Package (SSP), which have some functions similar to ESSL--the ESSL calling sequences differ from the calling sequences you are currently using. Your program must be modified to add the ESSL calling sequences and make the other ESSL-related coding changes.

If you are migrating from the Basic Linear Algebra Subroutine Library provided with the RS/6000 basic operating system, your calling sequences do not need to be changed.

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Guide and Reference

Migrating from LAPACK

ESSL contains a few subroutines that conform to the LAPACK interface (see Appendix B. "LAPACK"). If you are using these subroutines, no coding changes are needed to migrate to ESSL.

Additionally, you may be interested in using the Call Conversion Interface (CCI) that is available with LAPACK. The CCI substitutes a call to an ESSL subroutine in place of an LAPACK subroutine whenever an ESSL subroutine provides either functional or near-functional equivalence. Using the CCI allows LAPACK users to obtain the optimized performance of ESSL for an additional subset of LAPACK subroutines. For details, see reference [40].

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Guide and Reference

Migrating from a Non-IBM Subroutine Library

If you are using a non-IBM library, ESSL may provide subroutines corresponding to those you are currently using. You may choose to migrate your program to benefit from the increased performance offered by the ESSL subroutines. In this case, you may have to recode your program to use the ESSL calling sequences, because the names and arguments used by ESSL may be different from those used by the non-IBM library. On the other hand, if you are using any of the standard Level 1, 2, and 3 BLAS or LAPACK routines that correspond to ESSL subroutines, you do not need to recode the calling sequences. The ESSL calling sequences are the same as the public domain code.

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Guide and Reference

Handling Problems

This chapter provides the following information for your use when dealing with errors:

  • How to obtain IBM support.

  • What to do about NLS (National Language Support) problems.

  • A description of the different types of errors that can occur in ESSL. It explains what happens when an error occurs and, in some instances, how you can use error handling to obtain further information.

  • All of the ESSL error messages are categorized into the different error types. There is also a description of the error message format.

Where to Find More Information About Errors

Specific errors associated with each ESSL subroutine are listed under "Error Conditions" in each subroutine description in Part 3 of this book.

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Guide and Reference

Getting Help from IBM Support

Should you require help from IBM in resolving an ESSL problem, report it and provide the following information, if available and appropriate.

  1. Your customer number

  2. The ESSL program number:
    • 5765-C42

    This is important information that speeds up the correct routing of your call.

  3. The version and release of the operating system that you are running on. To get this information on AIX, enter the following command: 
       oslevel

  4. The names and versions of key products being run. To get this information on AIX, enter the following command: 
       lslpp  -h product

    where the appropriate values of product for AIX Version 4 are listed in Table 30.


    Table 30. Product File Set Names when Using AIX Version 4
    Product File Sets Descriptive Name
    essl.* ESSL
    xlfrte XL Fortran Run-Time Environment
    xlhpf.rte XL HPF Run-Time Environment
    xlsmp.rte SMP Run-Time Environment
    xlfcmp XL Fortran Compiler
    vac.C C for AIX Compiler
    ibmcxx.cmp IBM C, C++ Version 3.6 Compilers

  5. The message that is returned when an error is detected.

  6. Any error message relating to core dumps.

  7. The compiler listings, including compiler options in effect, and any run-time listings produced

  8. Program changes made in comparison with a previous successful run

  9. A small test case demonstrating the problem using the minimum number of statements and variables, including input data

Consult your IBM Service representative for more assistance.

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Guide and Reference

National Language Support

For National Language Support (NLS), all ESSL subroutines display messages located in externalized message catalogs. English versions of the message catalogs are shipped with the IBM Engineering and Scientific Subroutine Library for AIX product, but your site maybe using its own translated message catalogs. The AIX environment variable NLSPATH is used by the various ESSL subroutines to find the appropriate message catalog. NLSPATH specifies a list of directories to search for message catalogs. The directories are searched, in the order listed, to locate the message catalog. In resolving the path to the message catalog, NLSPATH is affected by the value of the environment variables LC_MESSAGES and LANG. If you get an error saying that a message catalog is not found and want the default message catalog, enter the following: 

        export NLSPATH = /usr/lib/nls/msg/%L/%N
 
        export LANG = C

The ESSL message catalogs are in English, and are located in the following directories: 

/usr/lib/nls/msg/C
/usr/lib/nls/msg/En_US
/usr/lib/nls/msg/en_US

If your site is using its own translations of the message catalogs, consult your system administrator for the appropriate value of NLSPATH or LANG. For additional information on NLS and message catalogs, see IBM AIX Version 4 for RISC System/6000 General Programming Concepts.

If ESSL cannot successfully find a message, ESSL returns message 2799, indicating which message could not be located. Message 2799 is described in "Miscellaneous Error Messages".

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Dealing with Errors

At run time, you can encounter a number of different types of errors that are specifically related to the use of the ESSL subroutines:

  • Program exceptions

  • Input-argument errors (2001-2099) and (2801-2899)

  • Computational errors (2100-2199)

  • Resource errors (2401-2499)

  • Informational and Attention messages (2600-2699)

  • Miscellaneous errors (2700-2799)

Program Exceptions

The program exceptions you can encounter in ESSL are described in the RS/6000 architecture manuals. For details, see:

  • ANSI/IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985.

  • RS/6000 POWERstation and POWERserver Hardware Technical Reference Information--General Architectures.

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ESSL Input-Argument Error Messages

If you receive an error message in the form 2538-20nn, you have an input-argument error in the calling sequence for an ESSL subroutine. Your program terminated at this point unless you did one of the following:

  • Specified the ESSL user exit routine, ENOTRM, with ERRSET to determine the correct input argument values in your program for the optionally-recoverable ESSL errors 2015 or 2030. For details on how to do this, see "Coding Your Program".

  • Reset the number of allowable errors (2099) during ESSL installation or using ERRSET in your program. This is not recommended for input-argument errors.

Note:For many of the ESSL subroutines requiring auxiliary storage, you can avoid program termination due to error 2015 by allowing ESSL to dynamically allocate auxiliary storage for you. You do this by setting naux = 0 and making error 2015 unrecoverable. For details on which aux arguments allow dynamic allocation and how to specify them, see the subroutine descriptions in Part 2 of this book.

The name of the ESSL subroutine detecting the error is listed as part of the message. The argument number(s) involved in the error appears in the message text. See "Input-Argument Error Messages" for a complete description of the information contained in each message and for an indication of which messages correspond to optionally-recoverable errors. Regardless of whether the name in the message is a user-callable ESSL subroutine or an internal ESSL routine, the message-text and its unique parts apply to the user-callable ESSL subroutine. Return code values are described under "Error Conditions" for each ESSL subroutine in Part 2 of this book.

You may get more than one error message, because most of the arguments are checked by ESSL for possible errors during each call to the subroutine. The ESSL subroutine returns as many messages as there are errors detected. As a result, fewer runs are necessary to diagnose your program.

Fix the error(s), recompile, relink, and rerun your program.

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ESSL Computational Error Messages

If you receive an error message in the form 2538-21nn, you have a computational error in the ESSL subroutine. A computational error is any error occurring in the ESSL subroutine while using the computational data (that is, scalar and array data). The name of the ESSL subroutine detecting the error is listed as part of the message. Regardless of whether the name in the message is a user-callable ESSL subroutine or an internal ESSL routine, the message-text and its unique parts apply to the user-callable ESSL subroutine. A nonzero return code is returned when the ESSL subroutine encounters a computational error. See "Computational Error Messages" for a complete description of the information in each message. Return code values are described under "Error Conditions" for each ESSL subroutine in Part 2 of this book.

Your program terminates for some computational errors unless you have called ERRSET to reset the number of allowable errors for that particular error, and the number has not been exceeded. A message is issued for each computational error. You should use the message to determine where the error occurred in your program.

If you called ERRSET and you have not reached the limit of errors you had set, you can check the return code. If it is not 0, you should call the EINFO subroutine to obtain information about the data involved in the error. EINFO provides the same information provided in the messages; however, it is provided to your program so your program can check the information during run time. Depending on what you want to do, you may choose to continue processing or terminate your program after the error occurs. For information on how to make these changes in your program to reset the number of allowable errors, how to diagnose the error, and how to decide whether to continue or terminate your program, see "Coding Your Program".

If you are unable to solve the problem, report it and provide the following information, if available and appropriate:

  • The message number and the module that detected an error

  • The system dump, system error code, and system log of this job

  • The compiler listings, including compiler options in effect, and any run-time listings produced

  • Program changes made in comparison with a previous successful run

  • A small test case demonstrating the problem using the minimum number of statements and variables, including input data

  • A brief description of the problem

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Guide and Reference

ESSL Resource Error Messages

If you receive a message in the form 2538-24nn, it means that ESSL issued a resource error message.

A resource error occurs when a buffer storage allocation request fails in a ESSL subroutine. In general, the ESSL subroutines allocate internal auxiliary storage dynamically as needed. Without sufficient storage, the subroutine cannot complete the computation.

When a buffer storage allocation request fails, a resource error message is issued, and the application program is terminated. You need to reduce the memory constraint on the system or increase the amount of memory available before rerunning the application program.

The following ways may reduce memory constraints:

  • Investigate the load of your process and run in a more dedicated environment.

  • Increase your processor's paging space.

  • Select a machine with more memory.

  • Consider specifying the -bmaxdata binder option when linking your program. For details see the Fortran publications.

  • Check the setting of your user ID's user limit (ulimit). (see the IBM AIX Version 4 Commands Reference).

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ESSL Informational and Attention Messages

If you receive a message in the form 2538-26nn, it means that ESSL issued an informational or attention message.

Informational Messages

When you receive an informational message, check your application to determine why the condition was detected.

ESSL Attention Messages

An attention message is issued to describe a condition that occurred. ESSL is able to continue processing, but performance may be degraded.

One condition that may produce an attention message is when enough work area was available to continue processing, but was not the amount initially requested. ESSL does not terminate your application program, but performance may be degraded. If you want to reduce the memory constraint on the system or increase the amount of memory available to eliminate the attention message, see the suggestions in "ESSL Resource Error Messages". For a list of subroutines that may generate this type of attention message, see Table 31.

Table 31. ESSL Subroutines
Subroutine Names 
Matrix-Vector Linear Algebra Subprograms:
  _GEMV, _GER,  _SPMV, _SYMV, _SPR, _SYR, _SPR2, _SYR2
  _GERC, _GERU, _HPMV, _HEMV, _HPR, _HER, _HPR2, _HER2
  _SBMV, _TBMV, SGBMV, DGBMV, CGBMV
  _TPMV
  _TRMV


Matrix Operations:
  _GEMM, _GEMUL
  _SYMM, _SYR2K, _TRMM
  _HEMM, _HER2K


Dense Linear Algebraic Equations:
  _POF, _POICD, _PPICD
  _GEICD, _TPI, _TRI
  _TRSM, _TPSV
  _TRSV


Banded Linear Algebraic Equations:
  STBSV, DTBSV


Linear Least Squares:
  _GESVS


Fourier Transforms:
  _CFT, _CFT3
  _RCFT, _RCFT3
  _CRFT, _CRFT3

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Guide and Reference

Miscellaneous Error Messages

If you receive a message in the form 2538-27nn, it means that ESSL issued a miscellaneous error message.

A miscellaneous error is an error that does not fall under any other categories.

When ESSL detects a miscellaneous error, you receive an error message with information on how to proceed and your application program is terminated.

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Guide and Reference

Messages

This section explains the conventions used for the ESSL messages and lists all the ESSL messages. For a description of each of the four types of ESSL messages, see "Dealing with Errors".

Message Conventions

This section describes the message conventions for the ESSL product.

About Upper- and Lowercase

The literals, such as, 'N', 'T', 'U', and so forth, appear in the messages in this book in uppercase; however, they may be specified in your ESSL calling sequence in either upper- or lowercase, for example, 'n', 't', and 'u'.

Message Format

The ESSL messages are issued in your output in the following format:

Figure 10. Message Format


rtn-name : 2538-mmnn
message-text

The parts of the ESSL message are as follows:

rtn-name

gives the name of the ESSL subroutine that encountered the error.

2538

is the ESSL component identification number.

mm

indicates the type of ESSL error message:

20--Input-argument error message
21--Computational error message
24--Resource error message
26--Information and attention message
27--Miscellaneous error message

nn

is the message identification number.

message-text

describes the nature of the error. Where one of several possible message-texts can be issued for a particular ESSL error, they are listed in this book with an "or" between them. The possible unique parts are:

  • The argument number of each argument involved in the error is included in the message description as (ARG NO. _ )

  • Additional information about the error is included in the message. The placement of this information is shown in the messages as (_)

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Guide and Reference

Resource Error Messages

RTN_NAME : 2538-2400
An internal buffer allocation has failed due to insufficient memory.

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Informational and Attention Error Messages

RTN_NAME : 2538-2600
Performance may be degraded due to limited buffer space availability.
 
RTN_NAME : 2538-2601
Execution terminating due to error count for error number (_)
Message summary: Message number - Count
 
RTN_NAME : 2538-2602
User error corrective routine entered.
User corrective action taken.  Execution continuing.
 
RTN_NAME : 2538-2603
Standard corrective action taken.  Execution continuing.
 
RTN_NAME : 2538-2604
Execution terminating due to error count for error number _.
 
RTN_NAME : 2538-2605
Message summary:      _     - _
 
RTN_NAME : 2538-2606
Serial execution is taking place since the input array is equal
to the output array and either:
INC2X (ARG NO. _) is not equal to 2 times INC2Y (ARG NO. _) or
INC3X (ARG NO. _) is not equal to 2 times INC3Y (ARG NO. _).
 
RTN_NAME : 2538-2607
Serial execution is taking place since the input array is equal
to the output array and either:
INC2X (ARG NO. _) is not equal to INC2Y (ARG NO. _) or
INC3X (ARG NO. _) is not equal to INC3Y (ARG NO. _).
 
RTN_NAME : 2538-2608
Performance may be improved by using a larger work array. For best
performance, specify the number of elements (ARG NO. _) in the work array
to be greater than or equal to (_).
 
RTN_NAME : 2538-2609
Performance may be improved by specifying a larger value for (ARG NO. _).
(_) compressions were performed.

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Miscellaneous Error Messages

RTN_NAME : 2538-2700
Internal ESSL error number (_).
Contact your IBM service representative.
 
RTN_NAME : 2538-2703
Internal ESSL error: message number requested (_) is outside of the
valid range.  Contact your IBM service representative.

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Guide and Reference

Matrix-Vector Linear Algebra Subprograms

The matrix-vector linear algebra subprograms operate on a higher-level data structure--matrix-vector rather than vector-scalar--using optimized algorithms to improve performance. These subprograms represent a subset of the Level 2 BLAS described in references [34] and [35]. Both real and complex versions of the subprograms are provided.

Table 34. List of Matrix-Vector Linear Algebra Subprograms
Descriptive Name Short- Precision Subprogram Long- Precision Subprogram Page
Matrix-Vector Product for a General Matrix, Its Transpose, or Its Conjugate Transpose 
SGEMVø
CGEMVø
SGEMX§
SGEMTX§


DGEMVø
ZGEMVø
DGEMX§
DGEMTX§

SGEMV, DGEMV, CGEMV, ZGEMV, SGEMX, DGEMX, SGEMTX, and DGEMTX--Matrix-Vector Product for a General Matrix, Its Transpose, or Its Conjugate Transpose
Rank-One Update of a General Matrix 
SGERø
CGERUø
CGERCø


DGERø
ZGERUø
ZGERCø

SGER, DGER, CGERU, ZGERU, CGERC, and ZGERC--Rank-One Update of a General Matrix
Matrix-Vector Product for a Real Symmetric or Complex Hermitian Matrix 
SSPMVø
CHPMVø
SSYMVø
CHEMVø
SSLMX§


DSPMVø
ZHPMVø
DSYMVø
ZHEMVø
DSLMX§

SSPMV, DSPMV, CHPMV, ZHPMV, SSYMV, DSYMV, CHEMV, ZHEMV, SSLMX, and DSLMX--Matrix-Vector Product for a Real Symmetric or Complex Hermitian Matrix
Rank-One Update of a Real Symmetric or Complex Hermitian Matrix 
SSPRø
CHPRø
SSYRø
CHERø
SSLR1§


DSPRø
ZHPRø
DSYRø
ZHERø
DSLR1§

SSPR, DSPR, CHPR, ZHPR, SSYR, DSYR, CHER, ZHER, SSLR1, and DSLR1 --Rank-One Update of a Real Symmetric or Complex Hermitian Matrix
Rank-Two Update of a Real Symmetric or Complex Hermitian Matrix 
SSPR2ø
CHPR2ø
SSYR2ø
CHER2ø
SSLR2§


DSPR2ø
ZHPR2ø
DSYR2ø
ZHER2ø
DSLR2§

SSPR2, DSPR2, CHPR2, ZHPR2, SSYR2, DSYR2, CHER2, ZHER2, SSLR2, and DSLR2--Rank-Two Update of a Real Symmetric or Complex Hermitian Matrix
Matrix-Vector Product for a General Band Matrix, Its Transpose, or Its Conjugate Transpose 
SGBMVø
CGBMVø


DGBMVø
ZGBMVø

SGBMV, DGBMV, CGBMV, and ZGBMV--Matrix-Vector Product for a General Band Matrix, Its Transpose, or Its Conjugate Transpose
Matrix-Vector Product for a Real Symmetric or Complex Hermitian Band Matrix 
SSBMVø
CHBMVø


DSBMVø
ZHBMVø

SSBMV, DSBMV, CHBMV, and ZHBMV--Matrix-Vector Product for a Real Symmetric or Complex Hermitian Band Matrix
Matrix-Vector Product for a Triangular Matrix, Its Transpose, or Its Conjugate Transpose 
STRMVø
CTRMVø
STPMVø
CTPMVø


DTRMVø
ZTRMVø
DTPMVø
ZTPMVø

STRMV, DTRMV, CTRMV, ZTRMV, STPMV, DTPMV, CTPMV, and ZTPMV--Matrix-Vector Product for a Triangular Matrix, Its Transpose, or Its Conjugate Transpose
Matrix-Vector Product for a Triangular Band Matrix, Its Transpose, or Its Conjugate Transpose 
STBMVø
CTBMVø


DTBMVø
ZTBMVø

STBMV, DTBMV, CTBMV, and ZTBMV--Matrix-Vector Product for a Triangular Band Matrix, Its Transpose, or Its Conjugate Transpose

ø Level 2 BLAS

§ These subroutines are provided only for migration from earlier releases of ESSL and are not intended for use in new programs.

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Guide and Reference

Sparse Matrix-Vector Linear Algebra Subprograms

The sparse matrix-vector linear algebra subprograms operate on sparse matrices using optimized storage techniques; that is, only the nonzero elements of the vector are stored. These subprograms provide similar functions to the matrix-vector subprograms.

