%!PS-Adobe-1.0 %%Creator: anl-mcs:dongarra (Jack &,C232,7246,3129635225) %%Title: stdin Troff %%CreationDate: Mon Nov 16 13:04:27 1987 %%DocumentFonts: Times-Roman Times-Italic Times-Bold Symbol Times-Roman %%Pages: (atend) %%EndComments % lib/pscat.pro -- prolog for pscat (troff) files % Copyright (C) 1985 Adobe Systems, Inc. save /pscatsave exch def /$pscat 50 dict def $pscat begin /fm [1 0 0 1 0 0] def /xo 0 def /yo 0 def /M /moveto load def /R /show load def /S {exch currentpoint exch pop moveto show}def /T {exch currentpoint pop exch moveto show}def /U {3 1 roll moveto show}def /siz 0 def /font 0 def /Z {/siz exch def SF}def /F {/font exch def SF}def /SF{font 0 ne {catfonts font 1 sub get fm 0 siz put fm 3 siz neg put fm makefont setfont}if}def /BP{save/catsv exch def 0 792 translate 72 432 div dup neg scale xo yo translate 0 0 moveto}def /EP{catsv restore showpage}def % definitions for PPROC callback functions % each PPROC is called with the following number on the stack: % pointsize charcode railmag pswidth pschar x y wid /$pprocs 50 dict def /fractm [.65 0 0 .6 0 0] def % fractions /PS1{gsave $pprocs begin /wid exch def pop pop pop pop pop /ch exch def /size exch def /pair $pprocs ch get def /cf currentfont def cf fractm makefont setfont 0 .3 size mul 6 mul 2 copy neg rmoveto pair 0 get show rmoveto currentfont cf setfont (\244) show setfont pair 1 get show grestore wid .06 div 0 rmoveto end}def $pprocs begin 8#34 [(1)(4)] def 8#36 [(1)(2)] def 8#46 [(3)(4)] def end % boxes /PS2{gsave /wid exch def pop pop /char exch def pop pop pop /size exch def /len size 3.5 mul def % length of a side len 0 rlineto 0 len neg rlineto len neg 0 rlineto closepath char 3 eq {fill}{size 5 mul .07 mul setlinewidth stroke}ifelse grestore wid .06 div 0 rmoveto}def /PS3/PS2 load def % boxes are the same... % circle /PS4{gsave /wid exch def pop pop pop pop pop pop /size exch def wid .8333 mul size 2.5 mul neg rmoveto currentpoint % center newpath size 1.8 mul 0 360 arc size .2 mul setlinewidth stroke grestore wid .06 div 0 rmoveto}def /bb{$pprocs begin /wid exch def pop pop pop pop pop pop /size exch 6 mul def /s2 size 2 div def /s4 size 4 div def gsave currentpoint newpath transform round exch round exch itransform translate size 16 div setlinewidth 2 setlinejoin 0 setgray}def $pprocs begin /mrr{moveto rlineto rlineto}def /be{stroke grestore wid .06 div 0 rmoveto end}def end % leftfloor /PS6 {bb s4 0 0 size s4 size -.8 mul mrr be}def % rightfloor /PS7 {bb s4 neg 0 0 size s4 size -.8 mul mrr be}def % leftceil /PS8 {bb s4 0 0 size neg s4 size .2 mul mrr be}def % rightceil /PS9 {bb s4 neg 0 0 size neg s4 size .2 mul mrr be}def % boldvert /PS5 {bb 0 0 0 size neg s4 size .2 mul mrr be}def % box rule /PS32 {bb /sw size 24 div def sw 2 div size 4.5 div moveto 0 size neg rlineto sw setlinewidth be}def % rule (roman, bold and italic) /PS16 {gsave $pprocs begin /wid exch def pop pop pop pop pop pop /size exch 6 mul def /sw size 14 div def currentpoint exch sw 2 div sub exch newpath transform round exch round exch itransform translate 0 0 moveto size 2 div 0 rlineto sw setlinewidth be}def % lefttopcurl /PS20 {bb s4 size .2 mul moveto 0 size -.55 mul rlineto currentpoint pop size -.8 mul 2 copy exch s4 add exch s4 arcto pop pop pop pop be}def % leftbotcurl /PS21 {bb s4 size -.8 mul moveto 0 size .55 mul rlineto currentpoint pop size .2 mul 2 copy exch s4 add exch s4 arcto pop pop pop pop be}def % righttopcurl /PS22 {bb s4 size .2 mul moveto 0 size -.55 mul rlineto currentpoint pop size -.8 mul 2 copy exch s4 sub exch s4 arcto pop pop pop pop be}def % rightbotcurl /PS23 {bb s4 size -.8 mul moveto 0 size .55 mul rlineto currentpoint pop size .2 mul 2 copy exch s4 sub exch s4 arcto pop pop pop pop be}def % rightmidcurl /PS25 {bb /s3 size -.3 mul def s4 size -.8 mul moveto s4 s3 s2 s3 s4 arcto pop pop size add s4 s3 4 2 roll s4 arcto pop pop pop pop s4 size .2 mul lineto be}def % leftmidcurl /PS24 {bb /s3 size -.3 mul def s4 size -.8 mul moveto s4 s3 0 s3 s4 arcto pop pop size add s4 s3 4 2 roll s4 arcto pop pop pop pop s4 size .2 mul lineto be}def /catfonts [ /Times-Roman findfont /Times-Italic findfont /Times-Bold findfont /Symbol findfont /Times-Roman findfont ] def %%EndProlog %%Page: ? 30 BP 1 F 66 Z 1775 270(-)U 1819(28)S 1907(-)S 3 F 448 462(Appendix)U 746(C)S 1 F 598 594(This)U 737(appendix)S 1000(contains)S 1241(the)S 1343(calling)S 1543(sequences)S 1832(for)S 1931(all)S 2018(the)S 2120(Level)S 2291(3)S 2346(BLAS.)S 48 Z 3204 786(x)U 5 F 452 978(_)U 1 F 484 786(n)U 517(a)S 542(me)S 932(o)S 964(p)S 1001(t)S 1033(i)S 1060(o)S 1092(n)S 1126(s)S 1892(d)S 1929(i)S 1950(m)S 2118(s)S 2149(c)S 2181(a)S 2217(l)S 2245(a)S 2280(r)S 2334(ma)S 2409(t)S 2440(r)S 2473(i)S 2500(x)S 2590(ma)S 2665(t)S 2696(r)S 2729(i)S 2756(x)S 2822(s)S 2853(c)S 2885(a)S 2921(l)S 2949(a)S 2984(r)S 3038(ma)S 3113(t)S 3144(r)S 3177(i)S 479 978(GE)U 539(M)S 571(M)S 648(\()S 1057(TRA)S 1151(NS)S 1215(A)S 1258(,)S 1313(TRA)S 1407(NS)S 1472(B)S 1514(,)S 1819(M)S 1866(,)S 1919(N)S 1962(,)S 2015(K)S 2058(,)S 2111(AL)S 2178(P)S 2207(H)S 2239(A)S 2282(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2592(B)S 2634(,)S 2689(LDB)S 2794(,)S 2848(BE)S 2913(TA)S 2986(,)S 3040(C)S 3082(,)S 3137(LDC)S 3240(\))S 3240 1074(\))U 5 F 452 1170(_)U 452 1074(_)U 1 F 482(S)S 511(Y)S 539(M)S 571(M)S 648(\()S 674(S)S 712(I)S 735(DE)S 810(,)S 863(UP)S 929(LO)S 1002(,)S 1313(TRA)S 1407(NS)S 1472(B)S 1514(,)S 1819(M)S 1866(,)S 1919(N)S 1962(,)S 2111(AL)S 2178(P)S 2207(H)S 2239(A)S 2282(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2592(B)S 2634(,)S 