Table 35. List of Sparse Matrix-Vector Linear Algebra Subprograms
Descriptive Name Long- Precision Subprogram Page
Matrix-Vector Product for a Sparse Matrix in Compressed-Matrix Storage Mode DSMMX DSMMX--Matrix-Vector Product for a Sparse Matrix in Compressed-Matrix Storage Mode
Transpose a Sparse Matrix in Compressed-Matrix Storage Mode DSMTM DSMTM--Transpose a Sparse Matrix in Compressed-Matrix Storage Mode
Matrix-Vector Product for a Sparse Matrix or Its Transpose in Compressed-Diagonal Storage Mode DSDMX DSDMX--Matrix-Vector Product for a Sparse Matrix or Its Transpose in Compressed-Diagonal Storage Mode

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Guide and Reference

Use Considerations

If your program uses a sparse matrix stored by rows, as defined in "Storage-by-Rows", you should first convert your sparse matrix to compressed-matrix storage mode by using the subroutine DSRSM on page DSRSM--Convert a Sparse Matrix from Storage-by-Rows to Compressed-Matrix Storage Mode. DSRSM converts a matrix to compressed-matrix storage mode. To convert your sparse matrix to compressed-diagonal storage mode, you need to perform this conversion in your application program before calling the ESSL subroutine.

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Guide and Reference

Performance and Accuracy Considerations

  1. In ESSL, the SSCAL and DSCAL subroutines provide the fastest way to zero out contiguous (stride 1) arrays, by specifying incx = 1 and alpha = 0.

  2. Where possible, use the matrix-vector linear algebra subprograms, rather than the vector-scalar, to optimize performance. Because data is presented in matrices rather than vectors, multiple operations can be performed by a single ESSL subprogram.

  3. Where possible, use subprograms that do multiple computations, such as SNDOT and SNAXPY, rather than individual computations, such as SDOT and SAXPY. You get better performance.

  4. Many of the short-precision subprograms provide increased accuracy by accumulating results in long precision. This is noted in the functional description of each subprogram.

  5. In some of the subprograms, because implementation techniques vary to optimize performance, accuracy of the results may vary for different array sizes. In the subprograms in which this occurs, a general description of the implementation techniques is given in the functional description for each subprogram.

  6. To select the sparse matrix subroutine that gives you the best performance, you must consider the layout of the data in your matrix. From this, you can determine the most efficient storage mode for your sparse matrix. ESSL provides two versions of each of its sparse matrix-vector subroutines that you can use. One operates on sparse matrices stored in compressed-matrix storage mode, and the other operates on sparse matrices stored in compressed-diagonal storage mode. These two storage modes are described in "Sparse Matrix".

    Compressed-matrix storage mode is generally applicable. It should be used when each row of the matrix contains approximately the same number of nonzero elements. However, if the matrix has a special form--that is, where the nonzero elements are concentrated along a few diagonals--compressed-diagonal storage mode gives improved performance.

  7. There are some ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Vector-Scalar Subprograms

This section contains the vector-scalar subprogram descriptions.

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Guide and Reference

ISAMIN and IDAMIN--Position of the First or Last Occurrence of the Vector Element Having Minimum Absolute Value

These subprograms find the position i of the first or last occurrence of a vector element having the minimum absolute value.

You get the position of the first or last occurrence of an element by specifying positive or negative stride, respectively, for vector x. Regardless of the stride, the position i is always relative to the location specified in the calling sequence for vector x (in argument x).

Table 37. Data Types
x Subprogram
Short-precision real ISAMIN
Long-precision real IDAMIN

Syntax

Fortran ISAMIN | IDAMIN (n, x, incx)
C and C++ isamin | idamin (n, x, incx);
PL/I ISAMIN | IDAMIN (n, x, incx);

On Entry

n
is the number of elements in vector x. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 37.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

On Return

Function value
 

is the position i of the element in the array, where:

If incx >= 0, i is the position of the first occurrence.

If incx < 0, i is the position of the last occurrence.

Returned as: a fullword integer; 0 <= i <= n.

Note

Declare the ISAMIN and IDAMIN functions in your program as returning a fullword integer value.

Function

These subprograms find the first element xk, where k is defined as the smallest index k, such that:

|xk| = min{|xj| for j = 1, n}

By specifying a positive or negative stride for vector x, the first or last occurrence, respectively, is found in the array. The position i, returned as the value of the function, is always figured relative to the location specified in the calling sequence for vector x (in argument x). Therefore, depending on the stride specified for incx, i has the following values:

For incx >= 0, i = k
For incx < 0, i = n-k+1

See reference [73]. The result is returned as a function value. If n is 0, then 0 is returned as the value of the function.

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows a vector, x, with a stride of 1.

Function Reference and Input
               N   X   INCX
               |   |    |
IMIN = ISAMIN( 6 , X ,  1  )
 
X        =  (3.0, 4.0, 1.0, 8.0, 1.0, 3.0)

Output
IMIN     =  3

Example 2

This example shows a vector, x, with a stride greater than 1.

Function Reference and Input
               N   X   INCX
               |   |    |
IMIN = ISAMIN( 4 , X ,  2  )
 
X        =  (-3.0, . , -9.0, . , -8.0, . , 3.0)

Output
IMIN     =  1

Example 3

This example shows a vector, x, with a positive stride and two elements with the minimum absolute value. The position of the first occurrence is returned.

Function Reference and Input
               N   X   INCX
               |   |    |
IMIN = ISAMIN( 4 , X ,  2  )
 
X        =  (2.0, . , -1.0, . , 4.0, . , 1.0)

Output
IMIN     =  2

Example 4

This example shows a vector, x, with a negative stride and two elements with the minimum absolute value. The position of the last occurrence is returned. Processing begins at element X(7), which is 1.0.

Function Reference and Input
               N   X   INCX
               |   |    |
IMIN = ISAMIN( 4 , X , -2  )
 
X        =  (2.0, . , -1.0, . , 4.0, . , 1.0)

Output
IMIN     =  4

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Guide and Reference

ISMAX and IDMAX--Position of the First or Last Occurrence of the Vector Element Having the Maximum Value

These subprograms find the position i of the first or last occurrence of a vector element having the maximum value.

You get the position of the first or last occurrence of an element by specifying positive or negative stride, respectively, for vector x. Regardless of the stride, the position i is always relative to the location specified in the calling sequence for vector x (in argument x).

Table 38. Data Types
x Subprogram
Short-precision real ISMAX
Long-precision real IDMAX

Syntax

Fortran ISMAX | IDMAX (n, x, incx)
C and C++ ismax | idmax (n, x, incx);
PL/I ISMAX | IDMAX (n, x, incx);

On Entry

n
is the number of elements in vector x. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 38.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

On Return

Function value
 

is the position i of the element in the array, where:

If incx >= 0, i is the position of the first occurrence.

If incx < 0, i is the position of the last occurrence.

Returned as: a fullword integer; 0 <= i <= n.

Note

Declare the ISMAX and IDMAX functions in your program as returning a fullword integer value.

Function

These subprograms find the first element xk, where k is defined as the smallest index k, such that:

xk = max{xj for j = 1, n}

By specifying a positive or negative stride for vector x, the first or last occurrence, respectively, is found in the array. The position i, returned as the value of the function, is always figured relative to the location specified in the calling sequence for vector x (in argument x). Therefore, depending on the stride specified for incx, i has the following values:

For incx >= 0, i = k
For incx < 0, i = n-k+1

See reference [73]. The result is returned as a function value. If n is 0, then 0 is returned as the value of the function.

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows a vector, x, with a stride of 1.

Function Reference and Input
              N   X  INCX
              |   |   |
IMAX = ISMAX( 6 , X , 1  )
 
X        =  (3.0, 4.0, 1.0, 8.0, 1.0, 8.0)

Output
IMAX     =  4

Example 2

This example shows a vector, x, with a stride greater than 1.

Function Reference and Input
              N   X  INCX
              |   |   |
IMAX = ISMAX( 4 , X , 2  )
 
X        =  (-3.0, . , 9.0, . , -8.0, . , 3.0)

Output
IMAX     =  2

Example 3

This example shows a vector, x, with a positive stride and two elements with the maximum value. The position of the first occurrence is returned.

Function Reference and Input
              N   X  INCX
              |   |   |
IMAX = ISMAX( 4 , X , 2  )
 
X        =  (2.0, . , 4.0, . , 4.0, . , 1.0)

Output
IMAX     =  2

Example 4

This example shows a vector, x, with a negative stride and two elements with the maximum value. The position of the last occurrence is returned. Processing begins at element X(7), which is 1.0.

Function Reference and Input
              N   X   INCX
              |   |    |
IMAX = ISMAX( 4 , X , -2  )
 
X        =  (2.0, . , 4.0, . , 4.0, . , 1.0)

Output
IMAX     =  3

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Guide and Reference

ISMIN and IDMIN--Position of the First or Last Occurrence of the Vector Element Having Minimum Value

These subprograms find the position i of the first or last occurrence of a vector element having the minimum value.

You get the position of the first or last occurrence of an element by specifying positive or negative stride, respectively, for vector x. Regardless of the stride, the position i is always relative to the location specified in the calling sequence for vector x (in argument x).

Table 39. Data Types
x Subprogram
Short-precision real ISMIN
Long-precision real IDMIN

Syntax

Fortran ISMIN | IDMIN (n, x, incx)
C and C++ ismin | idmin (n, x, incx);
PL/I ISMIN | IDMIN (n, x, incx);

On Entry

n
is the number of elements in vector x. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 39.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

On Return

Function value
 

is the position i of the element in the array, where:

If incx >= 0, i is the position of the first occurrence.

If incx < 0, i is the position of the last occurrence.

Returned as: a fullword integer; 0 <= i <= n.

Note

Declare the ISMIN and IDMIN functions in your program as returning a fullword integer value.

Function

These subprograms find the first element xk, where k is defined as the smallest index k, such that:

xk = min{xj for j = 1, n}

By specifying a positive or negative stride for vector x, the first or last occurrence, respectively, is found in the array. The position i, returned as the value of the function, is always figured relative to the location specified in the calling sequence for vector x (in argument x). Therefore, depending on the stride specified for incx, i has the following values:

For incx >= 0, i = k
For incx < 0, i = n-k+1

See reference [73]. The result is returned as a function value. If n is 0, then 0 is returned as the value of the function.

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows a vector, x, with a stride of 1.

Function Reference and Input
              N   X  INCX
              |   |   |
IMIN = ISMIN( 6 , X , 1  )
 
X        =  (3.0, 4.0, 1.0, 8.0, 1.0, 3.0)

Output
IMIN     =  3

Example 2

This example shows a vector, x, with a stride greater than 1.

Function Reference and Input
              N   X  INCX
              |   |   |
IMIN = ISMIN( 4 , X , 2  )
 
X        =  (-3.0, . , -9.0, . , -8.0, . , 3.0)

Output
IMIN     =  2

Example 3

This example shows a vector, x, with a positive stride and two elements with the minimum value. The position of the first occurrence is returned. Processing begins at element X(7), which is 1.0.

Function Reference and Input
              N   X  INCX
              |   |   |
IMIN = ISMIN( 4 , X , 2  )
 
X        =  (2.0, . , 1.0, . , 4.0, . , 1.0)

Output
IMIN     =  2

Example 4

This example shows a vector, x, with a negative stride and two elements with the minimum value. The position of the last occurrence is returned. Processing begins at element X(7), which is 1.0.

Function Reference and Input
              N   X   INCX
              |   |    |
IMIN = ISMIN( 4 , X , -2  )
 
X        =  (2.0, . , 1.0, . , 4.0, . , 1.0)

Output
IMIN     =  4

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Guide and Reference

SASUM, DASUM, SCASUM, and DZASUM--Sum of the Magnitudes of the Elements in a Vector

SASUM and DASUM compute the sum of the absolute values of the elements in vector x. SCASUM and DZASUM compute the sum of the absolute values of the real and imaginary parts of the elements in vector x.

Table 40. Data Types
x Result Subprogram
Short-precision real Short-precision real SASUM
Long-precision real Long-precision real DASUM
Short-precision complex Short-precision real SCASUM
Long-precision complex Long-precision real DZASUM

Syntax

Fortran SASUM | DASUM | SCASUM | DZASUM (n, x, incx)
C and C++ sasum | dasum | scasum | dzasum (n, x, incx);
PL/I SASUM | DASUM | SCASUM | DZASUM (n, x, incx);

On Entry

n
is the number of elements in vector x. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 40.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

On Return

Function value
is the result of the summation. Returned as: a number of the data type indicated in Table 40.

Note

Declare this function in your program as returning a value of the type indicated in Table 40.

Function

SASUM and DASUM compute the sum of the absolute values of the elements of x, which is expressed as follows:



Figure ESYGR54 not displayed.

SCASUM and DZASUM compute the sum of the absolute values of the real and imaginary parts of the elements of x, which is expressed as follows:



Figure ESYGR55 not displayed.

See reference [73]. The result is returned as a function value. If n is 0, then 0.0 is returned as the value of the function. For SASUM and SCASUM, intermediate results are accumulated in long precision.

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows a vector, x, with a stride of 1.

Function Reference and Input
              N   X  INCX
              |   |   |
SUMM = SASUM( 7 , X , 1  )
 
X        =  (1.0, -3.0, -6.0, 7.0, 5.0, 2.0, -4.0)

Output
SUMM     =  28.0

Example 2

This example shows a vector, x, with a stride greater than 1.

Function Reference and Input
              N   X  INCX
              |   |   |
SUMM = SASUM( 4 , X , 2  )
 
X        =  (1.0, . , -6.0, . , 5.0, . , -4.0)

Output
SUMM     =  16.0

Example 3

This example shows a vector, x, with negative stride. Processing begins at element X(7), which is -4.0.

Function Reference and Input
              N   X   INCX
              |   |    |
SUMM = SASUM( 4 , X , -2  )
 
X        =  (1.0, . , -6.0, . , 5.0, . , -4.0)

Output
SUMM     =  16.0

Example 4

This example shows a vector, x, with a stride of 0. The result in SUMM is nx1.

Function Reference and Input
              N   X  INCX
              |   |   |
SUMM = SASUM( 7 , X , 0  )
 
X        =  (-2.0, . , . , . , . , . , .)

Output
SUMM     =  14.0

Example 5

This example shows a vector, x, containing complex numbers and having a stride of 1.

Function Reference and Input
               N   X  INCX
               |   |   |
SUMM = SCASUM( 5 , X , 1  )
 
X        =  ((1.0, 2.0), (-3.0, 4.0), (5.0, -6.0 ), (-7.0, -8.0),
             (9.0, 10.0))

Output
SUMM     =  55.0

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Guide and Reference

SCOPY, DCOPY, CCOPY, and ZCOPY--Copy a Vector

These subprograms copy vector x to another vector, y:

y <-- x

Table 42. Data Types
x, y Subprogram
Short-precision real SCOPY
Long-precision real DCOPY
Short-precision complex CCOPY
Long-precision complex ZCOPY

Syntax

Fortran CALL SCOPY | DCOPY | CCOPY | ZCOPY (n, x, incx, y, incy)
C and C++ scopy | dcopy | ccopy | zcopy (n, x, incx, y, incy);
PL/I CALL SCOPY | DCOPY | CCOPY | ZCOPY (n, x, incx, y, incy);

On Entry

n
is the number of elements in vectors x and y. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 42.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

y
See "On Return".

incy
is the stride for vector y. Specified as: a fullword integer. It can have any value.

On Return

y
is the vector y of length n. Returned as: a one-dimensional array of (at least) length 1+(n-1)|incy|, containing numbers of the data type indicated in Table 42.

Notes

  1. If you specify the same vector for x and y, incx and incy must be equal; otherwise, results are unpredictable.

  2. If you specify different vectors for x and y, they must have no common elements; otherwise, results are unpredictable. See "Concepts".

Function

The copy is expressed as follows:



Figure ESYGR57 not displayed.


See reference [73]. If n is 0, no copy is performed.

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows input vector x and output vector y with positive strides.

Call Statement and Input
            N   X  INCX  Y  INCY
            |   |   |    |   |
CALL SCOPY( 5 , X , 1  , Y , 2  )
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)

Output
Y        =  (1.0, . , 2.0, . , 3.0, . , 4.0, . , 5.0)

Example 2

This example shows how to obtain a reverse copy of the input vector x by specifying strides with the same absolute value, but with opposite signs, for input vector x and output vector y. For y, which has a negative stride, results are stored beginning at element Y(5).

Call Statement and Input
            N   X  INCX  Y   INCY
            |   |   |    |    |
CALL SCOPY( 5 , X , 1  , Y , -1  )
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)

Output
Y        =  (5.0, 4.0, 3.0, 2.0, 1.0)

Example 3

This example shows an input vector, x, with 0 stride. Vector x is treated like a vector of length n, all of whose elements are the same as the single element in x. This is a technique for replicating an element of a vector.

Call Statement and Input
            N   X  INCX  Y  INCY
            |   |   |    |   |
CALL SCOPY( 5 , X , 0  , Y , 1  )
 
X        =  (13.0)

Output
Y        =  (13.0, 13.0, 13.0, 13.0, 13.0)

Example 4

This example shows input vector x and output vector y, containing complex numbers and having positive strides.

Call Statement and Input
            N   X  INCX  Y  INCY
            |   |   |    |   |
CALL CCOPY( 4 , X , 1  , Y , 2  )
 
X        =  ((1.0, 1.0), (2.0, 2.0), (3.0, 3.0), (4.0, 4.0))

Output
Y        =  ((1.0, 1.0), . , (2.0, 2.0), . , (3.0, 3.0), . ,
             (4.0, 4.0))

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Guide and Reference

SNORM2, DNORM2, CNORM2, and ZNORM2--Euclidean Length of a Vector with No Scaling of Input

These subprograms compute the euclidean length (l2 norm) of vector x with no scaling of input.

Table 47. Data Types
x Result Subprogram
Short-precision real Short-precision real SNORM2
Long-precision real Long-precision real DNORM2
Short-precision complex Short-precision real CNORM2
Long-precision complex Long-precision real ZNORM2

Syntax

Fortran SNORM2 | DNORM2 | CNORM2 | ZNORM2 (n, x, incx)
C and C++ snorm2 | dnorm2 | cnorm2 | znorm2 (n, x, incx);
PL/I SNORM2 | DNORM2 | CNORM2 | ZNORM2 (n, x, incx);

On Entry

n
is the number of elements in vector x. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n, whose euclidean length is to be computed. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 47.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

On Return

Function value
is the euclidean length (l2 norm) of the vector x. Returned as: a number of the data type indicated in Table 47.

Notes

  1. This subroutine does not underflow or overflow if the values of the elements in vector x conform to the following conditions. If these conditions are violated, overflow or destructive underflow may occur:

    • For short-precision numbers:
      Any valid short-precision number.

    • For long-precision numbers:
      |xi| = 0 or 0.10010E-145 < |xi| < 0.13408E+155 for i = 1, n

  2. Declare this function in your program as returning a value of the data type indicated in Table 47.

Function

The euclidean length (l2 norm) of vector x is expressed as follows with no scaling of input:



Figure ESYGR64 not displayed.

See reference [73]. The result is returned as the function value. If n is 0, then 0.0 is returned as the value of the function.

For SNORM2 and CNORM2, the sum of the squares of the absolute values of the elements is accumulated in long-precision. The square root of this long-precision sum is then computed.

This subroutine should not be used if the values in vector x do not conform to the restriction given in "Notes".

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows a vector, x, with a stride of 1.

Function Reference and Input
                N   X  INCX
                |   |   |
SNORM = SNORM2( 6 , X , 1  )
 
X        =  (3.0, 4.0, 1.0, 8.0, 1.0, 3.0)

Output
SNORM    =  10.0

Example 2

This example shows a vector, x, with a stride greater than 1.

Function Reference and Input
                N   X  INCX
                |   |   |
SNORM = SNORM2( 6 , X , 2  )
 
X        =  (3.0, . , 4.0, . , 1.0, . , 8.0, . , 1.0, . , 3.0)

Output
SNORM    =  10.0

Example 3

This example shows a vector, x, with a stride of 0. The result in SNORM is:



Figure ESYGR65 not displayed.

Function Reference and Input
                N   X  INCX
                |   |   |
SNORM = SNORM2( 4 , X , 0  )
 
X        =  (4.0)

Output
SNORM    =  8.0

Example 4

This example shows a vector, x, containing complex numbers and having a stride of 1.

Function Reference and Input
                N   X  INCX
                |   |   |
CNORM = CNORM2( 3 , X , 1  )
 
X        =  ((3.0, 4.0), (1.0, 8.0), (-1.0, 3.0))

Output
CNORM    =  10.0

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Guide and Reference

SROT, DROT, CROT, ZROT, CSROT, and ZDROT--Apply a Plane Rotation

SROT and DROT apply a real plane rotation to real vectors; CROT and ZROT apply a complex plane rotation to complex vectors; and CSROT and ZDROT apply a real plane rotation to complex vectors. The plane rotation is applied to n points, where the points to be rotated are contained in vectors x and y, and where the cosine and sine of the angle of rotation are c and s, respectively.

Table 49. Data Types
x, y c s Subprogram
Short-precision real Short-precision real Short-precision real SROT
Long-precision real Long-precision real Long-precision real DROT
Short-precision complex Short-precision real Short-precision complex CROT
Long-precision complex Long-precision real Long-precision complex ZROT
Short-precision complex Short-precision real Short-precision real CSROT
Long-precision complex Long-precision real Long-precision real ZDROT

Syntax

Fortran CALL SROT | DROT | CROT | ZROT | CSROT | ZDROT (n, x, incx, y, incy, c, s)
C and C++ srot | drot | crot | zrot | csrot | zdrot (n, x, incx, y, incy, c, s);
PL/I CALL SROT | DROT | CROT | ZROT | CSROT | ZDROT (n, x, incx, y, incy, c, s);

On Entry

n
is the number of points to be rotated--that is, the number of elements in vectors x and y. Specified as: a fullword integer; n >= 0.

x
is the vector x of length n, containing the xi coordinates of the points to be rotated. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx|, containing numbers of the data type indicated in Table 49.

incx
is the stride for vector x. Specified as: a fullword integer. It can have any value.

y
is the vector y of length n, containing the yi coordinates of the points to be rotated. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incy|, containing numbers of the data type indicated in Table 49.

incy
is the stride for vector y. Specified as: a fullword integer. It can have any value.

c
the cosine, c, of the angle of rotation. Specified as: a number of the data type indicated in Table 49.

s
the sine, s, of the angle of rotation. Specified as: a number of the data type indicated in Table 49.

On Return

x
is the vector x of length n, containing the rotated xi coordinates, where:
xi <-- cxi+syi    for i = 1,

Returned as: a one-dimensional array, containing numbers of the data type indicated in Table 49.

y
is the vector y of length n, containing the rotated yi coordinates, where:

For SROT, DROT, CSROT, and ZDROT:

yi <-- -sxi+cyi    for i = 1, n

For CROT and ZROT:



Figure ESYGR79 not displayed.

Returned as: a one-dimensional array, containing numbers of the data type indicated in Table 49.

Note

The vectors x and y must have no common elements; otherwise, results are unpredictable. See "Concepts".

Function

Applying a plane rotation to n points, where the points to be rotated are contained in vectors x and y, is expressed as follows, where c and s are the cosine and sine of the angle of rotation, respectively. For SROT, DROT, CSROT, and ZDROT:



Figure ESYGR80 not displayed.


For CROT and ZROT:



Figure ESYGR81 not displayed.


See references [54] and [73]. No computation is performed if n is 0 or if c is 1.0 and s is zero. For SROT, CROT, and CSROT, intermediate results are accumulated in long precision.

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows how to apply a real plane rotation to real vectors x and y having positive strides.

Call Statement and Input
           N   X  INCX  Y  INCY   C    S
           |   |   |    |   |     |    |
CALL SROT( 5 , X , 1  , Y , 2  , 0.5 , S )
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)
Y        =  (-1.0, . , -2.0, . , -3.0, . , -4.0, . , -5.0)



Figure ESYGR82 not displayed.

Output
X        =  (-0.366, -0.732, -1.098, -1.464, -1.830)
Y        =  (-1.366, -2.732, -4.098, -5.464, -6.830)

Example 2

This example shows how to apply a real plane rotation to real vectors x and y having strides of opposite sign.

Call Statement and Input
           N   X  INCX  Y   INCY   C    S
           |   |   |    |    |     |    |
CALL SROT( 5 , X , 1  , Y , -1  , 0.5 , S )
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)
Y        =  (-5.0, -4.0, -3.0, -2.0, -1.0)



Figure ESYGR82 not displayed.

Output
X        =(same as output X in Example 1)
Y        =  (-6.830, -5.464, -4.098, -2.732, -1.366)

Example 3

This example shows how scalar values in vectors x and y can be processed by specifying 0 strides and the number of elements to be processed, n, equal to 1.

Call Statement and Input
           N   X  INCX  Y  INCY   C    S
           |   |   |    |   |     |    |
CALL SROT( 1 , X , 0  , Y , 0  , 0.5 , S )
 
X        =  (1.0)
Y        =  (-1.0)



Figure ESYGR82 not displayed.

Output
X        =  (-0.366)
Y        =  (-1.366)

Example 4

This example shows how to apply a complex plane rotation to complex vectors x and y having positive strides.

Call Statement and Input
           N   X  INCX  Y  INCY   C    S
           |   |   |    |   |     |    |
CALL CROT( 3 , X , 1  , Y , 2  , 0.5 , S )
 
X        =  ((1.0, 2.0), (2.0, 3.0), (3.0, 4.0))
Y        =  ((-1.0, 5.0), . , (-2.0, 4.0), . , (-3.0, 3.0))
S        =  (0.75, 0.50)

Output
X        =  ((-2.750, 4.250), (-2.500, 3.500), (-2.250, 2.750))
Y        =  ((-2.250, 1.500), . , (-4.000, 0.750), . ,
             (-5.750, 0.000))

Example 5

This example shows how to apply a real plane rotation to complex vectors x and y having positive strides.

Call Statement and Input
            N   X  INCX  Y  INCY   C    S
            |   |   |    |   |     |    |
CALL CSROT( 3 , X , 1  , Y , 2  , 0.5 , S )
 
X        =  ((1.0, 2.0), (2.0, 3.0), (3.0, 4.0))
Y        =  ((-1.0, 5.0), . , (-2.0, 4.0), . , (-3.0, 3.0))



Figure ESYGR82 not displayed.

Output
X        =  ((-0.366, 5.330), (-0.732, 4.964), (-1.098, 4.598))
Y        =  ((-1.366, 0.768), . , (-2.732, -0.598), . ,
             (-4.098, -1.964))

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Guide and Reference

Sparse Vector-Scalar Subprograms

This section contains the sparse vector-scalar subprogram descriptions.

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Guide and Reference

SSCTR, DSCTR, CSCTR, ZSCTR--Scatter the Elements of a Sparse Vector X in Compressed-Vector Storage Mode into Specified Elements of a Sparse Vector Y in Full-Vector Storage Mode

These subprograms scatter the elements of sparse vector x, stored in compressed-vector storage mode, into specified elements of sparse vector y, stored in full-vector storage mode.

Table 57. Data Types
x, y Subprogram
Short-precision real SSCTR
Long-precision real DSCTR
Short-precision complex CSCTR
Long-precision complex ZSCTR

Syntax

Fortran CALL SSCTR | DSCTR | CSCTR | ZSCTR (nz, x, indx, y)
C and C++ ssctr | dsctr | csctr | zsctr (nz, x, indx, y);
PL/I CALL SSCTR | DSCTR | CSCTR | ZSCTR (nz, x, indx, y);

On Entry

nz
is the number of elements in sparse vector x, stored in compressed-vector storage mode. Specified as: a fullword integer; nz >= 0.

x
is the sparse vector x, containing nz elements, stored in compressed-vector storage mode in an array, referred to as X. Specified as: a one-dimensional array of (at least) length nz, containing numbers of the data type indicated in Table 57.

indx
is the array, referred to as INDX, containing the nz indices that indicate the positions of the elements of the sparse vector x when in full-vector storage mode. They also indicate the positions in vector y into which the elements are copied.

Specified as: a one-dimensional array of (at least) length nz, containing fullword integers.

y
See "On Return".

On Return

y
is the sparse vector y, stored in full-vector storage mode, of (at least) length max(INDX(i)) for i = 1, nz, into which nz elements of vector x are copied at positions indicated by the indices array INDX.

Returned as: a one-dimensional array of (at least) length max(INDX(i)) for i = 1, nz, containing numbers of the data type indicated in Table 57.

Notes

  1. Each value specified in array INDX must be unique; otherwise, results are unpredictable.

  2. Vectors x and y must have no common elements; otherwise, results are unpredictable. See "Concepts".

  3. For a description of how sparse vectors are stored, see "Sparse Vector".

Function

The copy is expressed as follows:

yINDX(i) <-- xi    for i = 1, nz

where:

x is a sparse vector, stored in compressed-vector storage mode.
INDX is the indices array for sparse vector x.
y is a sparse vector, stored in full-vector storage mode.

See reference [29]. If nz is 0, no copy is performed.

Error Conditions

Computational Errors

None

Input-Argument Errors

nz < 0

Example 1

This example shows how to use SSCTR to copy a sparse vector x of length 5 into the following vector y, where the elements of array INDX are in ascending order: 

          Y = (6.0, 2.0, 4.0, 7.0, 6.0, 10.0, -2.0, 8.0, 9.0, 0.0 )

Call Statement and Input
            NZ  X   INDX   Y
            |   |    |     |
CALL SSCTR( 5 , X , INDX , Y )
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)
INDX     =  (1, 3, 4, 7, 10)

Output
Y        =  (1.0, 2.0, 2.0, 3.0, 6.0, 10.0, 4.0, 8.0, 9.0, 5.0)

Example 2

This example shows how to use SSCTR to copy a sparse vector x of length 5 into the following vector y, where the elements of array INDX are in random order: 

         Y = (6.0, 2.0, 4.0, 7.0, 6.0, 10.0, -2.0, 8.0, 9.0, 0.0 )

Call Statement and Input
            NZ  X   INDX   Y
            |   |    |     |
CALL SSCTR( 5 , X , INDX , Y )
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)
INDX     =  (4, 3, 1, 10, 7)

Output
Y        =  (3.0, 2.0, 2.0, 1.0, 6.0, 10.0, 5.0, 8.0, 9.0, 4.0)

Example 3

This example shows how to use CSCTR to copy a sparse vector x of length 3 into the following vector y, where the elements of array INDX are in random order: 

         Y = ((6.0, 5.0), (-2.0, 3.0), (15.0, 4.0), (9.0, 0.0))

Call Statement and Input
            NZ  X   INDX   Y
            |   |    |     |
CALL CSCTR( 3 , X , INDX , Y )
 
X        =  ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))
INDX     =  (4, 1, 3)

Output
Y        =  ((3.0, 4.0), (-2.0, 3.0), (5.0, 6.0), (1.0, 2.0))

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Guide and Reference

SGTHR, DGTHR, CGTHR, and ZGTHR--Gather Specified Elements of a Sparse Vector Y in Full-Vector Storage Mode into a Sparse Vector X in Compressed-Vector Storage Mode

These subprograms gather specified elements of vector y, stored in full-vector storage mode, into sparse vector x, stored in compressed-vector storage mode.

Table 58. Data Types
x, y Subprogram
Short-precision real SGTHR
Long-precision real DGTHR
Short-precision complex CGTHR
Long-precision complex ZGTHR

Syntax

Fortran CALL SGTHR | DGTHR | CGTHR | ZGTHR (nz, y, x, indx)
C and C++ sgthr | dgthr | cgthr | zgthr (nz, y, x, indx);
PL/I CALL SGTHR | DGTHR | CGTHR | ZGTHR (nz, y, x, indx);

On Entry

nz
is the number of elements in sparse vector x, stored in compressed-vector storage mode. Specified as: a fullword integer; nz >= 0.

y
is the sparse vector y, stored in full-vector storage mode, of (at least) length max(INDX(i)) for i = 1, nz, from which nz elements are copied from positions indicated by the indices array INDX.

Specified as: a one-dimensional array of (at least) length max(INDX(i)) for i = 1, nz, containing numbers of the data type indicated in Table 58.

x
See "On Return".

indx
is the array, referred to as INDX, containing the nz indices that indicate the positions of the elements of the sparse vector x when in full-vector storage mode. They also indicate the positions in vector y from which elements are copied.

Specified as: a one-dimensional array of (at least) length nz, containing fullword integers.

On Return

x
is the sparse vector x, containing nz elements, stored in compressed-vector storage mode in an array, referred to as X, into which are copied the elements of vector y from positions indicated by the indices array INDX.

Returned as: a one-dimensional array of (at least) length nz, containing numbers of the data type indicated in Table 58.

Notes

  1. Vectors x and y must have no common elements; otherwise, results are unpredictable. See "Concepts".

  2. For a description of how sparse vectors are stored, see "Sparse Vector".

Function

The copy is expressed as follows:

xi <-- yINDX(i)    for i = 1, nz

where:

x is a sparse vector, stored in compressed-vector storage mode.
INDX is the indices array for sparse vector x.
y is a sparse vector, stored in full-vector storage mode.

See reference [29]. If nz is 0, no copy is performed.

Error Conditions

Computational Errors

None

Input-Argument Errors

nz < 0

Example 1

This example shows how to use SGTHR to copy specified elements of a vector y into a sparse vector x of length 5, where the elements of array INDX are in ascending order.

Call Statement and Input
            NZ  Y   X   INDX
            |   |   |    |
CALL SGTHR( 5 , Y , X , INDX )
 
Y        =  (6.0, 2.0, 4.0, 7.0, 6.0, 10.0, -2.0, 8.0, 9.0, 0.0)
INDX     =  (1, 3, 4, 7, 9)

Output
X        =  (6.0, 4.0, 7.0, -2.0, 9.0)

Example 2

This example shows how to use SGTHR to copy specified elements of a vector y into a sparse vector x of length 5, where the elements of array INDX are in random order. (Note that the element 0.0 occurs in output vector x. This does not produce an error.)

Call Statement and Input
            NZ  Y   X   INDX
            |   |   |    |
CALL SGTHR( 5 , Y , X , INDX )
 
Y        =  (6.0, 2.0, 4.0, 7.0, 6.0, 10.0, -2.0, 8.0, 9.0, 0.0)
INDX     =  (4, 3, 1, 10, 7)

Output
X        =  (7.0, 4.0, 6.0, 0.0, -2.0)

Example 3

This example shows how to use CGTHR to copy specified elements of a vector, y, into a sparse vector, x, of length 3, where the elements of array INDX are in random order.

Call Statement and Input
            NZ  Y   X   INDX
            |   |   |    |
CALL CGTHR( 3 , Y , X , INDX )
 
Y        =  ((6.0, 5.0), (-2.0, 3.0), (15.0, 4.0), (9.0, 0.0))
INDX     =  (4, 1, 3)

Output
X        =  ((9.0, 0.0), (6.0, 5.0), (15.0, 4.0))

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Guide and Reference

Matrix-Vector Subprograms

This section contains the matrix-vector subprogram descriptions.

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A?>|<s/Cχ=>xŸ>aς8K=G |ФNТ]' LD̜`l$ (#ßppS>:UV=k,h:OXl bjBM bjTw8$(I']8s:p]r@ Guide and Reference

Guide and Reference

Sparse Matrix-Vector Subprograms

This section contains the sparse matrix-vector subprogram descriptions.

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Guide and Reference

Matrix Operations

The matrix operation subroutines are described in this chapter.

Overview of the Matrix Operation Subroutines

Some of the matrix operation subroutines were designed in accordance with the Level 3 BLAS de facto standard. If these subroutines do not comply with the standard as approved, IBM will consider updating them to do so. If IBM updates these subroutines, the updates could require modifications of the calling application program. For details on the Level 3 BLAS, see reference [32]. The matrix operation subroutines also include the commonly used matrix operations: addition, subtraction, multiplication, and transposition (Table 71).

Table 71. List of Matrix Operation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Matrix Addition for General Matrices or Their Transposes 
SGEADD
CGEADD


DGEADD
ZGEADD

SGEADD, DGEADD, CGEADD, and ZGEADD--Matrix Addition for General Matrices or Their Transposes
Matrix Subtraction for General Matrices or Their Transposes 
SGESUB
CGESUB


DGESUB
ZGESUB

SGESUB, DGESUB, CGESUB, and ZGESUB--Matrix Subtraction for General Matrices or Their Transposes
Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes 
SGEMUL
CGEMUL


DGEMUL
ZGEMUL
DGEMLP§

SGEMUL, DGEMUL, CGEMUL, and ZGEMUL--Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes
Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes Using Winograd's Variation of Strassen's Algorithm 
SGEMMS
CGEMMS


DGEMMS
ZGEMMS

SGEMMS, DGEMMS, CGEMMS, and ZGEMMS--Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes Using Winograd's Variation of Strassen's Algorithm
Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes 
SGEMM¢
CGEMM¢


DGEMM¢
ZGEMM¢

SGEMM, DGEMM, CGEMM, and ZGEMM--Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes
Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian 
SSYMM¢
CSYMM¢
CHEMM¢


DSYMM¢
ZSYMM¢
ZHEMM¢

SSYMM, DSYMM, CSYMM, ZSYMM, CHEMM, and ZHEMM--Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian
Triangular Matrix-Matrix Product 
STRMM¢
CTRMM¢


DTRMM¢
ZTRMM¢

STRMM, DTRMM, CTRMM, and ZTRMM--Triangular Matrix-Matrix Product
Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix 
SSYRK¢
CSYRK¢
CHERK¢


DSYRK¢
ZSYRK¢
ZHERK¢

SSYRK, DSYRK, CSYRK, ZSYRK, CHERK, and ZHERK--Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix 
SSYR2K¢
CSYR2K¢
CHER2K¢


DSYR2K¢
ZSYR2K¢
ZHER2K¢

SSYR2K, DSYR2K, CSYR2K, ZSYR2K, CHER2K, and ZHER2K--Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
General Matrix Transpose (In-Place) 
SGETMI
CGETMI


DGETMI
ZGETMI

SGETMI, DGETMI, CGETMI, and ZGETMI--General Matrix Transpose (In-Place)
General Matrix Transpose (Out-of-Place) 
SGETMO
CGETMO


DGETMO
ZGETMO

SGETMO, DGETMO, CGETMO, and ZGETMO--General Matrix Transpose (Out-of-Place)

¢ Level 3 BLAS

§ This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use in new programs. Documentation for this subroutine is no longer provided.