2689(LDB)S 2794(,)S 2848(BE)S 2913(TA)S 2986(,)S 3040(C)S 3082(,)S 3137(LDC)S 479 1170(HE)U 539(M)S 571(M)S 648(\()S 674(S)S 712(I)S 735(DE)S 810(,)S 863(UP)S 929(LO)S 1002(,)S 1313(TRA)S 1407(NS)S 1472(B)S 1514(,)S 1819(M)S 1866(,)S 1919(N)S 1962(,)S 2111(AL)S 2178(P)S 2207(H)S 2239(A)S 2282(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2592(B)S 2634(,)S 2689(LDB)S 2794(,)S 2848(BE)S 2913(TA)S 2986(,)S 3040(C)S 3082(,)S 3137(LDC)S 3240(\))S 3240 1266(\))U 5 F 452 1362(_)U 452 1266(_)U 1 F 482(S)S 511(YR)S 575(K)S 648(\()S 863(UP)S 929(LO)S 1002(,)S 1057(TRA)S 1151(NS)S 1215(A)S 1258(,)S 1919(N)S 1962(,)S 2015(K)S 2058(,)S 2111(AL)S 2178(P)S 2207(H)S 2239(A)S 2282(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2848(BE)S 2913(TA)S 2986(,)S 3040(C)S 3082(,)S 3137(LDC)S 479 1362(HERK)U 648(\()S 863(UP)S 929(LO)S 1002(,)S 1057(TRA)S 1151(NS)S 1215(A)S 1258(,)S 1919(N)S 1962(,)S 2015(K)S 2058(,)S 2111(AL)S 2178(P)S 2207(H)S 2239(A)S 2282(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2848(BE)S 2913(TA)S 2986(,)S 3040(C)S 3082(,)S 3137(LDC)S 3240(\))S 5 F 452 1554(_)U 452 1458(_)U 1 F 481(TR)S 539(M)S 571(M)S 648(\()S 674(S)S 712(I)S 735(DE)S 810(,)S 863(UP)S 929(LO)S 1002(,)S 1057(TRA)S 1151(NS)S 1215(A)S 1258(,)S 1567(D)S 1608(I)S 1631(A)S 1663(G)S 1706(,)S 1819(M)S 1866(,)S 1919(N)S 1962(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2592(B)S 2634(,)S 2689(LDB)S 2792(\))S 481 1554(TR)U 546(SM)S 648(\()S 674(S)S 712(I)S 735(DE)S 810(,)S 863(UP)S 929(LO)S 1002(,)S 1057(TRA)S 1151(NS)S 1215(A)S 1258(,)S 1567(D)S 1608(I)S 1631(A)S 1663(G)S 1706(,)S 1819(M)S 1866(,)S 1919(N)S 1962(,)S 2335(A)S 2378(,)S 2433(LD)S 2495(A)S 2538(,)S 2592(B)S 2634(,)S 2689(LDB)S 2792(\))S 60 Z 3279 4704(-)U 16(--)S 3259(-)S EP %%Page: ? 29 BP 3 F 66 Z 448 462(Appendix)U 746(B)S 1 F 1775 270(-)U 1819(27)S 1907(-)S 598 690(A)U 676(model)S 870(implement)S 1150(ation,)S 1328(in)S 1409(Fortran)S 1633(77,)S 1746(of)S 1831(the)S 1940(Level)S 2118(3)S 2180(BLAS)S 2378(routines,)S 2636(together)S 2880(with)S 3026(test)S 3146(pro-)S 448 882(e)U 448 786(grams)U 635(and)S 756(timing)S 953(programs,)S 1245(is)S 1315(available)S 1577(via)S 1683(netlib.)S 1897(To)S 1996(obtain)S 2186(the)S 2292(Level)S 2467(3)S 2526(BLAS)S 2721(from)S 2874(netlib)S 3048(send)S 3194(an)S 477 882(lectronic)U 728(mail)S 866(message)S 1111(to)S 448 978(n)U (etlib)R 5 F (@)R 1 F (anl-m)R 811(cs.arpa)S 1018(or)S 1095(research!netli)S 1452(b.)S 1546(The)S 1670(message)S 1915(should)S 2113(be)S 2197(of)S 2274(the)S 2376(form:)S 448 1458(w)U 778 1170(send)U 921(sblas3)S 1108(from)S 1258(blas3)S 496 1458(hich)U 631(would)S 818(return)S 997(the)S 1099(single)S 1278(precision)S 1541(real)S 1661(version)S 1877(of)S 1954(the)S 2056(Level)S 2227(3)S 2282(BLAS.)S 2512(A)S 2582(message)S 2827(of)S 2904(the)S 3006(form:)S 448 1842(w)U 778 1650(send)U 921(index)S 1089(from)S 1239(blas3)S 496 1842(ould)U 642(return)S 828(an)S 919(index)S 1094(of)S 1178(information)S 1517(available)S 1753(.)S 1821(If)S 1894(you)S 2022(do)S 2117(not)S 2229(have)S 2381(access)S 2577(to)S 2656(electronic)S 2942(mail)S 3086(then)S 3227(a)S 3227 1938(e)U 448 2034(a)U 448 1938(copy)U 603(of)S 685(the)S 792(routines)S 1031(can)S 1149(be)S 1238(obtained)S 1491(by)S 1584(sending)S 1816(a)S 1872(request,)S 2106(along)S 2279(with)S 2423(a)S 2479(magnetic)S 2745(tape,)S 2897(to)S 2974(any)S 3095(of)S 3176(th)S 477 2034(uthors.)U 598 2166(N)U (ote)R 749(that)S 870(the)S 973(model)S 1159(implement)S 1439(ation)S 1592(is)S 1658(not)S 1764(claimed)S 1993(to)S 2066(be)S 2150(ef\256cient)S 2387(on)S 2475(any)S 2592(particular)S 2865(types)S 3026(of)S 3103(archi-)S 448 2358(B)U 448 2262(tecture)U 658(\(the)S 792(current)S 1010(version)S 1236(uses)S 1382(repeated)S 1636(calls)S 1788(to)S 1871(Level)S 2052(2)S 2117(BLAS\).)S 2356(Note)S 2515(also)S 2652(that)S 2781(after)S 2932(the)S 3043(Level)S 3223(3)S 492 2358(LAS)U 639(have)S 785(been)S 931(exposed)S 1169(to)S 1242(public)S 1428(review,)S 1646(their)S 1788(speci\256cation)S 2142(is)S 2208(liable)S 2375(to)S 2448(be)S 2532(changed.)S 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 28 BP 1 F 66 Z 1731 240(-)U 1813(2)S 1857(6)S 1951(-)S 60 Z 2098 432(\))U 453 564(*)U 453 498(*)U 691 432(S)U 727(U)S 768(BRO)S 887(UT)S 978(I)S 1007(NE)S 1127(DP)S 1209(LU)S 1298(\()S 1322(M)S 1380(,)S 1407(N)S 1460(,)S 1487(A)S 1540(,)S 1569(LD)S 1647(A)S 1700(,)S 1738(I)S 1771(P)S 1818(I)S 1847(V)S 1900(,)S 1938(I)S 1967(NF)S 2047(O)S 688 564(C)U 733(o)S 765(mp)S 853(u)S 899(t)S 934(e)S 976(s)S 1054(a)S 1093(n)S 1169(LU)S 1298(f)S 1334(a)S 1374(c)S 1419(t)S 1453(o)S 1498(r)S 1539(i)S 1574(z)S 1614(a)S 1659(t)S 1699(i)S 1733(o)S 1773(n)S 1853(o)S 1898(f)S 1974(a)S 2013(n)S 2085(m)S 2138(-)S 2173(b)S 2213(y)S 2258(-)S 2293(n)S 2365(m)S 2414(a)S 2459(t)S 2498(r)S 2539(i)S 2573(x)S 2647(A)S 2700(,)S 2773(u)S 2816(s)S 2859(i)S 2893(n)S 2933(g)S 3013(p)S 3054(a)S 3098(r)S 3139(t)S 3179(i)S 3214(a)S 3259(l)S 453 696(*)U 453 630(*)U 693(p)S 739(i)S 773(v)S 813(o)S 859(t)S 899(i)S 933(n)S 973(g)S 1047(w)S 1099(i)S 1139(t)S 1173(h)S 1258(r)S 1293(o)S 1327(w)S 1419(i)S 1453(n)S 1499(t)S 1534(e)S 1578(r)S 1614(c)S 1653(h)S 1694(a)S 1733(n)S 1773(g)S 1814(e)S 1856(s)S 1900(.)S 698 762(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1407(NF)S 1487(O)S 1540(,)S 1609(LD)S 1687(A)S 1740(,)S 1802(M)S 1860(,)S 1927(N)S 687 828(D)U 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1367(A)S 1418(\()S 1449(LD)S 1527(A)S 1580(,)S 1613(*)S 1658(\))S 698 894(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1411(P)S 1458(I)S 1487(V)S 1538(\()S 1567(N)S 1618(\))S 687 960(D)U 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1369(T)S 698 1026(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1420(,)S 1498(I)S 1531(P)S 1580(,)S 1656(J)S 1700(,)S 1776(J)S 1811(P)S 689 1158(E)U 698 1092(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1407(D)S 1447(A)S 1482(M)S 1527(A)S 1567(X)S 727 1158(XT)U 809(E)S 848(RN)S 927(AL)S 1378(I)S 1407(D)S 1447(A)S 1482(M)S 1527(A)S 1567(X)S 2367 1224(V)U 453 1290(*)U 689 1224(EXT)U 809(E)S 848(RN)S 927(AL)S 1367(D)S 1407(GE)S 1482(M)S 1527(V)S 1580(,)S 1647(DS)S 1728(CA)S 1809(L)S 1860(,)S 1927(DS)S 2000(W)S 2047(AP)S 2140(,)S 2207(DTR)S 2331(S)S 698 1356(I)U 727(NF)S 807(O)S 891(=)S 973(0)S 1207 1422(N)U 453 1554(*)U 453 1488(*)U 687 1422(D)U 727(O)S 813(4)S 853(0)S 936(J)S 1011(=)S 1093(1)S 1140(,)S 807 1554(A)U 853(p)S 893(p)S 939(l)S 973(y)S 1053(p)S 1098(r)S 1134(e)S 1173(v)S 1219(i)S 1253(o)S 1293(u)S 1336(s)S 1419(i)S 1453(n)S 1499(t)S 1534(e)S 1578(r)S 1614(c)S 1653(h)S 1694(a)S 1733(n)S 1773(g)S 1814(e)S 1856(s)S 1939(t)S 1973(o)S 2059(j)S 2099(t)S 2133(h)S 2214(c)S 2253(o)S 2299(l)S 2333(u)S 2365(mn)S 2460(.)S 453 1620(*)U 807 1686(D)U 847(O)S 933(1)S 973(0)S 1058(I)S 1131(=)S 1213(1)S 1260(,)S 1336(J)S 1418(-)S 1493(1)S 929 1818(T)U 938 1752(I)U 971(P)S 1051(=)S 1138(I)S 1171(P)S 1218(I)S 1247(V)S 1298(\()S 1338(I)S 1378(\))S 1011 1818(=)U 1087(A)S 1138(\()S 1178(I)S 1220(,)S 1256(J)S 1298(\))S 1538 1884(\))U 927 1950(A)U 927 1884(A)U 978(\()S 1018(I)S 1060(,)S 1096(J)S 1138(\))S 1211(=)S 1287(A)S 1338(\()S 1378(I)S 1411(P)S 1460(,)S 1496(J)S 978 1950(\()U 1018(I)S 1051(P)S 1100(,)S 1136(J)S 1178(\))S 1251(=)S 1329(T)S 453 2148(*)U 453 2082(*)U 573 2016(1)U 613(0)S 808(CO)S 887(NT)S 978(I)S 1007(N)S 1047(UE)S 808 2148(C)U 853(o)S 885(mp)S 973(u)S 1019(t)S 1054(e)S 1134(e)S 1179(l)S 1214(e)S 1245(m)S 1294(e)S 1333(n)S 1379(t)S 1416(s)S 1493(1)S 1539(:)S 1579(j)S 1618(-)S 1653(1)S 1733(o)S 1778(f)S 1859(j)S 1898(-)S 1939(t)S 1973(h)S 2054(c)S 2093(o)S 2139(l)S 2173(u)S 2205(mn)S 2300(.)S 453 2214(*)U 808 2280(CA)U 889(L)S 929(L)S 1007(DTR)S 1131(S)S 1167(V)S 1218(\()S 1258(')S 1289(L)S 1333(o)S 1367(w)S 1414(e)S 1458(r)S 1498(')S 1540(,)S 1578(')S 1607(N)S 1653(o)S 1739(t)S 1778(r)S 1814(a)S 1853(n)S 1896(s)S 1933(p)S 1973(o)S 2016(s)S 2054(e)S 2098(')S 2140(,)S 2178(')S 2207(U)S 2253(n)S 2299(i)S 2339(t)S 2378(')S 2420(,)S 2456(J)S 2498(-)S 2533(1)S 2580(,)S 2607(A)S 2658(\()S 2693(1)S 2740(,)S 2773(1)S 2818(\))S 2860(,)S 2889(LD)S 2967(A)S 3020(,)S 3047(A)S 3098(\()S 3133(1)S 3180(,)S 3216(J)S 3258(\))S 3300(,)S 453 2478(*)U 453 2412(*)U 653 2346($)U 1253(1)S 1298(\))S 807 2478(U)U 853(p)S 893(d)S 934(a)S 979(t)S 1014(e)S 1094(e)S 1139(l)S 1174(e)S 1205(m)S 1254(e)S 1293(n)S 1339(t)S 1376(s)S 1459(j)S 1499(:)S 1533(n)S 1613(o)S 1658(f)S 1739(j)S 1778(-)S 1819(t)S 1853(h)S 1934(c)S 1973(o)S 2019(l)S 2053(u)S 2085(mn)S 2180(.)S 453 2544(*)U 808 2610(CA)U 889(L)S 929(L)S 1007(D)S 1047(GE)S 1122(M)S 1167(V)S 1218(\()S 1258(')S 1287(N)S 1333(o)S 1419(t)S 1458(r)S 1494(a)S 1533(n)S 1576(s)S 1613(p)S 1653(o)S 1696(s)S 1734(e)S 1778(')S 1820(,)S 1842(M)S 1898(-)S 1936(J)S 1971(+)S 2013(1)S 2060(,)S 2096(J)S 2138(-)S 2173(1)S 2220(,)S 2258(-)S 2293(1)S 2340(.)S 2373(0)S 2407(D)S 2453(0)S 2500(,)S 2527(A)S 2578(\()S 2616(J)S 2660(,)S 2693(1)S 2738(\))S 2780(,)S 2809(LD)S 2887(A)S 2940(,)S 2967(A)S 3018(\()S 3053(1)S 3100(,)S 3136(J)S 3178(\))S 3220(,)S 3253(1)S 3300(,)S 453 2808(*)U 453 2742(*)U 653 2676($)U 1253(1)S 1300(.)S 1333(0)S 1367(D)S 1413(0)S 1460(,)S 1487(A)S 1538(\()S 1576(J)S 1620(,)S 1656(J)S 1698(\))S 1740(,)S 1773(1)S 1818(\))S 811 2808(F)U 859(i)S 893(n)S 933(d)S 1013(p)S 1059(i)S 1093(v)S 1133(o)S 1179(t)S 1254(a)S 1293(n)S 1333(d)S 1419(t)S 1454(e)S 1496(s)S 1539(t)S 1618(f)S 1653(o)S 1698(r)S 1776(s)S 1819(i)S 1853(n)S 1893(g)S 1933(u)S 1979(l)S 2014(a)S 2058(r)S 2099(i)S 2139(t)S 2173(y)S 2220(.)S 453 2874(*)U 816 2940(J)U 851(P)S 931(=)S 1016(J)S 1098(-)S 1173(1)S 1251(+)S 1338(I)S 1367(D)S 1407(A)S 1442(M)S 1487(A)S 1527(X)S 1578(\()S 1602(M)S 1658(-)S 1696(J)S 1731(+)S 1773(1)S 1820(,)S 1847(A)S 1898(\()S 1936(J)S 1980(,)S 2016(J)S 2058(\))S 2100(,)S 2133(1)S 2178(\))S 818 3072(I)U 818 3006(I)U 851(P)S 898(I)S 927(V)S 978(\()S 1016(J)S 1058(\))S 1131(=)S 1216(J)S 1251(P)S 851 3072(F)U 938(\()S 967(A)S 1018(\()S 1056(J)S 1091(P)S 1140(,)S 1176(J)S 1218(\))S 1260(.)S 1289(EQ)S 1380(.)S 1413(0)S 1460(.)S 1493(0)S 1527(D)S 1573(0)S 1618(\))S 1687(G)S 1727(O)S 1809(TO)S 1933(5)S 1973(0)S 453 3204(*)U 453 3138(*)U 807 3204(A)U 853(p)S 893(p)S 939(l)S 973(y)S 1059(i)S 1093(n)S 1139(t)S 1174(e)S 1218(r)S 1254(c)S 1293(h)S 1334(a)S 1373(n)S 1413(g)S 1454(e)S 1539(t)S 1573(o)S 1654(c)S 1693(o)S 1739(l)S 1773(u)S 1805(mn)S 1896(s)S 1973(1)S 2019(:)S 2059(j)S 2100(.)S 453 3270(*)U 818 3336(I)U 851(F)S 938(\()S 976(J)S 1011(P)S 1060(.)S 1087(NE)S 1180(.)S 1216(J)S 1258(\))S 1328(CA)S 1409(L)S 1449(L)S 1527(DS)S 1600(W)S 1647(AP)S 1738(\()S 1776(J)S 1820(,)S 1847(A)S 1898(\()S 1936(J)S 1980(,)S 2013(1)S 2058(\))S 2100(,)S 2129(LD)S 2207(A)S 2260(,)S 2287(A)S 2338(\()S 2376(J)S 2411(P)S 2460(,)S 2493(1)S 2538(\))S 2580(,)S 2609(LD)S 2687(A)S 2738(\))S 453 3468(*)U 453 3402(*)U 808 3468(C)U 853(o)S 885(mp)S 973(u)S 1019(t)S 1054(e)S 1134(e)S 1179(l)S 1214(e)S 1245(m)S 1294(e)S 1333(n)S 1379(t)S 1416(s)S 1499(j)S 1531(+)S 1573(1)S 1619(:)S 1645(m)S 1733(o)S 1778(f)S 1859(j)S 1898(-)S 1939(t)S 1973(h)S 2054(c)S 2093(o)S 2139(l)S 2173(u)S 2205(mn)S 2300(.)S 453 3534(*)U 818 3600(I)U 851(F)S 938(\()S 976(J)S 1020(.)S 1049(L)S 1089(T)S 1140(.)S 1162(M)S 1218(\))S 1288(CA)S 1369(L)S 1409(L)S 1487(DS)S 1568(CA)S 1649(L)S 1698(\()S 1722(M)S 1778(-)S 1816(J)S 1860(,)S 1893(1)S 1940(.)S 1973(0)S 2007(D)S 2053(0)S 2099(/)S 2127(A)S 2178(\()S 2216(J)S 2260(,)S 2296(J)S 2338(\))S 2380(,)S 2407(A)S 2458(\()S 2496(J)S 2531(+)S 2573(1)S 2620(,)S 2656(J)S 2698(\))S 2740(,)S 2773(1)S 2818(\))S 573 3666(4)U 613(0)S 688(CO)S 767(NT)S 858(I)S 887(N)S 927(UE)S 688 3732(RE)U 769(TUR)S 887(N)S 453 3798(*)U 573 3864(5)U 613(0)S 698(I)S 727(NF)S 807(O)S 891(=)S 976(J)S 688 3930(RE)U 769(TUR)S 887(N)S 689 3996(EN)U 767(D)S 16 4710(-)U (-)R 3259(--)S EP %%Page: ? 