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Use Considerations

This section describes some key points about using the matrix operations subroutines.

Specifying Normal, Transposed, or Conjugate Transposed Input Matrices

On each invocation, the matrix operation subroutines can perform one of several possible computations, using different forms of the input matrices A and B. For the real and complex versions of the subroutines, there are four and nine combinations, respectively, depending on the characters specified for the transa and transb arguments:

'N'

Normal form

'T'

Transposed form subroutines)

'C'

Conjugate transposed form

The four and nine possible combinations are defined as follows:
Real Combinations Complex Combinations
AB AB
ATB ATB

AHB
ABT ABT
ATBT ATBT

AHBT

ABH

ATBH

AHBH

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Transposing or Conjugate Transposing:

This section describes some key points about using transposed and conjugate transposed matrices.

On Input

In every case, the input arrays for the matrix, its transpose, or its conjugate transpose should be stored in the original untransposed form. You then specify the desired form of the matrix to be used in the computation in the transa or transb arguments. For a description of matrix transpose and matrix conjugate transpose, see "Matrices".

On Output

If you want to compute the transpose or the conjugate transpose of a matrix operation--that is, the output stored in matrix C--you should use the matrix identities described in "Special Usage" for each subroutine description. Examples are provided in the subroutine descriptions to show the use of these matrix identities. This accomplishes the transpose or conjugate transpose as part of the multiply operation.

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Performance and Accuracy Considerations

This section describes some key points about performance and accuracy in the matrix operations subroutines.

In General

  1. The matrix operation subroutines use algorithms that are tuned specifically to the workstation processors they run on. The techniques involve using any one of several computational methods, based on certain operation counts and sizes of data.

  2. The short-precision multiplication subroutines provide increased accuracy by partially accumulating results in long precision.

  3. Strassen's method is not stable for certain row or column scalings of the input matrices A and B. Therefore, for matrices A and B with divergent exponent values, Strassen's method may give inaccurate results. For these cases, you should use the _GEMUL or _GEMM subroutines.

  4. There are ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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For Large Matrices

If you are using large square matrices in your matrix multiplication operations, you get better performance by using SGEMMS, DGEMMS, CGEMMS, and ZGEMMS. These subroutines use Winograd's variation of Strassen's algorithm for both real and complex matrices.

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For Combined Operations

If you want to perform a combined matrix multiplication and addition with scaling, SGEMM, DGEMM, CGEMM, and ZGEMM provide better performance than if you perform the parts of the computation separately in your program. See references [32] and [35].

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Matrix Operation Subroutines

This section contains the matrix operation subroutine descriptions.

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Guide and Reference

SGETMI, DGETMI, CGETMI, and ZGETMI--General Matrix Transpose (In-Place)

These subroutines transpose an n by n matrix A in place--that is, in matrix A:

A <-- AT

Table 81. Data Types
A Subroutine
Short-precision real SGETMI
Long-precision real DGETMI
Short-precision complex CGETMI
Long-precision complex ZGETMI

Syntax

Fortran CALL SGETMI | DGETMI | CGETMI | ZGETMI (a, lda, n)
C and C++ sgetmi | dgetmi | cgetmi | zgetmi (a, lda, n);
PL/I CALL SGETMI | DGETMI | CGETMI | ZGETMI (a, lda, n);

On Entry

a
is the matrix A having n rows and n columns. Specified as: an lda by (at least) n array, containing numbers of the data type indicated in Table 81.

lda
is the leading dimension of the array specified for a. Specified as: a fullword integer; lda > 0 and lda >= n.

n
is the number of rows and columns in matrix A. Specified as: a fullword integer; n >= 0.

On Return

a
is the n by n matrix AT, containing the results of the matrix transpose operation Returned as: an lda by (at least) n array, containing numbers of the data type indicated in Table 81.

Notes

  1. To achieve optimal performance in these subroutines, specify an even value for lda. An odd value may degrade performance.

  2. To achieve optimal performance in CGETMI, align the array specified for a on a doubleword boundary.

Function

Matrix A is transposed in place; that is, the n rows and n columns in matrix A are exchanged. For matrix A with elements aij, where i, j = 1, n, the in-place transpose is expressed as aji = aij for i, j = 1, n.

For the following input matrix A:



Figure ESYGR116 not displayed.


the in-place matrix transpose operation A <-- AT is expressed as:



Figure ESYGR117 not displayed.


If n is 0, no computation is performed.

Error Conditions

Computational Errors

None

Input-Argument Errors
  1. n < 0 or n > lda
  2. lda <= 0

Example

This example shows an in-place matrix transpose of matrix A having 5 rows and 5 columns.

Call Statement and Input
               A     LDA   N
               |      |    |
CALL SGETMI( A(2,3) , 10 , 5 )

        *                                    *
        | .   .   .     .     .     .     .  |
        | .   .  1.0   6.0  11.0  16.0  21.0 |
        | .   .  2.0   7.0  12.0  17.0  22.0 |
        | .   .  3.0   8.0  13.0  18.0  23.0 |
A    =  | .   .  4.0   9.0  14.0  19.0  24.0 |
        | .   .  5.0  10.0  15.0  20.0  25.0 |
        | .   .   .     .     .     .     .  |
        | .   .   .     .     .     .     .  |
        | .   .   .     .     .     .     .  |
        | .   .   .     .     .     .     .  |
        *                                    *

Output
        *                                     *
        | .   .    .     .     .     .     .  |
        | .   .   1.0   2.0   3.0   4.0   5.0 |
        | .   .   6.0   7.0   8.0   9.0  10.0 |
        | .   .  11.0  12.0  13.0  14.0  15.0 |
A    =  | .   .  16.0  17.0  18.0  19.0  20.0 |
        | .   .  21.0  22.0  23.0  24.0  25.0 |
        | .   .    .     .     .     .     .  |
        | .   .    .     .     .     .     .  |
        | .   .    .     .     .     .     .  |
        | .   .    .     .     .     .     .  |
        *                                     *

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] integ l~SZj"NH6NH6NH6A./usr/lpp/essl.html.en_US/html/essl248.htmlen_U <>./ARe "0=BETVainorst&1;DGHILMNOPbdlmpu|#,-23568CFScfghx()49U_wy*7:Yv!QZj+WXKz8r1e2 %&HvD EECʎ*L(Q >*L":ϥHIKqd)M#:U$@ #Nbl3&)ه=$I$I$I$I$I$I$I$I$I$I$>@F,@:\q۳1鄐x @b\/`"dC>@F Eb{L4pwq(Z (3 "Ϣj+P ]٘C*QU *eD]-(BϢj+P b`0P HVTE`h@"}QXTB EdaJ*ER@L!YQpQGY DueGcÊ+$BϢ$IxHq$H!tAMsoZlÞI$I$I$I$I$I$I$I$I$I$I%YI lTHYq9u@e8cĦ@>]G)j$ =EIC!%Bn8eB^( ʴ 1 ʃ㇨2p&T]ʄ3ms95mQlp[] !S4$X3#uV}赪)7'pmufkᶜ=-Yѕ}-[5+.l @8Жl3ۛeDE'7waIn }%&Z!8uwDm:'b.ᱢ̺4 [%66#qwB֭@@ZϨ%grwQD d2Z3<4%_TnnT#uJ2Ji%gҲQIQ$(CZ$( ,wDn57PNN荧P#uđ@9B$5H R]'b XfuJH ET$DyDTd,@Q6XqۚO*:QIQ$(CZ$( ,QReO*: YY$(UI!DGIJB4ȘA"Ì42}AQJH ET$DyDTd,p&TJH ET$DyDTd,@Q6XqZ4lO*:QIQ$(CZ$( ,QReO*: YY$(UI!DGIJB4ȘA"Ì4"ѣff*}AQJH ET$DyDTd,QReO*: 2JYZϢwQD @t2X-WFNV}+)VCJϥGq$DyD$zkDARNmfPTuң"ѕee FVѕ0^e`QuetTwDGIJIQ$*Y ('gt$(U h"Y@29HQ$Msl00$) *Σ. 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Guide and Reference

Banded Linear Algebraic Equation Subroutines

The banded linear algebraic equation subroutines provide solutions to linear systems of equations for real general band matrices, real positive definite symmetric band matrices, real or complex general tridiagonal matrices, real positive definite symmetric tridiagonal matrices, and real or complex triangular band matrices.

Table 84. List of Banded Linear Algebraic Equation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
General Band Matrix Factorization SGBF DGBF SGBF and DGBF--General Band Matrix Factorization
General Band Matrix Solve SGBS DGBS SGBS and DGBS--General Band Matrix Solve
Positive Definite Symmetric Band Matrix Factorization 
SPBF
SPBCHF


DPBF
DPBCHF

SPBF, DPBF, SPBCHF, and DPBCHF--Positive Definite Symmetric Band Matrix Factorization
Positive Definite Symmetric Band Matrix Solve 
SPBS
SPBCHS


DPBS
DPBCHS

SPBS, DPBS, SPBCHS, and DPBCHS--Positive Definite Symmetric Band Matrix Solve
General Tridiagonal Matrix Factorization SGTF DGTF SGTF and DGTF--General Tridiagonal Matrix Factorization
General Tridiagonal Matrix Solve SGTS DGTS SGTS and DGTS--General Tridiagonal Matrix Solve
General Tridiagonal Matrix Combined Factorization and Solve with No Pivoting 
SGTNP
CGTNP


DGTNP
ZGTNP

SGTNP, DGTNP, CGTNP, and ZGTNP--General Tridiagonal Matrix Combined Factorization and Solve with No Pivoting
General Tridiagonal Matrix Factorization with No Pivoting 
SGTNPF
CGTNPF


DGTNPF
ZGTNPF

SGTNPF, DGTNPF, CGTNPF, and ZGTNPF--General Tridiagonal Matrix Factorization with No Pivoting
General Tridiagonal Matrix Solve with No Pivoting 
SGTNPS
CGTNPS


DGTNPS
ZGTNPS

SGTNPS, DGTNPS, CGTNPS, and ZGTNPS--General Tridiagonal Matrix Solve with No Pivoting
Positive Definite Symmetric Tridiagonal Matrix Factorization SPTF DPTF SPTF and DPTF--Positive Definite Symmetric Tridiagonal Matrix Factorization
Positive Definite Symmetric Tridiagonal Matrix Solve SPTS DPTS SPTS and DPTS--Positive Definite Symmetric Tridiagonal Matrix Solve
Triangular Band Equation Solve 
STBSVø
CTBSVø


DTBSVø
ZTBSVø

STBSV, DTBSV, CTBSV, and ZTBSV--Triangular Band Equation Solve

ø Level 2 BLAS

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] Solve< lꊍ~SZj QH6QH6QH60./usr/lpp/essl.html.en_US/html/essl251.htmlen_U    e"<=>ADLSTainorst /EFGHIMNPRlmp-COVbcdgv#&,.01239;BUfhuxy45:w!678KX_kqz)QY(h(32`@Yqx#. ,ciFOiB/)B1 ؏nG܇!I J*(03BBB ( csZ\%[:ؿLL~p'_xcl9("c@p0.)_^=UIx;:ؿhL~p'_G7չwb¢{/9[_Юk# v.xU{K:bľr<#uU9j6!)<䠉e:rң@+k}οksTlCb11KX{m #8*cfًwUzk:Fͅ;7k}/x5pjCx̀ظ?GZ׭չ'!\L Vk*͏cf,\V[\6jCx̀ظ85p+Z#¤5k"3R_3Q Is$ag9'2Sc=C s?ZT,4ʂBl|f- kLW[xTqPkLUeyοaN#yLt]"E(H$903BHA^xշb^y/%]؟|F׶*߼؇#]œ5݉S¤5kY|E]=5%M.{i9'2FFpIs(56>3Da͘oאsKo)/m#В$Hh rNeǶz~Xh %>*0>3Daz_V݉Gm{o伕wb^kYolCaNCIj%0kȋFͅ?KOxI#>]"E(H$903BHA^xշb^y/%]؟|F׶*߼؇#]œ5݉S¤5kY|E%^G|HBH# 8t$9a4Ez@má$I̠ÏP\֕ ʂ0>3Åοƭ3y*6æ*߼؇#]œ5݉O ծKaeבT/UA?KOxI#>]"E(H$903BHA}5WskS滱?Rp,Z"ѮaN)/m#В$Hh rNeh0A7kl9A7Is$ag9'2Sc=C s?ZT0$$ Kk* |T>3D1' 8X^5l)o]؟? Z䁮?K "-6/pG\SK^ESi9'2FFpIs(56<׮uw rv'<*OCV.^D\mRXU/)]"= rNd6В$Pjl|gSK^G$Ȋќ:@ Ml8 5iP-)P' 9pÅ;5#]œ5݉O ծHp,Z"jOxI#>]"E(H$903BHA ӂN`cSS{f 6 _ k_8'b>GuExGK5ݱ~+m]~_#ԗ?Bqkҡ`p-|'N ÏP\֕  meXS|I{K(, ͇*%1#1}0(PX;=C s?ZT ʂz#.`P)䠉`~6 /c\SK} k~SZje RH6RH6RH6e ./usr/lpp/essl.html.en_US/html/essl252.htmlen_U Guide and Reference

Guide and Reference

Linear Least Squares Subroutines

The linear least squares subroutines provide least squares solutions to linear systems of equations for real general matrices. Two methods are provided: one that uses the singular value decomposition and another that uses a QR decomposition with column pivoting.

Table 86. List of Linear Least Squares Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Singular Value Decomposition for a General Matrix SGESVF DGESVF SGESVF and DGESVF--Singular Value Decomposition for a General Matrix
Linear Least Squares Solution for a General Matrix Using the Singular Value Decomposition SGESVS DGESVS SGESVS and DGESVS--Linear Least Squares Solution for a General Matrix Using the Singular Value Decomposition
Linear Least Squares Solution for a General Matrix Using a QR Decomposition with Column Pivoting SGELLS DGELLS SGELLS and DGELLS--Linear Least Squares Solution for a General Matrix Using a QR Decomposition with Column Pivoting

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Guide and Reference

Dense and Banded Linear Algebraic Equation Considerations

This section provides some key points about using the dense and banded linear algebraic equation subroutines.

Use Considerations

  1. To solve a system of equations, you need to use both the factorization and solve subroutines for the type of matrix you have. Each factorization subroutine should be followed in your program by the corresponding solve subroutine. The output from the factorization subroutine should be used as input to the solve subroutine.

  2. To solve a system of equations with one or more right-hand sides, follow the call to the factorization subroutine with one or more calls to a solve subroutine or one call to a multiple solve subroutine.

  3. The ESSL naming conventions for the dense and banded linear algebraic equation subroutines are similar to those used in the LAPACK documentation. (LAPACK, as well as its documentation, is available from the sources listed in reference [8].) The following ESSL subroutines correspond to the LAPACK subroutines: SGETRF, DGETRF, CGETRF, ZGETRF, SGETRS, DGETRS, CGETRS, and ZGETRS.

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Guide and Reference

Performance and Accuracy Considerations

  1. Except in a few instances, the _GTNP subroutines provide better performance than the _GTNPF and _GTNPS subroutines. For details, see the subroutine descriptions.

  2. The general subroutines (dense and banded) use partial pivoting for accuracy and fast performance.

  3. The short-precision subroutines provide increased accuracy by accumulating intermediate results in long precision. Occasionally, for performance reasons, these intermediate results are stored.

  4. There are ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Sparse Matrix Direct Solver Considerations

This section provides some key points about using the sparse matrix direct solver subroutines.

Use Considerations

  1. To solve a sparse system of equations by a direct method, you must use both the factorization and solve subroutines. The factorization subroutine should be followed in your program by the corresponding solve subroutine; that is, the output from the factorization subroutine should be used as input to the solve subroutine.

  2. To solve a system of equations with one or more right-hand sides, follow the call to the factorization subroutine with one or more calls to the solve subroutine.

  3. The amount of storage required for the arrays depends on the sparsity pattern of the matrix. The requirement that lna > 2nz on entry to DGSF does not guarantee a successful run of the program. Some programs may be terminated because of the large number of fill-ins generated upon factorization. Fill-ins generated in a program depend on the structure of each matrix. If a large number of fill-ins is anticipated when factoring a matrix, the value of lna should be large enough to accommodate your problem.

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Guide and Reference

Performance and Accuracy Considerations

  1. To make the subroutine more efficient, an input matrix comprised of all nonzero elements is preferable. See the syntax description of each subroutine for details.

  2. DGSF optionally checks the validity of the indices and pointers of the input matrix. Use of this option is suggested; however, it may affect performance. For details, see the syntax description for DGSF.

  3. In DGSS, if there are multiple sparse right-hand sides to be solved, you should take advantage of the sparsity by selecting a proper value for jopt (such as jopt = 10 or 11). If there is only one right-hand side to be solved, it is suggested that you do not exploit the sparsity.

  4. In DGSF, the value you enter for the lower bound of all elements in the matrix (RPARM(1)) affects the accuracy of the result. Specifying a larger number allows you to gain some performance; however, you may lose some accuracy in the solution.

  5. In DGSF, the threshold pivot tolerance (RPARM(2)) is used to select pivots. A value that is close to 0.0 approaches no pivoting. A value close to 1.0 approaches partial pivoting. A value of 0.1 is considered to be a good compromise between numerical stability and sparsity.

  6. If the ESSL subroutine performs storage compressions, you receive an attention message. When this occurs, the performance of this subroutine is affected. You can improve the performance by increasing the value specified for lna.

  7. There are ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Sparse Matrix Skyline Solver Considerations

This section provides some key points about using the sparse matrix skyline solver subroutines.

Use Considerations

  1. To solve a system of equations with one or more right-hand sides, where the matrix is stored in skyline storage mode, you can use either of the following methods. The factored output matrix is the same for both of these methods.

    • Call the skyline subroutine with the combined factor-and-solve option.

    • Call the skyline subroutine with the factor-only option, followed in your program by a call to the same subroutine with the solve-only option. The factored output matrix resulting from the factorization should be used as input to the same subroutine to do the solve. You can solve for the right-hand sides in a single call or in individual calls.

    You also have the option of doing a partial factorization, where the subroutine assumes that the initial part of the input matrix is already factored. It then factors the remaining rows and columns. If you want, you can factor a very large matrix progressively by using this option.