27 BP 1 F 66 Z 1731 240(-)U 1813(2)S 1857(5)S 1951(-)S 60 Z 2258 432(\))U 453 564(*)U 453 498(*)U 691 432(S)U 727(U)S 768(BRO)S 887(UT)S 978(I)S 1007(NE)S 1127(DP)S 1209(LUB)S 1338(\()S 1362(M)S 1420(,)S 1447(N)S 1500(,)S 1527(A)S 1580(,)S 1609(LD)S 1687(A)S 1740(,)S 1778(I)S 1811(P)S 1858(I)S 1887(V)S 1940(,)S 1978(I)S 2007(NF)S 2087(O)S 2140(,)S 2167(N)S 2208(B)S 688 564(C)U 733(o)S 765(mp)S 853(u)S 899(t)S 934(e)S 976(s)S 1054(a)S 1093(n)S 1169(LU)S 1298(f)S 1334(a)S 1374(c)S 1419(t)S 1453(o)S 1498(r)S 1539(i)S 1574(z)S 1614(a)S 1659(t)S 1699(i)S 1733(o)S 1773(n)S 1853(o)S 1898(f)S 1974(a)S 2013(n)S 2085(m)S 2138(-)S 2173(b)S 2213(y)S 2258(-)S 2293(n)S 2365(m)S 2414(a)S 2459(t)S 2498(r)S 2539(i)S 2573(x)S 2647(A)S 2700(,)S 2773(u)S 2816(s)S 2859(i)S 2893(n)S 2933(g)S 3013(p)S 3054(a)S 3098(r)S 3139(t)S 3179(i)S 3214(a)S 3259(l)S 453 696(*)U 453 630(*)U 693(p)S 739(i)S 773(v)S 813(o)S 859(t)S 899(i)S 933(n)S 973(g)S 1047(w)S 1099(i)S 1139(t)S 1173(h)S 1258(r)S 1293(o)S 1327(w)S 1419(i)S 1453(n)S 1499(t)S 1534(e)S 1578(r)S 1614(c)S 1653(h)S 1694(a)S 1733(n)S 1773(g)S 1814(e)S 1856(s)S 1900(.)S 698 762(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1407(NF)S 1487(O)S 1540(,)S 1609(LD)S 1687(A)S 1740(,)S 1802(M)S 1860(,)S 1927(N)S 1980(,)S 2047(N)S 2088(B)S 687 828(D)U 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1367(A)S 1418(\()S 1449(LD)S 1527(A)S 1580(,)S 1613(*)S 1658(\))S 698 894(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1411(P)S 1458(I)S 1487(V)S 1538(\()S 1567(N)S 1618(\))S 1808 960(B)U 689 1026(E)U 698 960(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1420(,)S 1498(I)S 1531(P)S 1580(,)S 1656(J)S 1700(,)S 1776(J)S 727 1026(XT)U 809(E)S 848(RN)S 927(AL)S 1367(D)S 1407(GE)S 1482(M)S 1522(M)S 1580(,)S 1647(DP)S 1729(LU)S 1820(,)S 1887(DS)S 1960(W)S 2007(AP)S 2100(,)S 2167(DTR)S 2291(S)S 2322(M)S 453 1092(*)U 698 1158(I)U 727(NF)S 807(O)S 891(=)S 973(0)S 687 1224(D)U 727(O)S 813(4)S 853(0)S 936(J)S 1011(=)S 1093(1)S 1140(,)S 1207(N)S 1260(,)S 1327(N)S 1368(B)S 1498 1290(\))U 453 1422(*)U 453 1356(*)U 816 1290(J)U 848(B)S 931(=)S 1002(M)S 1058(I)S 1087(N)S 1138(\()S 1167(N)S 1218(-)S 1256(J)S 1291(+)S 1333(1)S 1380(,)S 1407(N)S 1448(B)S 807 1422(A)U 853(p)S 893(p)S 939(l)S 973(y)S 1053(p)S 1098(r)S 1134(e)S 1173(v)S 1219(i)S 1253(o)S 1293(u)S 1336(s)S 1419(i)S 1453(n)S 1499(t)S 1534(e)S 1578(r)S 1614(c)S 1653(h)S 1694(a)S 1733(n)S 1773(g)S 1814(e)S 1856(s)S 1939(t)S 1973(o)S 2054(c)S 2093(u)S 2138(r)S 2178(r)S 2214(e)S 2253(n)S 2299(t)S 2373(b)S 2419(l)S 2453(o)S 2494(c)S 2533(k)S 2580(.)S 453 1488(*)U 807 1554(D)U 847(O)S 933(1)S 973(0)S 1058(I)S 1131(=)S 1213(1)S 1260(,)S 1336(J)S 1418(-)S 1493(1)S 938 1686(I)U 938 1620(I)U 971(P)S 1051(=)S 1138(I)S 1171(P)S 1218(I)S 1247(V)S 1298(\()S 1338(I)S 1378(\))S 971 1686(F)U 1058(\()S 1098(I)S 1131(P)S 1180(.)S 1207(NE)S 1300(.)S 1338(I)S 1378(\))S 1448(CA)S 1529(L)S 1569(L)S 1647(DS)S 1720(W)S 1767(AP)S 1858(\()S 1896(J)S 1928(B)S 1980(,)S 2007(A)S 2058(\()S 2098(I)S 2140(,)S 2176(J)S 2218(\))S 2260(,)S 2289(LD)S 2367(A)S 2420(,)S 2447(A)S 2498(\()S 2538(I)S 2571(P)S 2620(,)S 2656(J)S 2698(\))S 2740(,)S 2769(LD)S 2847(A)S 2898(\))S 453 1884(*)U 453 1818(*)U 573 1752(1)U 613(0)S 808(CO)S 887(NT)S 978(I)S 1007(N)S 1047(UE)S 808 1884(C)U 853(o)S 885(mp)S 973(u)S 1019(t)S 1054(e)S 1136(s)S 1173(u)S 1213(p)S 1254(e)S 1298(r)S 1333(d)S 1379(i)S 1414(a)S 1453(g)S 1493(o)S 1533(n)S 1574(a)S 1619(l)S 1693(b)S 1739(l)S 1773(o)S 1814(c)S 1853(k)S 1933(o)S 1978(f)S 2047(U)S 2100(.)S 453 1950(*)U 808 2016(CA)U 889(L)S 929(L)S 1007(DTR)S 1131(S)S 1162(M)S 1218(\()S 1258(')S 1289(L)S 1334(e)S 1378(f)S 1419(t)S 1458(')S 1500(,)S 1538(')S 1569(L)S 1613(o)S 1647(w)S 1694(e)S 1738(r)S 1778(')S 1820(,)S 1858(')S 1887(N)S 1933(o)S 2019(t)S 2058(r)S 2094(a)S 2133(n)S 2176(s)S 2213(p)S 2253(o)S 2296(s)S 2334(e)S 2378(')S 2420(,)S 2458(')S 2487(U)S 2533(n)S 2579(i)S 2619(t)S 2658(')S 2700(,)S 2736(J)S 2778(-)S 2813(1)S 2860(,)S 2896(J)S 2928(B)S 2980(,)S 3007(A)S 3058(\()S 3093(1)S 3140(,)S 3173(1)S 3218(\))S 3260(,)S 453 2214(*)U 453 2148(*)U 653 2082($)U 1249(LD)S 1327(A)S 1380(,)S 1407(A)S 1458(\()S 1493(1)S 1540(,)S 1576(J)S 1618(\))S 1660(,)S 1689(LD)S 1767(A)S 1818(\))S 807 2214(U)U 853(p)S 893(d)S 934(a)S 979(t)S 1014(e)S 1093(d)S 1139(i)S 1174(a)S 1213(g)S 1253(o)S 1293(n)S 1334(a)S 1379(l)S 1454(a)S 1493(n)S 1533(d)S 1616(s)S 1653(u)S 1693(b)S 1733(d)S 1779(i)S 1814(a)S 1853(g)S 1893(o)S 1933(n)S 1974(a)S 2019(l)S 2093(b)S 2139(l)S 2173(o)S 2214(c)S 2253(k)S 2296(s)S 2340(.)S 453 2280(*)U 808 2346(CA)U 889(L)S 929(L)S 1007(D)S 1047(GE)S 1122(M)S 1162(M)S 1218(\()S 1258(')S 1287(N)S 1333(o)S 1419(t)S 1458(r)S 1494(a)S 1533(n)S 1576(s)S 1613(p)S 1653(o)S 1696(s)S 1734(e)S 1778(')S 1820(,)S 1858(')S 1887(N)S 1933(o)S 2019(t)S 2058(r)S 2094(a)S 2133(n)S 2176(s)S 2213(p)S 2253(o)S 2296(s)S 2334(e)S 2378(')S 2420(,)S 2442(M)S 2498(-)S 2536(J)S 2571(+)S 2613(1)S 2660(,)S 2696(J)S 2728(B)S 2780(,)S 2816(J)S 2858(-)S 2893(1)S 2940(,)S 2978(-)S 3013(1)S 3060(.)S 3093(0)S 3127(D)S 3173(0)S 3220(,)S 453 2544(*)U 453 2478(*)U 653 2412($)U 1247(A)S 1298(\()S 1336(J)S 1380(,)S 1413(1)S 1458(\))S 1500(,)S 1529(LD)S 1607(A)S 1660(,)S 1687(A)S 1738(\()S 1773(1)S 1820(,)S 1856(J)S 1898(\))S 1940(,)S 1969(LD)S 2047(A)S 2100(,)S 2133(1)S 2180(.)S 2213(0)S 2247(D)S 2293(0)S 2340(,)S 2367(A)S 2418(\()S 2456(J)S 2500(,)S 2536(J)S 2578(\))S 2620(,)S 2649(LD)S 2727(A)S 2778(\))S 811 2544(F)U 854(a)S 894(c)S 939(t)S 973(o)S 1018(r)S 1059(i)S 1094(z)S 1134(e)S 1213(d)S 1259(i)S 1294(a)S 1333(g)S 1373(o)S 1413(n)S 1454(a)S 1499(l)S 1574(a)S 1613(n)S 1653(d)S 1736(s)S 1773(u)S 1813(b)S 1853(d)S 1899(i)S 1934(a)S 1973(g)S 2013(o)S 2053(n)S 2094(a)S 2139(l)S 2213(b)S 2259(l)S 2293(o)S 2334(c)S 2373(k)S 2416(s)S 2494(a)S 2533(n)S 2573(d)S 2659(t)S 2694(e)S 2736(s)S 2779(t)S 2858(f)S 2893(o)S 2938(r)S 453 2676(*)U 453 2610(*)U 816(s)S 859(i)S 893(n)S 933(g)S 973(u)S 1019(l)S 1054(a)S 1098(r)S 1139(i)S 1179(t)S 1213(y)S 1260(.)S 808 2742(CA)U 889(L)S 929(L)S 1007(DP)S 1089(LU)S 1178(\()S 1202(M)S 1258(-)S 1296(J)S 1331(+)S 1373(1)S 1420(,)S 1456(J)S 1488(B)S 1540(,)S 1567(A)S 1618(\()S 1656(J)S 1700(,)S 1736(J)S 1778(\))S 1820(,)S 1849(LD)S 1927(A)S 1980(,)S 2018(I)S 2051(P)S 2098(I)S 2127(V)S 2178(\()S 2216(J)S 2258(\))S 2300(,)S 2338(I)S 2367(NF)S 2447(O)S 2498(\))S 807 2808(D)U 847(O)S 933(2)S 973(0)S 1058(I)S 1131(=)S 1216(J)S 1260(,)S 1336(J)S 1411(+)S 1496(J)S 1528(B)S 1618(-)S 1693(1)S 938 2874(I)U 971(P)S 1018(I)S 1047(V)S 1098(\()S 1138(I)S 1178(\))S 1251(=)S 1336(J)S 1418(-)S 1493(1)S 1571(+)S 1658(I)S 1691(P)S 1738(I)S 1767(V)S 1818(\()S 1858(I)S 1898(\))S 573 2940(2)U 613(0)S 808(CO)S 887(NT)S 978(I)S 1007(N)S 1047(UE)S 818 3006(I)U 851(F)S 938(\()S 978(I)S 1007(NF)S 1087(O)S 1140(.)S 1167(NE)S 1260(.)S 1293(0)S 1338(\))S 1407(G)S 1447(O)S 1529(TO)S 1653(5)S 1693(0)S 453 3138(*)U 453 3072(*)U 807 3138(A)U 853(p)S 893(p)S 939(l)S 973(y)S 1059(i)S 1093(n)S 1139(t)S 1174(e)S 1218(r)S 1254(c)S 1293(h)S 1334(a)S 1373(n)S 1413(g)S 1454(e)S 1496(s)S 1579(t)S 1613(o)S 1693(p)S 1738(r)S 1774(e)S 1813(v)S 1859(i)S 1893(o)S 1933(u)S 1976(s)S 2053(b)S 2099(l)S 2133(o)S 2174(c)S 2213(k)S 2256(s)S 2300(.)S 453 3204(*)U 807 3270(D)U 847(O)S 933(3)S 973(0)S 1058(I)S 1131(=)S 1216(J)S 1260(,)S 1336(J)S 1411(+)S 1496(J)S 1528(B)S 1618(-)S 1693(1)S 938 3402(I)U 938 3336(I)U 971(P)S 1051(=)S 1138(I)S 1171(P)S 1218(I)S 1247(V)S 1298(\()S 1338(I)S 1378(\))S 971 3402(F)U 1058(\()S 1098(I)S 1131(P)S 1180(.)S 1207(NE)S 1300(.)S 1338(I)S 1378(\))S 1448(CA)S 1529(L)S 1569(L)S 1647(DS)S 1720(W)S 1767(AP)S 1858(\()S 1896(J)S 1938(-)S 1973(1)S 2020(,)S 2047(A)S 2098(\()S 2138(I)S 2180(,)S 2213(1)S 2258(\))S 2300(,)S 2329(LD)S 2407(A)S 2460(,)S 2487(A)S 2538(\()S 2578(I)S 2611(P)S 2660(,)S 2693(1)S 2738(\))S 2780(,)S 2809(LD)S 2887(A)S 2938(\))S 573 3534(4)U 573 3468(3)U 613(0)S 808(CO)S 887(NT)S 978(I)S 1007(N)S 1047(UE)S 613 3534(0)U 688(CO)S 767(NT)S 858(I)S 887(N)S 927(UE)S 453 3666(*)U 688 3600(RE)U 769(TUR)S 887(N)S 573 3732(5)U 613(0)S 698(I)S 727(NF)S 807(O)S 891(=)S 978(I)S 1007(NF)S 1087(O)S 1171(+)S 1256(J)S 1338(-)S 1413(1)S 688 3798(RE)U 769(TUR)S 887(N)S 689 3864(EN)U 767(D)S 26 4710(-)U 66(-)S 3229(-)S 3269(-)S EP %%Page: ? 26 BP 1 F 66 Z 1731 240(-)U 1813(2)S 1857(4)S 1951(-)S 60 Z 1818 432(\))U 453 564(*)U 453 498(*)U 691 432(S)U 727(U)S 768(BRO)S 887(UT)S 978(I)S 1007(NE)S 1127(DL)S 1209(L)S 1249(T)S 1298(\()S 1327(N)S 1380(,)S 1407(A)S 1460(,)S 1489(LD)S 1567(A)S 1620(,)S 1658(I)S 1687(NF)S 1767(O)S 688 564(C)U 733(o)S 765(mp)S 853(u)S 899(t)S 934(e)S 976(s)S 1054(a)S 1093(n)S 1169(L)S 1213(*)S 1249(L)S 1293(*)S 1333(*)S 1369(T)S 1458(f)S 1494(a)S 1534(c)S 1579(t)S 1613(o)S 1658(r)S 1699(i)S 1734(z)S 1774(a)S 1819(t)S 1859(i)S 1893(o)S 1933(n)S 2013(o)S 2058(f)S 2134(a)S 2216(s)S 2253(y)S 2285(m)S 2325(m)S 2374(e)S 2419(t)S 2458(r)S 2499(i)S 2534(c)S 2613(p)S 2653(o)S 2696(s)S 2739(i)S 2779(t)S 2819(i)S 2853(v)S 2894(e)S 2938(-)S 2973(d)S 3014(e)S 3051(\256)S 3093(n)S 3139(i)S 3179(t)S 3214(e)S 453 696(*)U 453 630(*)U 685(m)S 734(a)S 779(t)S 818(r)S 859(i)S 893(x)S 967(A)S 1020(.)S 698 762(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1407(NF)S 1487(O)S 1540(,)S 1609(LD)S 1687(A)S 1740(,)S 1807(N)S 687 828(D)U 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1367(A)S 1418(\()S 1449(LD)S 1527(A)S 1580(,)S 1613(*)S 1658(\))S 698 894(I)U 727(NT)S 809(EGE)S 928(R)S 1376(J)S 1489 960(T)U 687(D)S 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1367(D)S 1407(D)S 1447(O)S 689 1026(EXT)U 809(E)S 848(RN)S 927(AL)S 1367(D)S 1407(D)S 1447(OT)S 1809 1092(L)U 453 1158(*)U 689 1092(EXT)U 809(E)S 848(RN)S 927(AL)S 1367(D)S 1407(GE)S 1482(M)S 1527(V)S 1580(,)S 1647(DS)S 1728(CA)S 698 1224(I)U 727(NF)S 807(O)S 891(=)S 973(0)S 1207 1290(N)U 453 1422(*)U 453 1356(*)U 687 1290(D)U 727(O)S 813(1)S 853(0)S 936(J)S 1011(=)S 1093(1)S 1140(,)S 807 1422(U)U 853(p)S 893(d)S 934(a)S 979(t)S 1014(e)S 1094(a)S 1138(\()S 1179(j)S 1220(,)S 1259(j)S 1298(\))S 1340(.)S 453 1488(*)U 807 1554(A)U 858(\()S 896(J)S 940(,)S 976(J)S 1018(\))S 1091(=)S 1167(A)S 1218(\()S 1256(J)S 1300(,)S 1336(J)S 1378(\))S 1458(-)S 1527(D)S 1567(D)S 1607(OT)S 1698(\()S 1736(J)S 1778(-)S 1813(1)S 1860(,)S 1887(A)S 1938(\()S 1976(J)S 2020(,)S 2053(1)S 2098(\))S 2140(,)S 2169(LD)S 2247(A)S 2300(,)S 2327(A)S 2378(\()S 2416(J)S 2460(,)S 2493(1)S 2538(\))S 2580(,)S 2609(LD)S 2687(A)S 2738(\))S 453 1686(*)U 453 1620(*)U 808 1686(C)U 853(o)S 885(mp)S 973(u)S 1019(t)S 1054(e)S 1139(l)S 1178(\()S 1219(j)S 1260(,)S 1299(j)S 1338(\))S 1414(a)S 1453(n)S 1493(d)S 1579(t)S 1614(e)S 1656(s)S 1699(t)S 