  2. Forward elimination can be done with or without scaling the right-hand side by the diagonal matrix elements. To perform the computation without scaling, call DGKFS with the normal solve-only option, and define the upper triangular skyline matrix (AU) as a diagonal. To perform the computation with scaling, call DGKFS with the transpose solve-only, option and define the lower triangular skyline matrix (AL) as a diagonal.

  3. Back substitution can be done with or without scaling the right-hand side by the diagonal matrix elements. To perform the computation without scaling, call DGKFS with the transpose solve-only option, and define the upper triangular skyline matrix (AU) as a diagonal. To perform the computation with scaling, call DGKFS with the normal solve-only option, and define the lower triangular skyline matrix (AL) as a diagonal.

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Guide and Reference

Performance and Accuracy Considerations

  1. For optimal performance, use diagonal-out skyline storage mode for both your input and output matrices. If you specify profile-in skyline storage mode for your input matrix, and either you do not plan to use the factored output or you plan to do a solve only, it is more efficient to specify diagonal-out skyline storage mode for your output matrix. These rules apply to all the computations.

  2. In some cases, elapsed time may be reduced significantly by using the combined factor-and-solve option to solve for all right-hand sides at once, in conjunction with the factorization, rather than doing the factorization and solve separately.

  3. If you do a solve only, and you solve for more than one right-hand side, it is most efficient to call the skyline subroutine once with all right-hand sides, rather than once for each right-hand side.

  4. The skyline subroutines allow some control over processing of the pivot (diagonal) elements of the matrix during the factorization phase. Pivot processing is controlled by IPARM(10) through IPARM(15) and RPARM(10) through RPARM(15). If a pivot occurs within a range that is designated to be fixed (IPARM(0) = 1, IPARM(10) = 1, and the appropriate element IPARM(11) through IPARM(15) = 1), it is replaced with the corresponding element of RPARM(11) through RPARM(15). Should this pivot fix-up occur, you receive an attention message. This message indicates that the matrix being factored may be unstable (singular or not definite). The results produced in this situation may be inaccurate, and you should review them carefully.

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Guide and Reference

Sparse Matrix Iterative Solver Considerations

This section provides some key points about using the sparse matrix iterative solver subroutines.

Use Considerations

If you need to solve linear systems with different right-hand sides but with the same matrix using the preconditioned algorithms, you can reuse the incomplete factorization computed during the first call to the subroutine.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] th k ~SZjE4 K64 K64 K6E./usr/lpp/essl.html.en_US/html/essl26.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui k~SZjVH6VH6VH6./usr/lpp/essl.html.en_US/html/essl260.htmlen_U Guide and Reference

Guide and Reference

Performance and Accuracy Considerations

  1. The DSMCG and DSMGCG subroutines are provided for migration purposes from earlier releases of ESSL. You get better performance and a wider choice of algorithms if you use the DSRIS subroutine.

  2. To select the sparse matrix subroutine that provides the best performance, you must consider the sparsity pattern of the matrix. From this, you can determine the most efficient storage mode for your sparse matrix. ESSL provides a number of versions of the sparse matrix iterative solve subroutines. They operate on sparse matrices stored in row-wise, diagonal, and compressed-matrix storage modes. These storage modes are described in "Sparse Matrix".

    Storage-by-rows is generally applicable. You should use this storage mode unless your matrices are already set up in one of the other storage modes. If, however, your matrix has a regular sparsity pattern--that is, where the nonzero elements are concentrated along a few diagonals--you may want to use compressed-diagonal storage mode. This can save some storage space. Compressed-matrix storage mode is provided for migration purposes from earlier releases of ESSL and is not intended for use. (You get better performance and a wider choice of algorithms if you use the DSRIS subroutine, which uses storage-by-rows.)

  3. The performance achieved in the sparse matrix iterative solver subroutines depends on the value specified for the relative accuracy epsilon. For details, see "Notes" for each subroutine.

  4. You can select the iterative algorithm you want to use to solve your linear system. The methods include conjugate gradient (CG), conjugate gradient squared (CGS), generalized minimum residual (GMRES), more smoothly converging variant of the CGS method (Bi-CGSTAB), or transpose-free quasi-minimal residual method (TFQMR).

  5. For a general sparse or positive definite symmetric matrix, the iterative algorithm may fail to converge for one of the following reasons:

    • The value of epsilon is too small, asking for too much precision.

    • The maximum number of iterations is too small, allowing too few iterations for the algorithm to converge.

    • The matrix is not positive real; that is, the symmetric part, (A+AT)/2, is not positive definite.

    • The matrix is ill-conditioned, which may cause overflows during the computation.

  6. These algorithms have a tendency to generate underflows that may hurt overall performance. The system default is to mask underflow, which improves the performance of these subroutines.

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Guide and Reference

Linear Least Squares Considerations

This section provides some key points about using the linear least squares subroutines.

Use Considerations

If you want to use a singular value decomposition method to compute the minimal norm linear least squares solution of AX  is congruent to  B, calls to SGESVF or DGESVF should be followed by calls to SGESVS or DGESVS, respectively.

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Guide and Reference

Performance and Accuracy Considerations

  1. Least squares solutions obtained by using a singular value decomposition require more storage and run time than those obtained using a QR decomposition with column pivoting. The singular value decomposition method, however, is a more reliable way to handle rank deficiency.

  2. The short-precision subroutines provide increased accuracy by accumulating intermediate results in long precision. Occasionally, for performance reasons, these intermediate results are stored.

  3. The accuracy of the resulting singular values and singular vectors varies between the short- and long-precision versions of each subroutine. The degree of difference depends on the size and conditioning of the matrix computation.

  4. There are ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Dense Linear Algebraic Equation Subroutines

This section contains the dense linear algebraic equation subroutine descriptions.

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Guide and Reference

SGES, DGES, CGES, and ZGES--General Matrix, Its Transpose, or Its Conjugate Transpose Solve

These subroutines solve the system Ax = b for x, where A is a general matrix and x and b are vectors. Using the iopt argument, they can also solve the real system ATx = b or the complex system AHx = b for x. These subroutines use the results of the factorization of matrix A, produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively.

Table 88. Data Types
A, b, x Subroutine
Short-precision real SGES
Long-precision real DGES
Short-precision complex CGES
Long-precision complex ZGES
Note:The input to these solve subroutines must be the output from the factorization subroutines SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, and ZGEF, respectively.

Syntax

Fortran CALL SGES | DGES | CGES | ZGES (a, lda, n, ipvt, bx, iopt)
C and C++ sges | dges | cges | zges (a, lda, n, ipvt, bx, iopt);
PL/I CALL SGES | DGES | CGES | ZGES (a, lda, n, ipvt, bx, iopt);

On Entry

a
is the factorization of matrix A, produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively. Specified as: an lda by (at least) n array, containing numbers of the data type indicated in Table 88.

lda
is the leading dimension of the array specified for a. Specified as: a fullword integer; lda > 0 and lda >= n.

n
is the order of matrix A. Specified as: a fullword integer; 0 <= n <= lda.

ipvt
is the integer vector ipvt of length n, containing the pivot indices produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively. Specified as: a one-dimensional array of (at least) length n, containing fullword integers.

bx
is the vector b of length n, containing the right-hand side of the system. Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 88.

iopt
determines the type of computation to be performed, where:

If iopt = 0, A is used in the computation.

If iopt = 1, AT is used in SGES and DGES. AH is used in CGES and ZGES.
Note:No data should be moved to form AT or AH; that is, the matrix A should always be stored in its untransposed form.

Specified as: a fullword integer; iopt = 0 or 1.

On Return

bx
is the solution vector x of length n, containing the results of the computation. Returned as: a one-dimensional array, containing numbers of the data type indicated in Table 88.

Notes

  1. The scalar data specified for input arguments lda and n for these subroutines must be the same as the corresponding input arguments specified for SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, and ZGEF, respectively.

  2. The array data specified for input arguments a and ipvt for these subroutines must be the same as the corresponding output arguments for SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, and ZGEF, respectively.

  3. The vectors and matrices used in this computation must have no common elements; otherwise, results are unpredictable. See "Concepts".

Function

The system Ax = b is solved for x, where A is a general matrix and x and b are vectors. Using the iopt argument, this subroutine can also solve the real system ATx = b or the complex system AHx = b for x. These subroutines use the results of the factorization of matrix A, produced by a preceding call to SGEF/SGEFCD, DGEF/DGEFP/DGEFCD, CGEF, or ZGEF, respectively. For a description of how A is factored, see SGEF, DGEF, CGEF, and ZGEF--General Matrix Factorization.

If n is 0, no computation is performed. See references [36] and [38].

Error Conditions

Computational Errors

None
Note:If the factorization performed by SGEF, DGEF, CGEF, ZGEF, SGEFCD, DGEFCD, or DGEFP failed because a pivot element is zero, the results returned by this subroutine are unpredictable, and there may be a divide-by-zero program exception message.

Input-Argument Errors
  1. lda <= 0
  2. n < 0
  3. n > lda
  4. iopt <> 0 or 1

Example 1

Part 1

This part of the example shows how to solve the system Ax = b, where matrix A is the same matrix factored in the "Example 1" for SGEF and DGEF.

Call Statement and Input
           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL SGES( A , 9 , 9 , IPVT , BX , 0  )

IPVT     =  (3, 4, 5, 6, 7, 8, 9, 8, 9)
BX       =  (4.0, 5.0, 9.0, 10.0, 11.0, 12.0, 12.0, 12.0, 33.0)
A        = (same as output A in
"Example 1")

Output
BX       =  (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)

Part 2

This part of the example shows how to solve the system ATx = b, where matrix A is the input matrix factored in "Example 1" for SGEF and DGEF. Most of the input is the same in Part 2 as in Part 1.

Call Statement and Input
           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL SGES( A , 9 , 9 , IPVT , BX , 1  )

IPVT     =  (3, 4, 5, 6, 7, 8, 9, 8, 9)
BX       =  (6.0, 8.0, 10.0, 12.0, 13.0, 14.0, 15.0, 15.0, 15.0)
A        = (same as output A in
"Example 1")

Output
BX       =  (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)

Example 2

Part 1

This part of the example shows how to solve the system Ax = b, where matrix A is the same matrix factored in the "Example 2" for CGEF and ZGEF.

Call Statement and Input
           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL CGES( A , 4 , 4 , IPVT , BX , 0  )

IPVT     =  (4, 4, 3, 4)
BX       =  ((-10.0, 85.0), (-6.0, 61.0), (10.0, 38.0),
             (58.0, 168.0))
A        = (same as output A in
"Example 1")

Output
BX       =  ((9.0, 0.0), (5.0, 1.0), (1.0, 6.0), (3.0, 4.0))

Part 2

This part of the example shows how to solve the system AHx = b, where matrix A is the input matrix factored in "Example 2" for CGEF and ZGEF. Most of the input is the same in Part 2 as in Part 1.

Call Statement and Input
           A  LDA  N   IPVT   BX  IOPT
           |   |   |    |     |    |
CALL CGES( A , 4 , 4 , IPVT , BX , 1  )

IPVT     =  (4, 4, 3, 4)
BX       =  ((71.0, 12.0), (61.0, -70.0), (123.0, -34.0),
             (68.0, 7.0))
A        = (same as output A in
"Example 1")

Output
BX       =  ((9.0, 0.0), (5.0, 1.0), (1.0, 6.0), (3.0, 4.0))

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Guide and Reference

Banded Linear Algebraic Equation Subroutines

This section contains the banded linear algebraic equation subroutine descriptions.

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Guide and Reference

SPTF and DPTF--Positive Definite Symmetric Tridiagonal Matrix Factorization

These subroutines factor symmetric tridiagonal matrix A, stored in symmetric-tridiagonal storage mode, using Gaussian elimination. To solve a tridiagonal system of linear equations with one or more right-hand sides, follow the call to these subroutines with one or more calls to SPTS or DPTS, respectively.

Table 111. Data Types
c, d Subroutine
Short-precision real SPTF
Long-precision real DPTF
Note:The output from these factorization subroutines should be used only as input to the solve subroutines SPTS and DPTS, respectively.

Syntax

Fortran CALL SPTF | DPTF (n, c, d, iopt)
C and C++ sptf | dptf (n, c, d, iopt);
PL/I CALL SPTF | DPTF (n, c, d, iopt);

On Entry

n
is the order n of tridiagonal matrix A. Specified as: a fullword integer; n >= 0.

c
is the vector c, containing the off-diagonal of matrix A in positions 2 through n in an array, referred to as C. Specified as: a one-dimensional array, of (at least) length n, containing numbers of the data type indicated in Table 111.

d
is the vector d, containing the main diagonal of matrix A in positions 1 through n in an array referred to as D. Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 111.

iopt
indicates the type of computation to be performed, where:

If iopt = 0 or 1, Gaussian elimination is used to factor the matrix.

Specified as: a fullword integer; iopt = 0 or 1.

On Return

c
is the vector c, containing part of the factorization of matrix A in an array, referred to as C. Returned as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 111.

d
is the vector d, containing part of the factorization of matrix A in positions 1 through n in an array, referred to as D. Returned as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 111. It has the same length as D on entry.

Note

For a description of how positive definite symmetric tridiagonal matrices are stored, see "Positive Definite Symmetric Tridiagonal Matrix".

Function

The factorization of positive definite symmetric tridiagonal matrix A is computed using Gaussian elimination. This factorization can then be used by SPTS or DPTS, respectively, to solve the tridiagonal systems of linear equations. See reference [71].

No pivoting is done. Therefore, these subroutines should not be used when pivoting is necessary to maintain the numerical accuracy of the solution. Overflow may occur if small pivots are generated.

For performance reasons, divides are done in a way that reduces the effective exponent range for which DPTF works properly; therefore, you may want to scale your problem, such that the diagonal elements are close to 1.0 for DPTF.

Error Conditions

Computational Errors

None
Note:There is no test for positive definiteness in these subroutines.

Input-Argument Errors
  1. n < 0
  2. iopt <> 0 or 1

Example

This example shows a factorization of the tridiagonal matrix A, of order 4: 

                          *                    *
                          | 1.0  1.0  0.0  0.0 |
                          | 1.0  2.0  1.0  0.0 |
                          | 0.0  1.0  3.0  1.0 |
                          | 0.0  0.0  1.0  1.0 |
                          *                    *

Call Statement and Input
           N   C   D  IOPT
           |   |   |   |
CALL DPTF( 4 , C , D , 0  )
 
C        =  ( . , 1.0, 1.0, 1.0)
D        =  (1.0, 2.0, 3.0, 1.0)

Output
C        =  ( . , -1.0, -1.0, -1.0)
D        =  (-1.0, -1.0, -1.0, -1.0)

Notes

  1. A "." means you do not have to store a value in that position in the array. However, these storage positions are required and may be overwritten during the computation.

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Guide and Reference

SPTS and DPTS--Positive Definite Symmetric Tridiagonal Matrix Solve

These subroutines solve a positive definite symmetric tridiagonal system of equations using the factorization of matrix A, stored in symmetric-tridiagonal storage mode, produced by SPTF and DPTF, respectively.

Table 112. Data Types
c, d, b, x Subroutine
Short-precision real SPTS
Long-precision real DPTS
Note:The input to these solve subroutines must be the output from the factorization subroutines SPTF and DPTF, respectively.

Syntax

Fortran CALL SPTS | DPTS (n, c, d, bx)
C and C++ spts | dpts (n, c, d, bx);
PL/I CALL SPTS | DPTS (n, c, d, bx);

On Entry

n
is the order n of tridiagonal matrix A. Specified as: a fullword integer; n >= 0.

c
is the vector c, containing part of the factorization of matrix A from SPTF or DPTF, respectively, in an array, referred to as C. Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 112.

d
is the vector d, containing part of the factorization of matrix A from SPTF or DPTF, respectively, in an array, referred to as D. Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 112.

bx
is the vector b, containing the right-hand side of the system in the first n positions in an array, referred to as BX. Specified as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 112.

On Return

bx
is the solution vector x of length n, containing the solution of the tridiagonal system in the first n positions in an array, referred to as BX. Returned as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 112.

Note

For a description of how tridiagonal matrices are stored, see "Positive Definite or Negative Definite Symmetric Matrix".

Function

The solution of positive definite symmetric tridiagonal system Ax = b is computed using the factorization produced by SPTF or DPTF, respectively. The factorization is based on Gaussian elimination. See reference [71].

Error Conditions

Computational Errors

None

Input-Argument Errors

n < 0

Example

This example finds the solution of tridiagonal system Ax = b, where matrix A is the same matrix factored in "Example" for SPTF and DPTF. b is: 

                  (2.0, 4.0, 5.0, 2.0)

and x is: 

                  (1.0, 1.0, 1.0, 1.0)

Call Statement and Input
           N   C   D   BX
           |   |   |   |
CALL DPTS( 4 , C , D , BX )
 
C        =  ( . , -1.0, -1.0, -1.0)
D        =  (-1.0, -1.0, -1.0, -1.0)
BX       =  (2.0, 4.0, 5.0, 2.0)

Output
BX       =  (1.0, 1.0, 1.0, 1.0)

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Guide and Reference

Sparse Linear Algebraic Equation Subroutines

This section contains the sparse linear algebraic equation subroutine descriptions.

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Guide and Reference

Linear Least Squares Subroutines

This section contains the linear least squares subroutine descriptions.

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Guide and Reference

Performance and Accuracy Considerations

  1. The accuracy of the resulting eigenvalues and eigenvectors varies between the short- and long-precision versions of each subroutine. The degree of difference depends on the size and conditioning of the matrix computation. Some of the subroutines have examples illustrating this difference.

  2. The short precision subroutines provide increased accuracy by accumulating intermediate results in long precision. Occasionally, for performance reasons, these intermediate results are stored.

  3. If you want to compute 10% or fewer eigenvalues only, or you want to compute 30% or fewer eigenvalues and eigenvectors, you get better performance if you use _SPSV and _HPSV instead of _SPEV and _HPEV, respectively. For all other uses, you should use _SPEV and _HPEV.

  4. There are some ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] X:Vt_ kV~SZjH6H6H6./usr/lpp/essl.html.en_US/html/essl308.htmlen_U Guide and Reference

Guide and Reference

Eigensystem Analysis Subroutines

This section contains the eigensystem analysis subroutine

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Guide and Reference

Fourier Transforms, Convolutions and Correlations, and Related Computations

The signal processing subroutines, provided in three areas, are described in this chapter.

Overview of the Signal Processing Subroutines

This section describes the subroutines in each of the three signal processing areas:

  • Fourier transform subroutines (Table 123)
  • Convolution and correlation subroutines (Table 124)
  • Related-computation subroutines (Table 125)

Fourier Transforms Subroutines

The Fourier transform subroutines perform mixed-radix transforms in one, two, and three dimensions.

Table 123. List of Fourier Transform Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Complex Fourier Transform 
SCFT
SCFTP§

DCFT SCFT and DCFT--Complex Fourier Transform
Real-to-Complex Fourier Transform SRCFT DRCFT SRCFT and DRCFT--Real-to-Complex Fourier Transform
Complex-to-Real Fourier Transform SCRFT DCRFT SCRFT and DCRFT--Complex-to-Real Fourier Transform
Cosine Transform 
SCOSF
SCOSFT§

DCOSF SCOSF and DCOSF--Cosine Transform
Sine Transform SSINF DSINF SSINF and DSINF--Sine Transform
Complex Fourier Transform in Two Dimensions 
SCFT2
SCFT2P§

DCFT2 SCFT2 and DCFT2--Complex Fourier Transform in Two Dimensions
Real-to-Complex Fourier Transform in Two Dimensions SRCFT2 DRCFT2 SRCFT2 and DRCFT2--Real-to-Complex Fourier Transform in Two Dimensions
Complex-to-Real Fourier Transform in Two Dimensions SCRFT2 DCRFT2 SCRFT2 and DCRFT2--Complex-to-Real Fourier Transform in Two Dimensions
Complex Fourier Transform in Three Dimensions 
SCFT3
SCFT3P§

DCFT3 SCFT3 and DCFT3--Complex Fourier Transform in Three Dimensions
Real-to-Complex Fourier Transform in Three Dimensions SRCFT3 DRCFT3 SRCFT3 and DRCFT3--Real-to-Complex Fourier Transform in Three Dimensions
Complex-to-Real Fourier Transform in Three Dimensions SCRFT3 DCRFT3 SCRFT3 and DCRFT3--Complex-to-Real Fourier Transform in Three Dimensions

§ This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use in new programs. Documentation for this subroutine is no longer provided.

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Guide and Reference

Related-Computation Subroutines

The related-computation subroutines consist of a group of computations that can be used in general signal processing applications. They are similar to those provided on the IBM 3838 Array Processor; however, the ESSL subroutines generally solve a wider range of problems.

Table 125. List of Related-Computation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Polynomial Evaluation SPOLY DPOLY SPOLY and DPOLY--Polynomial Evaluation
I-th Zero Crossing SIZC DIZC SIZC and DIZC--I-th Zero Crossing
Time-Varying Recursive Filter STREC DTREC STREC and DTREC--Time-Varying Recursive Filter
Quadratic Interpolation SQINT DQINT SQINT and DQINT--Quadratic Interpolation
Wiener-Levinson Filter Coefficients SWLEV DWLEV SWLEV, DWLEV, CWLEV, and ZWLEV--Wiener-Levinson Filter Coefficients

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Guide and Reference

Initializing Auxiliary Working Storage

In many of those subroutines requiring aux1 auxiliary working storage, two invocations of the subroutines are necessary. The first invocation initializes the working storage in aux1 for the subroutine, and the second performs the computations. (For an explanation of auxiliary working storage, see "Understanding Auxiliary Working Storage Requirements".) As a result, the working storage in aux1 should not be used by the calling program between the two calls to the subroutine. However, it can be reused after intervening calls to the subroutine with different arguments.

If you plan to repeat a computation many times using the same set of arguments, you only need to do one initialization of the aux1 array; that is, the initialized aux1 array can be saved and reused as many times as needed for the computation.

If you plan to perform different computations, with different sets of arguments (except for input argument x), you need to do an initialization for each different computation; that is, you initialize the various aux1 arrays for use with the different computations, saving and reusing them until they are not needed any more.

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Guide and Reference

Determining the Amount of Auxiliary Working Storage That You Need

To determine the size of auxiliary storage, you have several choices. First, you can use the formulas provided in each subroutine description. Second, ESSL's input-argument error recovery provides a means of determining the minimum size you need for auxiliary storage. It uses the optionally-recoverable error 2015. For details, see "Using Auxiliary Storage in ESSL". Third, you can have ESSL dynamically allocate aux and aux2. For details, see "Dynamic Allocation of Auxiliary Storage".

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] u k~SZjE4 K64 K64 K6E./usr/lpp/essl.html.en_US/html/essl32.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui k(n~SZjwH6H6H6w./usr/lpp/essl.html.en_US/html/essl320.htmlen_U Guide and Reference

Guide and Reference

Performance and Accuracy Considerations

The following sections explain the performance and accuracy considerations for the Fourier transforms, convolution, and correlation subroutines. For further details about performance and accuracy, see "Planning Your Program".

When Running on the Workstation Processors

There are ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Defining Arrays

The stride arguments, inc1h, inc1x, inc1y, inc2x, inc2y, inc3x, and inc3y, provide great flexibility in defining the input and output data arrays. The arrangement of data in storage, however, can have an effect upon cache performance. By using strides, you can have data scattered in storage. Best performance is obtained with data closely spaced in storage and with elements of the sequence in contiguous locations. The optimum values for inc1h, inc1x, and inc1y are 1.

In writing the calling program, you may find it convenient to declare X or Y as a two-dimensional array. For example, you can declare X in a DIMENSION statement as X(INC2X,M).

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] HD lr~SZjH6H6H6&./usr/lpp/essl.html.en_US/html/essl322.htmlen_U    e<=>ARVainorstu /Tbcdfhlmp"&,-.123;EFHgy#()045BCDILMNOPSv9:UX_kqwx!+68GQYjz7K`(x*W(9(I"(9( Ȓ5EO.yʀqfW8pDh*w$O#H((+$ #k.U`hx@PAH4`9bŋ,1ư@Hh%Jm M1'`065Im5+ֵkZֵkZֵkZֵkZֵkZֵx7h$Q0Hh0"3K|#8W7h,xpx^K4f TI&1ūOmϴteuiy'Z]=877h@X(0csk$U`P csk\z֝-6z#7wNZTG|ͧ7.lPsn@  s$U X ,|8[qz>X{qpW J}SSyj|ZTuú5U'ZF[[U'Z{ W86{T^io›Ӓ3Vx3ӭ;{Ά:ҡ}hΆ֛ђou+l{/;Dvul]]BWVb;I՞ \C~oy_T^WHsԝ/:iQemK{UCTϭ7Ay#X@ A>oyku+R)[|it_8^!CdSA o;jL_[|)|ʉz VnWVSO7 hͽ $AL,9p Tuc{u"PsHW^ԺdeDtwU^/%sO/DHXt_yQ.Ⱦ"eEqRU.Rmu՚2A]׶  h(ѴB#Dd_N2â8m6]rHwK}Sz*_M^vum/m6.isuLV^WG|) ʩt7Vq_EO\)!VJͧte򅫜Nu|pLD@㹚JTo o>}򅫜N^l oWZpSt_8pqҖ$CUeo/eN:|Wŏy^ln=SSoB'M芹KRcx,ћ{L AL,9p>F8XҥӭB'M{W_T1)Mm'[-*#tS~$-\{oJNX W86㹚JToێL:/d_Na|;j-H )Xq:oj*6RۣҚKδ)b^QmW6^MmZI_+ZFHs߅-=)I+W6ҥӭq:m͕Ic/DH o#>wHsju$Tk \Ⱦ a|Åݑ|6h(ѴB Ȏk":Xt_  (Dwd_ :/ Qh(; ;/IEpp„G*%h(ѴB Ȏ&â8h8XaB>)KnJkm/:J{GҚKδd_PȾ&&/MEw[j^tJkm/:JZ3y   w V;=yd_*â8$w)YʰD.F\":l:/&Et& !EpHȾ&â8!| a|@޴5;j\^J}S<w2U,j]>t4դ [\ud_MEpȾt_]| a|O7 ̱:w11y<p!u4m"â8h8XaB! ":tDEu 0PHȾ&â8!|Bl:/"@޴5q%}~V{ 8~p<DHsͷI`);;`}+ um/|0 â9 \mQG+S}Z3][i݇C<2/a|;Mq:mpꪵN=A!NȾ$â8mAh(ȾTHEt6h(4 ȔD.@Et&! a|CkJ5m=S6VyoWd/VT{$wn(OM6V{ 8Cn6N7Ǽ N4ΆdcXsƃ$S hͽ&H0+ms~+#Þ4 TFm2D)=ư绍"H,cx,ћ{HYPs1w>Zci绍"H,cx,ћ{D0YPr!DÞ4 L w5o*с$Q0Hh0( Guide and Reference

Guide and Reference

How the Fourier Transform Subroutines Achieve High Performance

There are two levels of optimization for the fast Fourier transforms (FFTs) in the ESSL library. For sequences with a large power of 2 length, we provide efficient radix-2 and radix-8 transform implementations where cache use is optimized. The cache optimization includes ordering of operations to maximize stride-1 data access and prefetching cache lines.

Similar optimization techniques are used for sequence lengths which are not a power of 2 and mixed-radix FFT's are performed. Many short sequence FFT's have sequence size specific optimizations. Some of these optimizations were originally developed for a vector machine and have been adapted for cache based RISC machines (see references [1], [5], and [7])

The other optimization in the FFT routine is to treat multiple sequences as efficiently as possible. Techniques here include blocking sequences to fit into available CPU cache and transposing sequences to ensure stride-1 access. Whenever possible, the highest performance can be obtained when multiple sequences are transformed in a single call.

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Guide and Reference

Convolution and Correlation Considerations

This section describes some ways to optimize performance in the convolution and correlation subroutines.

Performance Tradeoffs between Subroutines

The subroutines SCON, SCOR, SACOR, SCOND, SCORD, SDCON, SDCOR, DDCON, and DDCOR compute convolutions, correlations, and autocorrelations using essentially the same methods. They make a decision, based on estimated timings, to use one of two methods:

  • A direct method that is most efficient when one or both of the input sequences are short

  • A direct method that is most efficient when the output sequence is short

Using this approach has the following advantages:

  • In most cases, improved performance can be achieved for direct methods because:

    • No initialization is required.

    • No working storage or padding of sequences is necessary.

  • In some cases, greater accuracy may be available.

  • Negative strides can be used.

In general, using SCONF, SCORF, and SACORF provides the best performance, because the mixed-radix Fourier transform subroutines are used. However, if you can determine from your arguments that a direct method is preferred, you should use SCOND and SCORD instead. These give you better performance for the direct methods, and also give you additional capabilities.

In cases where there is doubt as to the best choice of a subroutine, perform timing experiments.

Special Uses of SCORD

The subroutine SCORD can perform the functions of SCON and SACOR; that is, it can compute convolutions and autocorrelations. To compute a convolution, you must specify a negative stride for h (see Example 4 in SCORD). To compute the autocorrelation, you must specify the two input sequences to be the same (see Example 5 in SCORD).

Special Uses of _DCON and _DCOR

The _DCON and _DCOR subroutines compute convolutions and correlations, respectively, by the direct method with decimated output. Setting the decimation interval id = 1 in SDCON and SDCOR provides the same function as SCOND and SCORD, respectively. Doing the same in DDCON and DDCOR provides long-precision versions of SCOND and SCORD, respectively, which are not otherwise available.

Accuracy When Direct Methods Are Used

The direct methods used by the convolution and correlation subroutines use vector operations to accumulate sums of products. The products are computed and accumulated in long precision. As a result, higher accuracy can be obtained in the final results for some types of data. For example, if input data consists only of integers, and if no intermediate and final numbers become too large (larger than 224-1 for short-precision computations and larger than 256-1 for long-precision computations), the results are exact.

Accuracy When Fourier Methods Are Used

The Fourier methods used by the convolution and correlation subroutines compute Fourier transforms of input data that is multiplied element-by-element in short-precision arithmetic. The inverse Fourier transform is then computed. There are internally generated rounding errors in the Fourier transforms. It has been shown in references [ 90] and [ 79] that, in the case of white noise data, the relative root mean square (RMS) error of the Fourier transform is proportional to log2n with a very small proportionality factor. In general, with random, evenly distributed data, this is better than the RMS error of the direct method. However, one must keep in mind the fact that, while the Fourier method may yield a smaller root mean square error, there can be points with large relative errors. Thus, it can happen that some points, usually at the ends of the output sequence, can be obtained with greater relative accuracy with direct methods.

Convolutions and Correlations by Fourier Methods

The convolution and correlation subroutines that use the Fourier methods determine a sequence length n, whose Fourier transform is computed using ESSL subroutines. In the simple case where iy0 = 0 for convolution or iy0 = -nh+1 for correlation, n is chosen as a value greater than or equal to the following, which is also acceptable to the Fourier tranform subroutines:

nt = min(nh+nx-1, ny) for convolution and correlation
nt = min(nx+nx-1, ny) for autocorrelation

which is also acceptable to the Fourier subroutines.

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Guide and Reference

Related Computation Considerations

This section describes some key points about using the related-computation subroutines.

Accuracy Considerations

  • Many of the subroutines performing short-precision computations provide increased accuracy by accumulating results in long precision. This is noted in the functional description for each subroutine.

  • There are ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Fourier Transform Subroutines

This section contains the Fourier transform subroutine descriptions.

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Guide and Reference

Convolution and Correlation Subroutines

This section contains the convolution and correlation subroutine descriptions.

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Guide and Reference

Related-Computation Subroutines

This section contains the related-computation subroutine descriptions.

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DHlw]] rMa֔].ٷ_շ/}tD؆L lf-(b.oA8DN/|%vevuw169)]]+!?ov7DJo0C H't U.aDN/$E_*֫ DoA8DN/|Vp3 0:$q}?N _*B}>/]h04 4 "j_$U"Z1kU"$MtD׮9&DkJml_DJoݍb02r.afL]pVPEi@\ƒ'Xll@ RO`d'XuփBXF%\[-w=EcZ b|U 80 w+|_PPXK(661ǵ\cv˃ xOr"jɚ%b:@˃`r>Ɲn \P2r. PLb86,@ k~SZjٸH6ٸH6ٸH6./usr/lpp/essl.html.en_US/html/essl352.htmlen_U Guide and Reference

Guide and Reference

Use Considerations

It is important to understand the concept of stride for sequences when using these subroutines. For example, in the sort subroutines, a negative stride causes a sequence to be sorted into descending order in an array. In the search subroutines, a negative stride reverses the direction of the search. See "How Stride Is Used for Vectors".

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Guide and Reference

Performance and Accuracy Considerations

  1. The binary search subroutines provide better performance than the sequential search subroutines because of the nature of the searching algorithms. However, the binary search subroutines require that, before the subroutine is called, the sequence to be searched is sorted into ascending order. Therefore, if your data is already sorted, a binary search subroutine is faster. On the other hand, if your data is in random order and the number of elements being searched for is small, a sequential search subroutine is faster than doing a sort and binary search.

  2. When doing multiple invocations of the binary search subroutines, you get better overall performance from the searching algorithms by doing fewer invocations and specifying larger search element arrays for argument x.

  3. If you do not need the results provided in array RC by these subroutine, you get better performance if you do not request it. That is, specify 0 for the iopt argument.

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Guide and Reference

Sorting and Searching Subroutines

This section contains the sorting and searching subroutine descriptions.

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Guide and Reference

ISORTS, SSORTS, and DSORTS--Sort the Elements of a Sequence Using a Stable Sort and Note the Original Element Positions

These subroutines sort the elements of sequence x using a stable sort; that is, where equal elements occur in the input sequence, they remain in the same relative order in the output sequence. The original positions of the elements in sequence x are returned in the indices array INDX.
Note:If you need a stable sort, then you should use these subroutines rather than ISORTX, SSORTX, or DSORTX.

Table 146. Data Types
x, work Subroutine
Integer ISORTS
Short-precision real SSORTS
Long-precision real DSORTS

Syntax

Fortran CALL ISORTS | SSORTS | DSORTS (x, incx, n, indx, work, lwork)
C and C++ isorts | ssorts | dsorts (x, incx, n, indx, work, lwork);
PL/I CALL ISORTS | SSORTS | DSORTS (x, incx, n, indx, work, lwork);

On Entry

x
is the sequence x of length n, to be sorted. Specified as: a one-dimensional array of (at least) length 1+(n-1)|incx| elements, containing numbers of the data type indicated in Table 146.

incx
is the stride for both the input sequence x and the output sequence x. If it is positive, elements are sorted into ascending order in the array, and if it is negative, elements are sorted into descending order in the array.

Specified as: a fullword integer. It can have any value.

n
is the number of elements in sequence x. Specified as: a fullword integer; n >= 0.

indx
See "On Return".

work
is the storage work area used by this subroutine. Its size is specified by lwork. Specified as: an area of storage, containing numbers of the data type indicated in Table 146.

lwork
is the size of the work area specified by work-- that is, the number of elements in work. Specified as: a fullword integer; lwork >= n/2.
Note:This is the value to achieve optimal performance. The sort is performed regardless of the value you specify for lwork, but you may receive an attention message.

On Return

x
is the sequence x of length n, with its elements sorted into designated order in the array. Returned as: a one-dimensional array, containing numbers of the data type indicated in Table 146.

indx
is the array, referred to as INDX, containing the n indices that indicate, for the elements in the sorted output sequence, the original positions of those elements in input sequence x.
Note:It is important to remember that when you specify a negative stride, ESSL assumes that the order of the input and output sequence elements in the X array is reversed; however, the elements in INDX are not reversed. See "Function".

Returned as: a one-dimensional array of length n, containing fullword integers; 1 <= (INDX elements) <= n.

Function

The elements of input sequence x are sorted into ascending order using a partition sort. The sorting is stable; that is, where equal elements occur in the input sequence, they remain in the same relative order in the output sequence. The elements of output sequence x can be expressed as follows:

x1 <= x2 <= x3 <= ... <= xn

By specifying a negative stride for x, the elements of input sequence x are assumed to be reversed in the array, (xn, xn-1, ... , x1), thus producing a sort into descending order within the array.

In addition, the INDX array contains the n indices that indicate, for the elements in the sorted output sequence, the original positions of those elements in input sequence x. (These are not the positions in the array, but rather the positions in the sequence.) For each element xj in the input sequence, becoming element xxk in the output sequence, the elements in INDX are defined as follows:

INDX(k) = j    for j = 1, n and k = 1, n
where xxk = xj

To understand INDX when you specify a negative stride, you should remember that both the input and output sequences, x, are assumed to be in reverse order in array X, but INDX is not affected by stride. The sequence elements of x are assumed to be stored in your input array as follows:

X = (xn, xn-1, ... , x1)

The sequence elements of x are stored in your output array by ESSL as follows:

X = (xxn, xxn-1, ... , xx1)

where the elements xxk are the elements xj, sorted into descending order in X. As an example of how INDX is calculated, if xx1 = xn-1, then INDX(1) = n-1.

If n is 0, no computation is performed. See references [28] and [69].

Error Conditions

Resource Errors

Unable to allocate internal work area.

Computational Errors

None

Input-Argument Errors

n < 0

Example 1

This example shows how to sort a sequence x into ascending order by specifying a positive stride. Because this is a stable sort, the -1 elements remain in the same relative order in the output sequence, indicated by INDX(2) = 2 and INDX(3) = 4.