1778(f)S 1813(o)S 1858(r)S 1933(n)S 1973(o)S 2013(n)S 2058(-)S 2093(p)S 2133(o)S 2176(s)S 2219(i)S 2259(t)S 2299(i)S 2333(v)S 2374(e)S 2418(-)S 2453(d)S 2494(e)S 2531(\256)S 2573(n)S 2619(i)S 2659(t)S 2694(e)S 2733(n)S 2774(e)S 2816(s)S 2856(s)S 2900(.)S 453 1752(*)U 818 1818(I)U 851(F)S 938(\()S 967(A)S 1018(\()S 1056(J)S 1100(,)S 1136(J)S 1178(\))S 1220(.)S 1249(L)S 1289(E)S 1340(.)S 1373(0)S 1420(.)S 1453(0)S 1487(D)S 1533(0)S 1578(\))S 1647(G)S 1687(O)S 1769(TO)S 1893(2)S 1933(0)S 453 1950(*)U 807 1884(A)U 858(\()S 896(J)S 940(,)S 976(J)S 1018(\))S 1091(=)S 1171(S)S 1207(Q)S 1248(RT)S 1338(\()S 1367(A)S 1418(\()S 1456(J)S 1500(,)S 1536(J)S 1578(\))S 1618(\))S 818 2016(I)U 851(F)S 938(\()S 976(J)S 1020(.)S 1049(L)S 1089(T)S 1140(.)S 1167(N)S 1218(\))S 1289(THEN)S 453 2148(*)U 453 2082(*)U 927 2148(U)U 973(p)S 1013(d)S 1054(a)S 1099(t)S 1134(e)S 1214(e)S 1259(l)S 1294(e)S 1325(m)S 1374(e)S 1413(n)S 1459(t)S 1496(s)S 1579(j)S 1611(+)S 1653(1)S 1699(:)S 1733(n)S 1813(o)S 1858(f)S 1939(j)S 1978(-)S 2019(t)S 2053(h)S 2134(c)S 2173(o)S 2219(l)S 2253(u)S 2285(mn)S 2380(.)S 453 2214(*)U 928 2280(CA)U 1009(L)S 1049(L)S 1127(D)S 1167(GE)S 1242(M)S 1287(V)S 1338(\()S 1378(')S 1407(N)S 1453(o)S 1539(t)S 1578(r)S 1614(a)S 1653(n)S 1696(s)S 1733(p)S 1773(o)S 1816(s)S 1854(e)S 1898(')S 1940(,)S 1967(N)S 2018(-)S 2056(J)S 2100(,)S 2136(J)S 2178(-)S 2213(1)S 2260(,)S 2298(-)S 2333(1)S 2380(.)S 2413(0)S 2447(D)S 2493(0)S 2540(,)S 2567(A)S 2618(\()S 2656(J)S 2691(+)S 2733(1)S 2780(,)S 2813(1)S 2858(\))S 2900(,)S 2929(LD)S 3007(A)S 3060(,)S 453 2478(*)U 453 2412(*)U 653 2346($)U 1367(A)S 1418(\()S 1456(J)S 1500(,)S 1533(1)S 1578(\))S 1620(,)S 1649(LD)S 1727(A)S 1780(,)S 1813(1)S 1860(.)S 1893(0)S 1927(D)S 1973(0)S 2020(,)S 2047(A)S 2098(\()S 2136(J)S 2171(+)S 2213(1)S 2260(,)S 2296(J)S 2338(\))S 2380(,)S 2413(1)S 2458(\))S 928 2478(C)U 973(o)S 1005(mp)S 1093(u)S 1139(t)S 1174(e)S 1254(e)S 1299(l)S 1334(e)S 1365(m)S 1414(e)S 1453(n)S 1499(t)S 1536(s)S 1619(j)S 1651(+)S 1693(1)S 1739(:)S 1773(n)S 1853(o)S 1898(f)S 1979(j)S 2018(-)S 2059(t)S 2093(h)S 2174(c)S 2213(o)S 2259(l)S 2293(u)S 2325(mn)S 2453(o)S 2498(f)S 2569(L)S 2620(.)S 453 2544(*)U 928 2610(CA)U 1009(L)S 1049(L)S 1127(DS)S 1208(CA)S 1289(L)S 1338(\()S 1367(N)S 1418(-)S 1456(J)S 1500(,)S 1533(1)S 1580(.)S 1613(0)S 1647(D)S 1693(0)S 1739(/)S 1767(A)S 1818(\()S 1856(J)S 1900(,)S 1936(J)S 1978(\))S 2020(,)S 2047(A)S 2098(\()S 2136(J)S 2171(+)S 2213(1)S 2260(,)S 2296(J)S 2338(\))S 2380(,)S 2413(1)S 2458(\))S 809 2676(EN)U 887(D)S 978(I)S 1011(F)S 969 2742(E)U 573(1)S 613(0)S 688(CO)S 767(NT)S 858(I)S 887(N)S 927(U)S 688 2808(RE)U 769(TUR)S 887(N)S 453 2874(*)U 573 2940(2)U 613(0)S 698(I)S 727(NF)S 807(O)S 891(=)S 976(J)S 688 3006(RE)U 769(TUR)S 887(N)S 689 3072(EN)U 767(D)S 26 4710(-)U 66(-)S 3229(-)S 3269(-)S EP %%Page: ? 25 BP 1 F 66 Z 1731 240(-)U 1813(2)S 1857(3)S 1951(-)S 60 Z 1978 432(\))U 453 564(*)U 453 498(*)U 691 432(S)U 727(U)S 768(BRO)S 887(UT)S 978(I)S 1007(NE)S 1127(DL)S 1209(L)S 1249(T)S 1288(B)S 1338(\()S 1367(N)S 1420(,)S 1447(A)S 1500(,)S 1529(LD)S 1607(A)S 1660(,)S 1698(I)S 1727(NF)S 1807(O)S 1860(,)S 1887(N)S 1928(B)S 688 564(C)U 733(o)S 765(mp)S 853(u)S 899(t)S 934(e)S 976(s)S 1054(a)S 1093(n)S 1169(L)S 1213(*)S 1249(L)S 1293(*)S 1333(*)S 1369(T)S 1458(f)S 1494(a)S 1534(c)S 1579(t)S 1613(o)S 1658(r)S 1699(i)S 1734(z)S 1774(a)S 1819(t)S 1859(i)S 1893(o)S 1933(n)S 2013(o)S 2058(f)S 2134(a)S 2216(s)S 2253(y)S 2285(m)S 2325(m)S 2374(e)S 2419(t)S 2458(r)S 2499(i)S 2534(c)S 2613(p)S 2653(o)S 2696(s)S 2739(i)S 2779(t)S 2819(i)S 2853(v)S 2894(e)S 2938(-)S 2973(d)S 3014(e)S 3051(\256)S 3093(n)S 3139(i)S 3179(t)S 3214(e)S 453 696(*)U 453 630(*)U 685(m)S 734(a)S 779(t)S 818(r)S 859(i)S 893(x)S 967(A)S 1020(.)S 698 762(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1407(NF)S 1487(O)S 1540(,)S 1609(LD)S 1687(A)S 1740(,)S 1807(N)S 1860(,)S 1927(N)S 1968(B)S 687 828(D)U 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1367(A)S 1418(\()S 1449(LD)S 1527(A)S 1580(,)S 1613(*)S 1658(\))S 698 894(I)U 727(NT)S 809(EGE)S 928(R)S 1376(J)S 1420(,)S 1496(J)S 1528(B)S 689 960(E)U (XT)R 809(E)S 848(RN)S 927(AL)S 1367(D)S 1407(GE)S 1482(M)S 1522(M)S 1580(,)S 1647(DL)S 1729(L)S 1769(T)S 1820(,)S 1887(DS)S 1967(Y)S 2008(RK)S 2100(,)S 2167(DTR)S 2291(S)S 2322(M)S 453 1026(*)U 698 1092(I)U 727(NF)S 807(O)S 891(=)S 973(0)S 687 1158(D)U 727(O)S 813(1)S 853(0)S 936(J)S 1011(=)S 1093(1)S 1140(,)S 1207(N)S 1260(,)S 1327(N)S 1368(B)S 1498 1224(\))U 453 1356(*)U 453 1290(*)U 816 1224(J)U 848(B)S 931(=)S 1002(M)S 1058(I)S 1087(N)S 1138(\()S 1167(N)S 1208(B)S 1260(,)S 1287(N)S 1338(-)S 1376(J)S 1411(+)S 1453(1)S 807 1356(U)U 853(p)S 893(d)S 934(a)S 979(t)S 1014(e)S 1093(d)S 1139(i)S 1174(a)S 1213(g)S 1253(o)S 1293(n)S 1334(a)S 1379(l)S 1453(b)S 1499(l)S 1533(o)S 1574(c)S 1613(k)S 1660(.)S 453 1422(*)U 808 1488(CA)U 889(L)S 929(L)S 1007(DS)S 1087(Y)S 1128(RK)S 1218(\()S 1258(')S 1289(L)S 1333(o)S 1367(w)S 1414(e)S 1458(r)S 1498(')S 1540(,)S 1578(')S 1607(N)S 1653(o)S 1739(t)S 1778(r)S 1814(a)S 1853(n)S 1896(s)S 1933(p)S 1973(o)S 2016(s)S 2054(e)S 2098(')S 2140(,)S 2176(J)S 2208(B)S 2260(,)S 2296(J)S 2338(-)S 2373(1)S 2420(,)S 2458(-)S 2493(1)S 2540(.)S 2573(0)S 2607(D)S 2653(0)S 2700(,)S 2727(A)S 2778(\()S 2816(J)S 2860(,)S 2893(1)S 2938(\))S 2980(,)S 3009(LD)S 3087(A)S 3140(,)S 453 1686(*)U 453 1620(*)U 653 1554($)U 1253(1)S 1300(.)S 1333(0)S 1367(D)S 1413(0)S 1460(,)S 1487(A)S 1538(\()S 1576(J)S 1620(,)S 1656(J)S 1698(\))S 1740(,)S 1769(LD)S 1847(A)S 1898(\))S 811 1686(F)U 854(a)S 894(c)S 939(t)S 973(o)S 1018(r)S 1059(i)S 1094(z)S 1134(e)S 