Call Statement and Input
             X  INCX  N   INDX   WORK  LWORK
             |   |    |    |      |      |
CALL ISORTS( X , 2  , 5 , INDX , WORK ,  5  )
 
X        =  (2, . , -1, . , 5, . , -1, . , -2)

Output
X        =  (-2, . , -1, . , -1, . , 2, . , 5)
INDX     =  (5, 2, 4, 1, 3)

Example 2

This example shows how to sort a sequence x into descending order by specifying a negative stride. Therefore, both the input and output sequences are assumed to be reversed in the array X. The input sequence is assumed to be stored as follows:

X = (x5, x4, x3, x2, x1) = (2, -1, 5, -1, -2)

The output sequence is stored by ESSL as follows:

X = (xx5, xx4, xx3, xx2, xx1) = (5, 2, -1, -1, -2)

As a result, INDX is defined as follows:

INDX = (indx1, indx2, indx3, indx4, indx5) = (1, 2, 4, 5, 3)

For example, because output sequence element xx4 = 2 is input sequence element x5, then INDX(4) = 5. Also, because this is a stable sort, the -1 elements remain in the same relative order in the output sequence, indicated by INDX(2) = 2 and INDX(3) = 4.

Call Statement and Input
             X   INCX  N   INDX   WORK  LWORK
             |    |    |    |      |      |
CALL ISORTS( X , -1  , 5 , INDX , WORK ,  5  )
 
X        =  (2, -1, 5, -1, -2)

Output
X        =  (5, 2, -1, -1, -2)
INDX     =  (1, 2, 4, 5, 3)

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Guide and Reference

Interpolation

The interpolation subroutines are described in this chapter.

Overview of the Interpolation Subroutines

The interpolation subroutines provide the capabilities of doing polynomial interpolation, local polynomial interpolation, and one- and two-dimensional cubic spline interpolation (Table 149).

Table 149. List of Interpolation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Polynomial Interpolation SPINT DPINT SPINT and DPINT--Polynomial Interpolation
Local Polynomial Interpolation STPINT DTPINT STPINT and DTPINT--Local Polynomial Interpolation
Cubic Spline Interpolation SCSINT DCSINT SCSINT and DCSINT--Cubic Spline Interpolation
Two-Dimensional Cubic Spline Interpolation SCSIN2 DCSIN2 SCSIN2 and DCSIN2--Two-Dimensional Cubic Spline Interpolation

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Guide and Reference

Use Considerations

Polynomial interpolation (SPINT and DPINT) is a global scheme. As the number of data points increases, the degree of the interpolating polynomial is raised; therefore, the graph of the interpolating polynomial tends to be oscillatory.

Local polynomial interpolation (STPINT and DTPINT) is a local scheme. The data generated is affected only by locally grouped data points. The degree of the local interpolating polynomial is usually lower than a global interpolating polynomial.

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Guide and Reference

Performance and Accuracy Considerations

  1. Doing extrapolation with SPINT and DPINT is not encouraged unless you know the consequences of doing polynomial extrapolation.

  2. If performance is the overriding consideration, you should investigate using the general signal processing subroutines, DQINT and SQINT.

  3. There are some ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Interpolation Subroutines

This section contains the interpolation subroutine descriptions.

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Guide and Reference

Use Considerations

This section contains some key points about using the numerical quadrature subroutines.

Choosing the Method

The theoretical aspects of choosing the method to use for integration can be found in the references [26], [58], and [86].

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] } kP~SZjE5 K65 K65 K6E./usr/lpp/essl.html.en_US/html/essl37.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui kR~SZj5H6H6H65./usr/lpp/essl.html.en_US/html/essl370.htmlen_U Guide and Reference

Guide and Reference

Performance and Accuracy Considerations

  1. There are n function evaluations for a method of order n. Because function evaluations are expensive in terms of computing time, you should weigh the considerations for computing time and accuracy in choosing a value for n.

  2. To achieve optimal performance in the _GLNQ2 subroutines, specify the first variable integrated to be the variable having more points. This allows both the subroutine and the function evaluation to achieve optimal performance. Details on how to do this are given in "Notes".

  3. There are some ESSL-specific rules that apply to the results of computations on the workstation processors using the ANSI/IEEE standards. For details, see "What Data Type Standards Are Used by ESSL, and What Exceptions Should You Know About?".

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Guide and Reference

Coding and Setting Up SUBF in Your Program

Examples of coding a subf subroutine in Fortran are provided for each subroutine in this chapter. Examples of coding a subf subroutine in C, C++, and PL/I are provided in "Example 1".

Depending on the programming language you use for your program that calls the numerical quadrature subroutines, you have a choice of one or more languages that you can use for writing subf. These rules and other langauge-related coding rules for setting up subf in your program are described in the following sections:

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] Tʴ kA~SZjH6H6H6./usr/lpp/essl.html.en_US/html/essl373.htmlen_U Guide and Reference

Guide and Reference

Numerical Quadrature Subroutines

This section contains the numerical quadrature subroutine descriptions.

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Guide and Reference

SGLNQ and DGLNQ--Numerical Quadrature Performed on a Function Using Gauss-Legendre Quadrature

These functions approximate the integral of a real valued function over a finite interval, using the Gauss-Legendre Quadrature method of specified order.

Table 156. Data Types
a, b, Result Subroutine
Short-precision real SGLNQ
Long-precision real DGLNQ

Syntax

Fortran SGLNQ | DGLNQ (subf, a, b, n)
C and C++ sglnq | dglnq (subf, a, b, n);
PL/I SGLNQ | DGLNQ (subf, a, b, n);

On Entry

subf
is the user-supplied subroutine that evaluates the integrand function. The subroutine should be defined with three arguments: t, y, and n. For details, see "Programming Considerations for the SUBF Subroutine".

Specified as: subf must be declared as an external subroutine in you application program. It can be whatever name you choose.

a
is the lower limit of integration, a. Specified as: a number of the data type indicated in Table 156.

b
is the upper limit of integration, b. Specified as: a number of the data type indicated in Table 156.

n
is the order of the quadrature method to be used. Specified as: a fullword integer; n = 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 24, 32, 40, 48, 64, 96, 128, or 256.

On Return

Function value
 

is the approximation of the integral. Returned as: a number of the data type indicated in Table 156.

Notes

  1. Declare the DGLNQ function in your program as returning a long-precision real number. Declare the SGLNQ, if necessary, as returning a short-precision real number.

  2. The subroutine specified for subf must be declared as external in your program. Also, data types used by subf must agree with the data types specified by this ESSL subroutine. The variable x, described under "Function", and the argument n correspond to the subf arguments t and n, respectively. For details on how to set up the subroutine, see "Programming Considerations for the SUBF Subroutine".

Function

The integral is approximated for a real valued function over a finite interval, using the Gauss-Legendre Quadrature method of specified order. The region of integration is from a to b. The method of order n is theoretically exact for integrals of the following form, where f is a polynomial of degree less than 2n:



Figure ESYGR150 not displayed.

The method of order n is a good approximation when your integrand is closely approximated by a function of the form f(x), where f is a polynomial of degree less than 2n. See references [26] and [86]. The result is returned as the function value.

Error Conditions

Computational Errors

None

Input-Argument Errors

n is not an allowable value, as listed in the syntax for this argument.

Example

This example shows how to compute the integral of the function f given by:

f(x) = x2+ex

over the interval (0.0, 2.0), using the Gauss-Legendre method with 10 points:



Figure ESYGR151 not displayed.

The user-supplied subroutine FUN1, which evaluates the integrand function, is coded in Fortran as follows: 

   SUBROUTINE FUN1 (T,Y,N)
   INTEGER*4 N
   REAL*4 T(*),Y(*)
   DO 1 I=1,N
1     Y(I)=T(I)**2+EXP(T(I))
   RETURN
   END

Program Statements and Input
EXTERNAL FUN1
        .
        .
        .
              SUBF    A     B    N
               |      |     |    |
XINT = SGLNQ( FUN1 , 0.0 , 2.0 , 10 )
        .
        .
        .

FUN1     =(see above)

Output
XINT     =  9.056

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IZsM=ʯ!$#~-J:%J;̕3pL zQ0hm>)X 4q R#%ĉnϠ϶lnv96>HPٱ* xvޕZUesi&П6lJ%y͉s{g}ϝmҧ_GNLB:9жН[AI5;ٙN}Wcb@ȿ%~%JQ M&z =C]4JPPGn8 ͑D G_iggirPh 04`80\:CE#Vx@rT(3qƂF?t .ixGB±(h@Fun:ݵfϓxv~rE|40~͞ ~-7!ՁDVNM<4+C[f!jBsdb;-Qڹ\u 6{nޕ҂ bKNs+yλSu83gҼv_'9i;~-7h@C.q`Qj?]U*@a=W z0a`PP02ø]1Y1 )p`q Guide and Reference

Guide and Reference

SGHMQ and DGHMQ--Numerical Quadrature Performed on a Function Using Gauss-Hermite Quadrature

These functions approximate the integral of a real valued function over the entire real line, using the Gauss-Hermite Quadrature method of specified order.

Table 161. Data Types
a, b, Result Subroutine
Short-precision real SGHMQ
Long-precision real DGHMQ

Syntax

Fortran SGHMQ | DGHMQ (subf, a, b, n)
C and C++ sghmq | dghmq (subf, a, b, n);
PL/I SGHMQ | DGHMQ (subf, a, b, n);

On Entry

subf
is the user-supplied subroutine that evaluates the integrand function. The subroutine should be defined with three arguments: t, y, and n. For details, see "Programming Considerations for the SUBF Subroutine".

Specified as: subf must be declared as an external subroutine in your application program. It can be whatever name you choose.

a
is the centering constant a for the exponential. Specified as: a number of the data type indicated in Table 161.

b
is the scaling constant b for the exponential. Specified as: a number of the data type indicated in Table 161; b > 0.

n
is the order of the quadrature method to be used. Specified as: a fullword integer; n = 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 24, 32, 40, 48, 64, or 96.

On Return

Function value
 

is the approximation of the integral. Returned as: a number of the data type indicated in Table 161.

Notes

  1. Declare the DGHMQ function in your program as returning a long-precision real number. Declare the SGHMQ function, if necessary, as returning a short-precision real number.

  2. The subroutine specified for subf must be declared as external in your program. Also, data types used by subf must agree with the data types specified by this ESSL subroutine. The variable x, described under "Function", and the argument n correspond to the subf arguments t and n, respectively. For details on how to set up the subroutine, see "Programming Considerations for the SUBF Subroutine".

Function

The integral is approximated for a real valued function over the entire real line, using the Gauss-Hermite Quadrature method of specified order. The region of integration is from -infinity to infinity. The method of order n is theoretically exact for integrals of the following form, where f is a polynomial of degree less than 2n:



Figure ESYGR164 not displayed.

The method of order n is a good approximation when your integrand is closely approximated by a function of the following form, where f is a polynomial of degree less than 2n:



Figure ESYGR165 not displayed.

See references [26] and [86]. The result is returned as the function value to a Fortran, C, C++, or PL/I program.

Error Conditions

Computational Errors

None

Input-Argument Errors
  1. b <= 0
  2. n is not an allowable value, as listed in the syntax for this argument.

Example

This example shows how to compute the integral of the function f given by:



Figure ESYGR166 not displayed.

over the interval (-infinity, infinity), using the Gauss-Hermite method with 4 points:



Figure ESYGR167 not displayed.

The user-supplied subroutine FUN1, which evaluates the integrand function, is coded in Fortran as follows: 

   SUBROUTINE FUN1 (T,Y,N)
   INTEGER*4 N
   REAL*4 T(*),Y(*)
   DO 1 I=1,N
1     Y(I)=T(I)**2*EXP(-2.0*(T(I)+5.0)**2)
   RETURN
   END

Program Statements and Input
EXTERNAL FUN1
        .
        .
        .
              SUBF     A     B    N
               |       |     |    |
XINT = SGHMQ( FUN1 , -5.0 , 2.0 , 4 )
        .
        .
        .

FUN1     =(see above)

Output
XINT     =  31.646

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Guide and Reference

Random Number Generation

The random number generation subroutines are described in this chapter.

Overview of the Random Number Generation Subroutines

Random number generation subroutines generate uniformly distributed random numbers or normally distributed random numbers (Table 162).

Table 162. List of Random Number Generation Subroutines
Descriptive Name Short- Precision Subroutine Long- Precision Subroutine Page
Generate a Vector of Uniformly Distributed Random Numbers SURAND DURAND SURAND and DURAND--Generate a Vector of Uniformly Distributed Random Numbers
Generate a Vector of Normally Distributed Random Numbers SNRAND DNRAND SNRAND and DNRAND--Generate a Vector of Normally Distributed Random Numbers
Generate a Vector of Long Period Uniformly Distributed Random Numbers SURXOR DURXOR SURXOR and DURXOR--Generate a Vector of Long Period Uniformly Distributed Random Numbers

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Guide and Reference

Use Considerations

If you need a very long period random number generator, you should use SURXOR and DURXOR, rather than SURAND and DURAND, respectively. The very long period of the generator used by SURXOR and DURXOR, 21279-1, makes it useful in modern statistical simulations in which the shorter period of other generators can be exhausted during a single run. As a result, if you need a large number of random numbers, you can use these subroutines, because with this generator, you are not be requesting more than a small percentage of the entire period of the generator.

Random Number Generation Subroutines

This section contains the random number generation subroutine descriptions.

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Guide and Reference

SURAND and DURAND--Generate a Vector of Uniformly Distributed Random Numbers

These subroutines generate vector x of uniform (0,1) pseudo-random numbers, using the multiplicative congruential method with a user-specified seed.

Table 163. Data Types
x seed Subroutine
Short-precision real Long-precision real SURAND
Long-precision real Long-precision real DURAND
Note:If you need a very long period random number generator, use SURXOR and DURXOR instead of these subroutines.

Syntax

Fortran CALL SURAND | DURAND (seed, n, x)
C and C++ surand | durand (seed, n, x);
PL/I CALL SURAND | DURAND (seed, n, x);

On Entry

seed
is the initial value used to generate the random numbers. Specified as: a number of the data type indicated in Table 163. It should be a whole number; that is, the fraction part should be 0. (If you specify a mixed number, it is truncated.) Its value must be 1.0 <= seed < (2147483647.0 = 231-1).

Note:seed is always a long-precision real number, even in SURAND.

n
is the number of random numbers to be generated. Specified as: a fullword integer; n >= 0.

x
See "On Return".

On Return

seed
is the new seed that is to be used to generate additional random numbers in subsequent invocations of SURAND or DURAND. Returned as: a number of the data type indicated in Table 163. It is a whole number whose value is 1.0 <= seed < (2147483647.0 = 231-1).

x
is a vector of length n, containing the uniform pseudo-random numbers with values between 0 and 1. Returned as: a one-dimensional array of (at least) length n, containing numbers of the data type indicated in Table 163.

Note

In your C program, argument seed must be passed by reference.

Function

The uniform (0,1) pseudo-random numbers are generated as follows, using the multiplicative congruential method:

si = (a(si-1)) mod(m) = (ais0) mod(m)
xi = si/m    for i = 1, 2, ..., n

where:

si is a random sequence.
xi is a random number.
s0 is the initial seed provided by the caller.
a = 75 = 16807.0
m = 231-1 = 2147483647.0
n is the number of random numbers to be generated.

See references [70] and [74]. If n is 0, no computation is performed, and the initial seed is unchanged.

Error Conditions

Computational Errors

None

Input-Argument Errors
  1. n < 0
  2. seed < 1.0 or seed >= 2147483647.0

Example 1

This example shows a call to SURAND to generate 10 random numbers.

Call Statement and Input
             SEED    N   X
               |     |   |
CALL SURAND( SEED , 10 , X )
 
SEED     =  80629.0
Note:It is important to note that SEED is a long-precision number, even though X contains short-precision numbers.

Output
SEED     =  759150100.0
 
X        =  (0.6310323,
             0.7603202,
             0.7015232,
             0.5014868,
             0.4895853,
             0.4602344,
             0.1603608,
             0.1832564,
             0.9899062,
             0.3535068)

Example 2

This example shows a call to DURAND to generate 10 random numbers.

Call Statement and Input
             SEED    N   X
               |     |   |
CALL DURAND( SEED , 10 , X )
 
SEED     =  80629.0

Output
SEED     =  759150100.0
 
X        =  (0.6310323270182275,
             0.7603201953509451,
             0.7015232633340746,
             0.5014868557925740,
             0.4895853057920864,
             0.4602344475967038,
             0.1603607578018497,
             0.1832563756887132,
             0.9899062002030695,
             0.3535068129904134)

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Guide and Reference

Use Considerations

This section describes what you use the utility subroutines for.

Determining the Level of ESSL Installed

IESSL gets the level of ESSL and returns it to your program. The level consists of the following: version number, release number, modification number, and number of the most recently installed ESSL PTF. You can use this function to verify that you are running on or using the capabilities of the desired level.

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Finding the Optimal Stride(s) for Your Fourier Transforms

STRIDE is used to determine optimal stride values for your Fourier transforms when using any of the Fourier transform subroutines, except _RCFT and _CRFT. You must invoke STRIDE for each optimal stride you want computed. Sometimes you need a separate stride for your input and output data. For the three-dimensional Fourier transforms, you need an optimal stride for both the second and third dimensions of the array. The examples provided for STRIDE explain how it is used for each of the subroutines listed above.

After obtaining the optimal strides from STRIDE, you should arrange your data using these stride values. After the data is set up, call the Fourier transform subroutine. For additional information on how to set up your data, see "Setting Up Your Data".

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Converting Sparse Matrix Storage

DSRSM is used to migrate your existing program from sparse matrices stored by rows to sparse matrices stored in compressed-matrix storage mode. This converts the matrices into a storage format that is compatible with the input requirements for some ESSL sparse matrix subroutines, such as DSMMX.

DGKTRN and DSKTRN are used to convert your sparse matrix from one skyline storage mode to another, if necessary, before calling the subroutines DGKFS/DGKFSP or DSKFS/DSKFSP, respectively.

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Guide and Reference

Utility Subroutines

This section contains the utility subroutine descriptions.

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Guide and Reference

ERRSAV--ESSL ERRSAV Subroutine for ESSL

The ERRSAV subroutine copies an ESSL error option table entry into an 8-byte storage area that is accessible to your program.

For multithreaded application programs, if you want the error handling capabilities that this subroutine provides to be implemented on each thread created by your program, this subroutine must be called from each thread. If your application creates multiple threads, the action performed by a call to this subroutine applies to the thread that this subroutine was invoked from. For an example, see "Example of Handling Errors in a Multithreaded Application Program".

Syntax

Fortran CALL ERRSAV (ierno, tabent)
C and C++ errsav (ierno, tabent);
PL/I CALL ERRSAV (ierno, tabent);

On Entry

ierno
is the error number in the option table. The entry for ierno in the ESSL error option table is stored in the 8-byte storage area tabent. Specified as: a fullword integer; ierno must be one of the error numbers in the option table. For a list of these numbers, see Table 26.

tabent
is the storage area where the option table entry is stored. Specified as: an area of storage of length 8-bytes.

Note

Examples of how to use ERRSAV are provided in "Coding Your Program".