1213(d)S 1259(i)S 1294(a)S 1333(g)S 1373(o)S 1413(n)S 1454(a)S 1499(l)S 1573(b)S 1619(l)S 1653(o)S 1694(c)S 1733(k)S 1814(a)S 1853(n)S 1893(d)S 1979(t)S 2014(e)S 2056(s)S 2099(t)S 2178(f)S 2213(o)S 2258(r)S 453 1818(*)U 453 1752(*)U 813(n)S 853(o)S 893(n)S 938(-)S 973(p)S 1013(o)S 1056(s)S 1099(i)S 1139(t)S 1179(i)S 1213(v)S 1254(e)S 1298(-)S 1333(d)S 1374(e)S 1411(\256)S 1453(n)S 1499(i)S 1539(t)S 1574(e)S 1613(n)S 1654(e)S 1696(s)S 1736(s)S 1780(.)S 808 1884(CA)U 889(L)S 929(L)S 1007(DL)S 1089(L)S 1129(T)S 1178(\()S 1216(J)S 1248(B)S 1300(,)S 1327(A)S 1378(\()S 1416(J)S 1460(,)S 1496(J)S 1538(\))S 1580(,)S 1609(LD)S 1687(A)S 1740(,)S 1778(I)S 1807(NF)S 1887(O)S 1938(\))S 453 2016(*)U 818 1950(I)U 851(F)S 938(\()S 978(I)S 1007(NF)S 1087(O)S 1140(.)S 1167(NE)S 1260(.)S 1293(0)S 1338(\))S 1407(G)S 1447(O)S 1529(TO)S 1653(2)S 1693(0)S 818 2082(I)U 851(F)S 938(\()S 976(J)S 1011(+)S 1056(J)S 1088(B)S 1140(.)S 1169(L)S 1209(E)S 1260(.)S 1287(N)S 1338(\))S 1409(THEN)S 453 2214(*)U 453 2148(*)U 927 2214(U)U 973(p)S 1013(d)S 1054(a)S 1099(t)S 1134(e)S 1216(s)S 1253(u)S 1293(b)S 1333(d)S 1379(i)S 1414(a)S 1453(g)S 1493(o)S 1533(n)S 1574(a)S 1619(l)S 1693(b)S 1739(l)S 1773(o)S 1814(c)S 1853(k)S 1900(.)S 453 2280(*)U 928 2346(CA)U 1009(L)S 1049(L)S 1127(D)S 1167(GE)S 1242(M)S 1282(M)S 1338(\()S 1378(')S 1407(N)S 1453(o)S 1539(t)S 1578(r)S 1614(a)S 1653(n)S 1696(s)S 1733(p)S 1773(o)S 1816(s)S 1854(e)S 1898(')S 1940(,)S 1978(')S 2009(T)S 2058(r)S 2094(a)S 2133(n)S 2176(s)S 2213(p)S 2253(o)S 2296(s)S 2334(e)S 2378(')S 2420(,)S 2447(N)S 2498(-)S 2536(J)S 2578(-)S 2616(J)S 2648(B)S 2691(+)S 2733(1)S 2780(,)S 2816(J)S 2848(B)S 2900(,)S 2936(J)S 2978(-)S 3013(1)S 3060(,)S 3260 2412(,)U 653 2478($)U 653 2412($)U 1378(-)S 1413(1)S 1460(.)S 1493(0)S 1527(D)S 1573(0)S 1620(,)S 1647(A)S 1698(\()S 1736(J)S 1771(+)S 1816(J)S 1848(B)S 1900(,)S 1933(1)S 1978(\))S 2020(,)S 2049(LD)S 2127(A)S 2180(,)S 2207(A)S 2258(\()S 2296(J)S 2340(,)S 2373(1)S 2418(\))S 2460(,)S 2489(LD)S 2567(A)S 2620(,)S 2653(1)S 2700(.)S 2733(0)S 2767(D)S 2813(0)S 2860(,)S 2887(A)S 2938(\()S 2976(J)S 3011(+)S 3056(J)S 3088(B)S 3140(,)S 3176(J)S 3218(\))S 1369 2478(LD)U 1447(A)S 1498(\))S 453 2610(*)U 453 2544(*)U 928 2610(C)U 973(o)S 1005(mp)S 1093(u)S 1139(t)S 1174(e)S 1256(s)S 1293(u)S 1333(b)S 1373(d)S 1419(i)S 1454(a)S 1493(g)S 1533(o)S 1573(n)S 1614(a)S 1659(l)S 1733(b)S 1779(l)S 1813(o)S 1854(c)S 1893(k)S 1973(o)S 2018(f)S 2089(L)S 2140(.)S 453 2676(*)U 928 2742(CA)U 1009(L)S 1049(L)S 1127(DTR)S 1251(S)S 1282(M)S 1338(\()S 1378(')S 1408(R)S 1459(i)S 1493(g)S 1533(h)S 1579(t)S 1618(')S 1660(,)S 1698(')S 1729(L)S 1773(o)S 1807(w)S 1854(e)S 1898(r)S 1938(')S 1980(,)S 2018(')S 2049(T)S 2098(r)S 2134(a)S 2173(n)S 2216(s)S 2253(p)S 2293(o)S 2336(s)S 2374(e)S 2418(')S 2460(,)S 2498(')S 2527(N)S 2573(o)S 2613(n)S 2658(-)S 2693(u)S 2733(n)S 2779(i)S 2819(t)S 2858(')S 2900(,)S 2927(N)S 2978(-)S 3016(J)S 3058(-)S 3096(J)S 3128(B)S 3171(+)S 3213(1)S 3260(,)S 653 2808($)U 1376(J)S 1408(B)S 1460(,)S 1487(A)S 1538(\()S 1576(J)S 1620(,)S 1656(J)S 1698(\))S 1740(,)S 1769(LD)S 1847(A)S 1900(,)S 1927(A)S 1978(\()S 2016(J)S 2051(+)S 2096(J)S 2128(B)S 2180(,)S 2216(J)S 2258(\))S 2300(,)S 2329(LD)S 2407(A)S 2458(\))S 809 2874(EN)U 887(D)S 978(I)S 1011(F)S 573 2940(1)U 613(0)S 688(CO)S 767(NT)S 858(I)S 887(N)S 927(UE)S 688 3006(RE)U 769(TUR)S 887(N)S 453 3072(*)U 573 3138(2)U 613(0)S 698(I)S 727(NF)S 807(O)S 891(=)S 978(I)S 1007(NF)S 1087(O)S 1171(+)S 1256(J)S 1338(-)S 1413(1)S 688 3204(RE)U 769(TUR)S 887(N)S 689 3270(EN)U 767(D)S 26 4710(-)U 66(-)S 3229(-)S 3269(-)S EP %%Page: ? 24 BP 3 F 66 Z 448 462(Appendix)U 746(A)S 1 F 1775 270(-)U 1819(22)S 1907(-)S 60 Z 691 624(S)U 727(U)S 768(BRO)S 887(UT)S 978(I)S 1007(NE)S 1127(D)S 1162(M)S 1202(M)S 1258(\()S 1282(M)S 1340(,)S 1367(N)S 1420(,)S 1447(K)S 1500(,)S 1527(A)S 1580(,)S 1609(LD)S 1687(A)S 1740(,)S 1768(B)S 1820(,)S 1849(LDB)S 1980(,)S 2008(C)S 2060(,)S 2089(LDC)S 2218(\))S 453 756(*)U 453 690(*)U 689 756(T)U 733(h)S 779(i)S 816(s)S 898(r)S 933(o)S 973(u)S 1019(t)S 1059(i)S 1093(n)S 1134(e)S 1213(p)S 1254(e)S 1298(r)S 1338(f)S 1373(o)S 1418(r)S 1445(m)S 1496(s)S 1574(a)S 1645(m)S 1694(a)S 1739(t)S 1778(r)S 1819(i)S 1853(x)S 1925(mu)S 2019(l)S 2059(t)S 2099(i)S 2133(p)S 2179(l)S 2213(y)S 2293(o)S 2338(f)S 2419(t)S 2453(h)S 2494(e)S 2578(f)S 2613(o)S 2658(r)S 2685(m)S 2768(C)S 2819(:)S 2851(=)S 2927(A)S 2973(*)S 3008(B)S 3060(.)S 3098 822(f)U 453 888(*)U 453 822(*)U 689(T)S 733(h)S 774(e)S 854(a)S 893(p)S 933(p)S 978(r)S 1013(o)S 1054(a)S 1094(c)S 1133(h)S 1219(i)S 1256(s)S 1333(b)S 1374(a)S 1416(s)S 1454(e)S 1493(d)S 1573(o)S 1613(n)S 1694(a)S 1773(b)S 1819(l)S 1853(o)S 1894(c)S 1933(k)S 2016(s)S 2054(c)S 2093(h)S 2134(e)S 2165(m)S 2214(e)S 2287(w)S 2333(h)S 2379(i)S 2414(c)S 2453(h)S 2531(\256)S 2573(x)S 2614(e)S 2656(s)S 2734(a)S 2813(b)S 2859(l)S 2893(o)S 2934(c)S 2973(k)S 3053(o)S 699 888(t)U 733(h)S 774(e)S 845(m)S 894(a)S 939(t)S 978(r)S 1019(i)S 1053(x)S 1127(A)S 1180(.)S 1249(T)S 1293(h)S 1334(e)S 1413(b)S 1459(l)S 1493(o)S 1534(c)S 1573(k)S 1656(s)S 1699(i)S 1734(z)S 1774(e)S 1859(i)S 1896(s)S 1976(s)S 2014(e)S 2059(t)S 2133(b)S 2173(y)S 2247(N)S 2288(R)S 2374(a)S 2413(n)S 2453(d)S 2527(N)S 2568(C)S 2620(.)S 453 1020(*)U 453 954(*)U 689 1020(T)U 733(h)S 774(e)S 853(v)S 894(a)S 939(l)S 973(u)S 1014(e)S 1093(o)S 1138(f)S 1207(N)S 1248(R)S 1325(m)S 1374(a)S 1413(y)S 1493(b)S 1534(e)S 1613(d)S 1654(e)S 1699(t)S 1734(e)S 1778(r)S 1805(m)S 1859(i)S 1893(n)S 1934(e)S 1973(d)S 2020(,)S 2096(s)S 2134(a)S 2173(y)S 2220(,)S 2293(b)S 2333(y)S 2419(t)S 2453(h)S 2494(e)S 2573(v)S 2614(e)S 2654(c)S 