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Guide and Reference

ERRSET--ESSL ERRSET Subroutine for ESSL

The ERRSET subroutine allows you to control execution when error conditions occur. It modifies the information in the ESSL error option table for the error number indicated. For a range of error messages, you can specify the following:

  • How many times a particular error is allowed to occur before the program is terminated

  • How many times a particular error message is printed before printing is suppressed

  • Whether the ESSL error exit routine is to be invoked

For multithreaded application programs, if you want the error handling capabilities that this subroutine provides to be implemented on each thread created by your program, this subroutine must be called from each thread. If your application creates multiple threads, the action performed by a call to this subroutine applies to the thread that this subroutine was invoked from. For an example, see "Example of Handling Errors in a Multithreaded Application Program".

Syntax

Fortran CALL ERRSET (ierno, inoal, inomes, itrace, iusadr, irange)
C and C++ errset (ierno, inoal, inomes, itrace, iusadr, irange);
PL/I CALL ERRSET (ierno, inoal, inomes, itrace, iusadr, irange);

On Entry

ierno
is the error number in the option table. The entry for ierno in the ESSL error option table is updated as indicated by the other arguments. Specified as: a fullword integer; ierno must be one of the error numbers in the option table. For a list of these numbers, see Table 26.

inoal
indicates the number of errors allowed before each execution is terminated, where:

If inoal <= 0, the specification is ignored, and the number-of-errors option is not changed.

If inoal = 1, execution is terminated after one error.

If 2 <= inoal <= 255, then inoal specifies the number of errors allowed before each execution is terminated.

If inoal > 255, an unlimited number of errors is allowed.

Specified as: a fullword integer, where:

If iusadr = ENOTRM, then 2 <= inoal <= 255.

inomes
indicates the number of messages to be printed, where:

If inomes < 0, all messages are suppressed.

If inomes = 0, the number-of-messages option is not changed.

If 0 < inomes <= 255, then inomes specifies the number of messages to be printed.

If inomes > 255, an unlimited number of error messages is allowed.

Specified as: a fullword integer.

itrace
this argument is ignored, but must be specified. Specified as: a fullword integer where, itrace = 0, 1, or 2 (for migration purposes).

iusadr
indicates whether or not the ESSL error exit routine is to be invoked, where:

If iusadr is zero, the option table is not altered.

If iusadr is one, the option table is set to show no exit routine. Therefore, standard corrective action is to be used when continuing execution.

If iusadr = ENOTRM, the option table entry is set to the ESSL error exit routine ENOTRM. Therefore, the ENOTRM subroutine is to be invoked after the occurrence of the indicated errors. (ENOTRM must appear in an EXTERNAL statement in your program.)

Specified as: a 32-bit integer in a 32-bit environment or the name of a subroutine; iusadr = 0, 1, or ENOTRM.

Specified as: a 64-bit integer in a 64-bit environment or the name of a subroutine; iusadr = 0_8, 1_8, or ENOTRM.

irange
indicates the range of errors to be updated in the ESSL error option table, where:

If irange < ierno, the parameter is ignored.

If irange >= ierno, the options specified for the other parameters are to be applied to the entire range of error conditions encompassed by ierno and irange.

Specified as: a fullword integer.

Notes

  1. Examples of how to use ERRSET are provided in "Coding Your Program".

  2. If you specify ENOTRM for iusadr, then inoal must be in the following range: 2 <= inoal <= 255.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] GN=" k2~SZj H6 H6 H6 ./usr/lpp/essl.html.en_US/html/essl393.htmlen_U Guide and Reference

Guide and Reference

ERRSTR--ESSL ERRSTR Subroutine for ESSL

The ERRSTR subroutine stores an entry in the ESSL error option table.

For multithreaded application programs, if you want the error handling capabilities that this subroutine provides to be implemented on each thread created by your program, this subroutine must be called from each thread. If your application creates multiple threads, the action performed by a call to this subroutine applies to the thread that this subroutine was invoked from. For an example, see "Example of Handling Errors in a Multithreaded Application Program".

Syntax

Fortran CALL ERRSTR (ierno, tabent)
C and C++ errstr (ierno, tabent);
PL/I CALL ERRSTR (ierno, tabent);

On Entry

ierno
is the error number in the option table. The information in the 8-byte storage area tabent is stored into the entry for ierno in the ESSL error option table. Specified as: a fullword integer; ierno must be one of the error numbers in the option table. For a list of these numbers, see Table 26.

tabent
is the storage area containing the table entry data. Specified as: an area of storage of length 8-bytes.

Note

Examples of how to use ERRSTR are provided in "Coding Your Program".

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Guide and Reference

Appendixes

Appendix A. Basic Linear Algebra Subprograms (BLAS)

This appendix lists the ESSL subprograms corresponding to a subprogram in the standard set of BLAS.

Level 1 BLAS


Table 168. Level 1 BLAS Included in ESSL
Descriptive Name Short- Precision Subprogram Long- Precision Subprogram
Position of the First or Last Occurrence of the Vector Element Having the Largest Magnitude 
ISAMAX
ICAMAX


IDAMAX
IZAMAX

Sum of the Magnitudes of the Elements in a Vector 
SASUM
SCASUM


DASUM
DZASUM

Multiply a Vector X by a Scalar, Add to a Vector Y, and Store in the Vector Y 
SAXPY
CAXPY


DAXPY
ZAXPY

Copy a Vector 
SCOPY
CCOPY


DCOPY
ZCOPY

Dot Product of Two Vectors 
SDOT
CDOTU
CDOTC


DDOT
ZDOTU
ZDOTC

Euclidean Length of a Vector with Scaling of Input to Avoid Destructive Underflow and Overflow 
SNRM2
SCNRM2


DNRM2
DZNRM2

Construct a Givens Plane Rotation 
SROTG
CROTG


DROTG
ZROTG

Apply a Plane Rotation 
SROT
CROT
CSROT


DROT
ZROT
ZDROT

Multiply a Vector X by a Scalar and Store in the Vector X 
SSCAL
CSCAL
CSSCAL


DSCAL
ZSCAL
ZDSCAL

Interchange the Elements of Two Vectors 
SSWAP
CSWAP


DSWAP
ZSWAP

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  k0~SZjE5 K65 K65 K6E./usr/lpp/essl.html.en_US/html/essl40.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui lꀠ~SZj#H6#H6#H6 ./usr/lpp/essl.html.en_US/html/essl400.htmlen_U    <>T "/=ADELPReBFGHIMNOSVainort-2Ccdlmpsux01349Zfg#&,.567:;UY_bhvy!8Xjk()QqwKHugʃJ^a}01};Me;>0o,7-q u__}' H**Ac-q4d!B!B!B!B!B!Dt'&Tą#X%vF@xCA X CkC3$O7@1"q Ap8 a"y|8oDNN$@jkZ(|8`Dc8`XX ( w7O 0i\$+Z(|9%`Dc8b`X`)>7f'\X`PqApPPPVasu2)Dm ,-81+ ,&TA|ҭ Hpۨ6e1v̄!B!B!B!B!B! '"|\=}T_nTmE!v. 颻cD`apuȑ(dx', wR+XU"=4W7cDH |yB':@( P(9ݯ7J7kWlq8;ʥH`>R+XU"Ćn*M$k*IHS7~r48 Lv'bqH{J\ ʡ+ nr48 Lv'bqH{J+ȜW[$ ,(?^W7_bu&_18=ݥuXDNUTaq+{_sw呡!Re8Q&%;ev2.4pAv1L2]a2¤s$sQb$#p6.]`="Ar7n2؜hqAލ]|ԌY=6itv2؜hqAލ]|3hy=6itvoۛ,1:/݉ƇeĢ@j0˭v n2؜hqAލ]|Ԍ,fNۯҭo#CBbq!z5w7kidd'`kC̝_[O[.~dhqAԙ~N48 FeIN] $1]S-ubA1 k :۰=VUQl=FXTu.ߖFICBjn椕OMͩ$Ǧfۯҭo#CBbq!z5w7kidOM$ǦfzmV˵n߷7~Ybu&_1;ѫ,Ek4.!eH&<}aQ-q\vʪ- n2؜hqAލ]|Ԓ,٩$E;7_[O[.ߖFICBjn*١͒@4<ٺ|*ݯv#CBbq!z5wYEq>Wi3|\C]AؐLx£1v0Z<6UT[Q&˷呡!Re8%[ hkaMI u4Yitv2؜hqAލ]|%[ hy u42_(,Ph368m\r$G, *"9Uo`(46.#BGw!rlT=P XflpL:H ݯ7^R2v>OTG k{~SZj $H6$H6$H6 ./usr/lpp/essl.html.en_US/html/essl401.htmlen_U Guide and Reference

Guide and Reference

Level 3 BLAS


Table 170. Level 3 BLAS Included in ESSL
Descriptive Name Short- Precision Subprogram Long- Precision Subprogram
Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes 
SGEMM
CGEMM


DGEMM
ZGEMM

Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian 
SSYMM
CSYMM
CHEMM


DSYMM
ZSYMM
ZHEMM

Triangular Matrix-Matrix Product 
STRMM
CTRMM


DTRMM
ZTRMM

Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix 
SSYRK
CSYRK
CHERK


DSYRK
ZSYRK
ZHERK

Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix 
SSYR2K
CSYR2K
CHER2K


DSYR2K
ZSYR2K
ZHER2K

Solution of Triangular Systems of Equations with Multiple Right-Hand Sides 
STRSM
CTRSM


DTRSM
ZTRSM

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  k{~SZj*$H6$H6$H6*./usr/lpp/essl.html.en_US/html/essl402.htmlen_U Guide and Reference

Guide and Reference

Appendix B. LAPACK

This appendix lists the ESSL subroutines corresponding to subroutines in the standard set of LAPACK.

LAPACK


Table 171. LAPACK Included in ESSL
Descriptive Name Short- Precision Subprogram Long- Precision Subprogram
General Matrix Factorization 
SGETRF
CGETRF


DGETRF
ZGETRF

General Matrix, Its Transpose, or Its Conjugate Transpose Multiple Right-Hand Side Solve 
SGETRS
CGETRS


DGETRS
ZGETRS

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Guide and Reference

ESSL Publications

This section lists the ESSL publications for each major task that you perform.

You can order the full set of ESSL for AIX publications through the Subscription Library Services System (SLSS) by specifying:

  • The subject code 82

  • The program number: 5765-C42

You can also order individual publications by specifying the individual order numbers. For example, to order a copy of the ESSL Version 3 Release 1.1 Guide and Reference, specify the order number SA22-7272. Contact your IBM Marketing Representative or Systems Engineer to order manuals through SLSS.

Evaluation and Planning

ESSL Products General Information, GC23-0529--provides detailed information helpful in evaluating and planning for ESSL: Parallel ESSL, ESSL for AIX, and ESSL/370.

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  kF~SZj:)H6)H6)H6:./usr/lpp/essl.html.en_US/html/essl406.htmlen_U Guide and Reference

Guide and Reference

Installation

ESSL Version 3 Release 1.1 Installation Memo, GI10-0604-01--describes how to install ESSL on AIX. It is a packing list for the ESSL product when it is shipped. (One copy is delivered with each ESSL product.)

[ Top of Page | Previous Page | Next Page | Table of Contents | Index ] >
  • T k~SZj}^H6^H6^H6}./usr/lpp/essl.html.en_US/html/essl407.htmlen_U Guide and Reference

    Guide and Reference

    Application Programming

    ESSL Version 3 Release 1.1 Guide and Reference, SA22-7272--contains ESSL guidance information for designing, coding, and running programs using ESSL, and contains complete reference information for coding calls to the ESSL subroutines. This manual is available in HTML and PostScript format on the product medium.

    ESSL Version 2 Release 2 Guide and Reference, SC23-0526--contains ESSL guidance information for designing, coding, and running programs using ESSL/370, and contains complete reference information for coding calls to the ESSL/370 subroutines.

    Parallel ESSL Version 2 Guide and Reference, SA22-7273--contains Parallel ESSL guidance information for designing, coding, and running programs using Parallel ESSL, and contains complete reference information for coding calls to the Parallel ESSL subroutines. This manual is available in HTML and PostScript format on the Parallel ESSL product medium.

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ] tml k2~SZj^H6^H6^H6./usr/lpp/essl.html.en_US/html/essl408.htmlen_U Guide and Reference

    Guide and Reference

    Related Publications

    The related publications listed below may be useful to you when using ESSL.

    AIX for the RS/6000

    IBM AIX Calls and Subroutines Reference for IBM RS/6000,(all volumes) SC23-2198

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ES k~ ~SZj,H6,H6,H6./usr/lpp/essl.html.en_US/html/essl409.htmlen_U Guide and Reference

    Guide and Reference

    AIX Version 4 Release 2 for the RS/6000

    IBM AIX Version 4.1 and 4.2 Commands and Reference, (all volumes) SBOF-1851

    IBM Version 4.1 and 4.2 General Programming Concepts: Writing and Debugging Programs, SC23-2533

    IBM AIX Version 4.1 and 4.2 System Management Guide: Operating System and Devices, SC23-2525

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ME="Bot kZ~SZjE5 K65 K65 K6E./usr/lpp/essl.html.en_US/html/essl41.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference Gui k꒔"~SZj-H6-H6-H6./usr/lpp/essl.html.en_US/html/essl410.htmlen_U Guide and Reference

    Guide and Reference

    AIX Version 4 Release 3 for the RS/6000

    For the latest updates, visit the RS/6000 web site at: http://www.rs6000.ibm.com/resource/aix_resource/Pubs

    AIX Version 4.3 Commands and Reference, (all volumes) SBOF-1877

    AIX Version 4.3 General Programming Concepts:Writing and Debugging Programs, SC23-4128

    AIX Version 4.3 System Guide: Operating System and Devices, SC23-4126

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ] Versi k\$~SZjJ-H6-H6-H6J./usr/lpp/essl.html.en_US/html/essl411.htmlen_U Guide and Reference

    Guide and Reference

    XL Fortran

    On the XL Fortran Version 5 Release 1.1 CD ROM and the XL High Performance Fortran Version 1 Release 3.1 tape, you may reference the following document available in both PostScript and HTML formats, which describes the changes for those releases: Changes to XL Fortran for AIX and XL High Performance Fortran for AIX

    IBM XL Fortran for AIX User's Guide Version 5.1, SC09-2606

    IBM XL Fortran for AIX Language Reference Version 5.1, SC09-2607

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ] ation  k&~SZj-H6-H6-H6./usr/lpp/essl.html.en_US/html/essl412.htmlen_U Guide and Reference

    Guide and Reference

    PL/I

    IBM PL/I Set for AIX Programming Guide, SC26-8456

    IBM PL/I Set for AIX Language Reference, SC26-8455

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ]  kz(~SZj.H6.H6.H6./usr/lpp/essl.html.en_US/html/essl413.htmlen_U Guide and Reference

    Guide and Reference

    Workstation Processors

    RS/6000 POWERstation and POWERserver Hardware Technical Reference Information--General Architectures, SA23-2643

    [ Top of Page | Previous Page | Next Page | Table of Contents | Index ] AIX  kr*~SZj.H6.H6.H6./usr/lpp/essl.html.en_US/html/essl414.htmlen_U Guide and Reference

    Guide and Reference

    IBM 3838 Array Processor

    IBM 3838 Array Processor Functional Characteristics GA24-3639

    OS/VS1 and OS/VS2 MVS Vector Processing Subsystem Programmer's Guide, GC24-5125

    MVS Extended Architecture Vector Processing Subsystem Application Programmer's Guide, SC28-1202

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幗ETjijEq:@z*\-̽r*Hj2P8J)+ulz*\O-̽r*SHD!+ǘF6=M]VNV.y-̽r*SHHy3HH|y@lz*\D^Q BrP5 SW!uʫ'+cs/\R2|xR!v|AW.h4d0c;H_ǘq]`m f`ʫ'+c8Ks/\搙y3BJ)+pPcUdlbyneU@ΗWO1_+E5vUY9[[zUF=!2P=!%q8B=M]VNV/3R-̽r*H @y( @|y!U=M]VNV.xt s/\%9y( )+pE 5vUY9[[zUF"%4DSW /cUdlbineUBh/% q\e><|6=M]VNV.|-̽r*HLP)+pE=M]VNV.y s/\t<|y!VA=M]VNV.yKs/\ l@Ҹ|y!VB=M]VNV.yKs/\LL|y!Q{ʫ'+c<:%ETiM6eq8Blz*\s/\9y("t)+pEʫ'+c>bKs/\8y(L)+pBWcUdlb2Ȫ їD2bT(PcUdlb 2Ȫ A+ǘ emSWeUAneUA/%O\e><|S"=M]VNV.xq%ETig9y(#9SWKj쪲r1s<2Ȫ h@+ǘdj쪲r1sΙneU@3fWO1_v=M]VNV.x幗ETi/%WO1_V5vUY9[D^Q\e><|j쪲r1sĉ-̽r*H=y(=|ypPcUdlb2"4>^J|yuE 5vUY9[[zUFPq:lz*\-̽r*Hy(Ҹ|yuY5vUY9[[zUF8Hy($A\e><|=SWeUETi /% WO1_'z*\-̽r*Hhy(h|yuY<z*\-̽r*H |䠁+ǘ5vUY9[42Ȫ %@J3+ǘlz*\-̽r*Hyy(r|yp+;ʫ'+c0SBh)T  VZ_SEo h k~SZjE5 K65 K65 K6E./usr/lpp/essl.html.en_US/html/essl42.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kȰ~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl43.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl44.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl45.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kX~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl46.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl47.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl48.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k ~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl49.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kh~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl50.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl51.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k(~SZjE6 K66 K66 K6E./usr/lpp/essl.html.en_US/html/essl52.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl53.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kx~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl54.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kH~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl55.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl56.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kx~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl57.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kp~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl58.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k"~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl59.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl60.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl61.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k"~SZjE7 K67 K67 K6E./usr/lpp/essl.html.en_US/html/essl62.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k8~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl63.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl64.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kꀘ~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl65.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kꠠ~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl66.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k(~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl67.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k ~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl68.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kҒ ~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl69.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kP~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl70.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl71.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k갠~SZjE8 K68 K68 K6E./usr/lpp/essl.html.en_US/html/essl72.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k&~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl73.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl74.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl75.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k6~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl76.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kꎽ~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl77.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl78.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k ~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl79.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k꾅"~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl80.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kꨇ$~SZjE9 K69 K69 K6E./usr/lpp/essl.html.en_US/html/essl81.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k감&~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl82.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k((~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl83.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kP*~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl84.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kȏ,~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl85.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kh.~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl86.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k0~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl87.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kh2~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl88.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  k4~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl89.htmlen_UObsolete. See ESSL V3 R1.1 Guide and Reference  kR6~SZjE: K6: K6: K6E./usr/lpp/essl.html.en_US/html/essl90.htmlen_UObsolete. 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    Version 3 - Release 1.1

    5765-C42


    • ESSL Version 3 - Release 1.1 README
      Last Updated: October 1998
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    • ESSL Version 3 - Release 1.1 Installation Memo
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    • ESSL Version 3 Release 1.1 Guide and Reference
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    • ESSL Version 3 Release 1.1 Guide and Reference PostScript
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