2699(t)S 2733(o)S 2778(r)S 2858(r)S 2894(e)S 2933(g)S 2979(i)S 3016(s)S 3059(t)S 3094(e)S 3138(r)S 453 1152(*)U 453 1086(*)U 699(l)S 734(e)S 773(n)S 813(g)S 859(t)S 893(h)S 940(,)S 1014(a)S 1053(n)S 1093(d)S 1179(t)S 1213(h)S 1254(e)S 1333(v)S 1374(a)S 1419(l)S 1453(u)S 1494(e)S 1573(o)S 1618(f)S 1687(N)S 1728(C)S 1813(b)S 1853(y)S 1939(t)S 1973(h)S 2014(e)S 2096(s)S 2139(i)S 2174(z)S 2214(e)S 2293(o)S 2338(f)S 2419(t)S 2453(h)S 2494(e)S 2574(c)S 2614(a)S 2654(c)S 2693(h)S 2734(e)S 2813(o)S 2858(r)S 2939(l)S 2973(o)S 3014(c)S 3054(a)S 3099(l)S 685 1152(m)U 734(e)S 765(mo)S 858(r)S 893(y)S 940(.)S 453 1218(*)U 698 1284(I)U 727(NT)S 809(EGE)S 928(R)S 1367(N)S 1408(R)S 1460(,)S 1527(N)S 1568(C)S 1858 1350(\))U 691(P)S 727(A)S 768(RA)S 842(M)S 889(E)S 929(T)S 969(E)S 1008(R)S 1378(\()S 1407(N)S 1448(R)S 1491(=)S 1533(6)S 1573(4)S 1620(,)S 1647(N)S 1688(C)S 1731(=)S 1773(3)S 1813(2)S 698 1416(I)U 727(NT)S 809(EGE)S 928(R)S 1367(K)S 1420(,)S 1489(LD)S 1567(A)S 1620(,)S 1689(LDB)S 1820(,)S 1889(LDC)S 2020(,)S 2082(M)S 2140(,)S 2207(N)S 2458 1482(\))U 687 1548(D)U 687 1482(D)U 727(O)S 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1367(A)S 1418(\()S 1449(LD)S 1527(A)S 1580(,)S 1613(*)S 1658(\))S 1700(,)S 1768(B)S 1818(\()S 1849(LDB)S 1980(,)S 2013(*)S 2058(\))S 2100(,)S 2168(C)S 2218(\()S 2249(LDC)S 2380(,)S 2413(*)S 727 1548(O)U 767(U)S 808(BL)S 889(E)S 971(P)S 1008(RE)S 1088(C)S 1138(I)S 1171(S)S 1218(I)S 1247(O)S 1287(N)S 1368(BE)S 1449(TA)S 1967 1614(D)U 689 1680(E)U 698 1614(I)U 727(NT)S 809(EGE)S 928(R)S 1378(I)S 1420(,)S 1498(I)S 1529(EN)S 1607(D)S 1660(,)S 1736(J)S 1780(,)S 1856(J)S 1889(EN)S 727 1680(XT)U 809(E)S 848(RN)S 927(AL)S 1367(D)S 1407(GE)S 1482(M)S 1522(M)S 453 1746(*)U 687 1812(D)U 727(O)S 813(2)S 853(0)S 936(J)S 1011(=)S 1093(1)S 1140(,)S 1207(N)S 1260(,)S 1327(N)S 1368(C)S 1407 1878(N)U 818(I)S 851(F)S 938(\()S 976(J)S 1020(.)S 1049(EQ)S 1140(.)S 1173(1)S 1218(\))S 1289(THE)S 928 1944(BE)U 1009(TA)S 1131(=)S 1213(0)S 1260(.)S 1293(0)S 1327(D)S 1373(0)S 809 2010(E)U 849(L)S 891(S)S 929(E)S 928 2076(BE)U 1009(TA)S 1131(=)S 1213(1)S 1260(.)S 1293(0)S 1327(D)S 1373(0)S 809 2142(EN)U 887(D)S 978(I)S 1011(F)S 816 2208(J)U 849(EN)S 927(D)S 1011(=)S 1082(M)S 1138(I)S 1167(N)S 1218(\()S 1256(J)S 1291(+)S 1327(N)S 1368(C)S 1418(-)S 1453(1)S 1500(,)S 1527(N)S 1578(\))S 807 2274(D)U 847(O)S 933(1)S 973(0)S 1058(I)S 1131(=)S 1213(1)S 1260(,)S 1322(M)S 1380(,)S 1447(N)S 1488(R)S 938 2340(I)U 969(EN)S 1047(D)S 1131(=)S 1202(M)S 1258(I)S 1287(N)S 1338(\()S 1378(I)S 1411(+)S 1447(N)S 1488(R)S 1538(-)S 1573(1)S 1620(,)S 1642(M)S 1698(\))S 928 2406(C)U (A)R 1009(L)S 1049(L)S 1127(D)S 1167(GE)S 1242(M)S 1282(M)S 1338(\()S 1378(')S 1407(N)S 1453(o)S 1539(t)S 1578(r)S 1614(a)S 1653(n)S 1696(s)S 1733(p)S 1773(o)S 1816(s)S 1854(e)S 1898(')S 1940(,)S 1978(')S 2007(N)S 2053(o)S 2139(t)S 2178(r)S 2214(a)S 2253(n)S 2296(s)S 2333(p)S 2373(o)S 2416(s)S 2454(e)S 2498(')S 2540(,)S 2578(I)S 2609(EN)S 2687(D)S 2738(-)S 2778(I)S 2811(+)S 2853(1)S 2900(,)S 2927(K)S 2980(,)S 3300 2472(,)U 653 2538($)U 653 2472($)U 1376(J)S 1409(EN)S 1487(D)S 1538(-)S 1576(J)S 1611(+)S 1653(1)S 1700(,)S 1733(1)S 1780(.)S 1813(0)S 1847(D)S 1893(0)S 1940(,)S 1967(A)S 2018(\()S 2058(I)S 2100(,)S 2136(J)S 2178(\))S 2220(,)S 2249(LD)S 2327(A)S 2380(,)S 2408(B)S 2458(\()S 2496(J)S 2540(,)S 2573(1)S 2618(\))S 2660(,)S 2689(LDB)S 2820(,)S 2848(BE)S 2929(TA)S 3020(,)S 3048(C)S 3098(\()S 3138(I)S 3180(,)S 3213(1)S 3258(\))S 1369 2538(LDC)U 1498(\))S 573 2670(2)U 573 2604(1)U 613(0)S 808(CO)S 887(NT)S 978(I)S 1007(N)S 1047(UE)S 613 2670(0)U 688(CO)S 767(NT)S 858(I)S 887(N)S 927(UE)S 453 2736(*)U 688 2802(RE)U 769(TUR)S 887(N)S 26 4710(-)U 689 2868(EN)U 767(D)S 66 4710(-)U 3229(-)S 3269(-)S EP %%Page: ? 23 BP 1 F 66 Z 448 462(A)U 1775 270(-)U 1819(21)S 1907(-)S 496 462(lgebra)U 711(Subprograms,'')S 1175(Argonne)S 1457(National)S 1733(Laboratory)S 2075(Report,)S 2321(ANL-MCS-TM-41)S 2878(\(Revision)S 3184(3\),)S 448 750(J)U 448 558(November)U 748(1986.)S 474 750(.)U 520(Dongarra,)S 815(J.)S 887(DuCroz,)S 1142(S.)S 1225(Hammarling,)S 1603(and)S 1727(R.)S 1817(Hanson,)S 2065(``An)S 2218(Extended)S 2494(Set)S 2606(of)S 2689(Fortran)S 2911(Basic)S 3085(Linear)S 448 942(R)U 448 846(Algebra)U 701(Subprograms:)S 1112(Model)S 1325(Implementa)S 1638(tion)S 1781(and)S 1917(Test)S 2071(Programs,'')S 2426(Argonne)S 2698(National)S 2964(Laboratory)S 492 942(eport,)U 666(ANL-MCS-TM-81,)S 1212(November,)S 1529(1986.)S 448 1134(J)U (.J.)R 575(Dongarra)S 865(and)S 1001(I.S.)S 1135(Duff,)S 1318(``Advanced)S 1669(Architecture)S 2037(Computers,'')S 2427(Argonne)S 2698(National)S 2964(Laboratory)S 448 1422(J)U 448 1230(Report,)U 666(ANL-MCS-TM-57)S 1195(\(Revision)S 1473(1\),)S 1567(January,)S 1811(1987.)S 474 1422(.)U 524(DuCroz,)S 783(S.)S 870(Nugent,)S 1114(J.)S 1190(Reid,)S 1364(and)S 1492(D.)S 1589(Taylor,)S 1813(``Solving)S 2094(Large)S 2279(Full)S 2417(Sets)S 2559(of)S 2646(Linear)S 2849(Equations)S 3144(in)S 3227(a)S 448 1710(I)U 448 1518(Paged)U 631(Virtual)S 839(Store,'')S 2 F 1061(TOMS)S 1 F 1256(,)S 1295(vol.)S 1418(7,4,)S 1540(pp.)S 1645(527-536,)S 1904(1981.)S 470 1710(.S.)U 565(Duff,)S 731(``Full)S 904(Matrix)S 1106(Techniques)S 1432(in)S 1506(Sparse)S 1705(Gaussian)S 1970(Eliminat)S 2195(ion,'')S 2 F 2363(Numerical)S 2664(Analysis)S 2910(Proceedings,)S 1 F 448 1998(A)U 2 F 448 1806(Dundee)U 675(1981,)S 846(Lecture)S 1069(Notes)S 1241(in)S 1314(Mathematics)S 1676(912)S 1 F (,)R 1814(pp.)S 1919(71-84,)S 2112(Springer-Verlag,)S 2579(1986.)S 496 1998(.)U 538(George)S 757(and)S 877(H.)S 967(Rashwan,)S 1251(``Auxiliary)S 1572(Storage)S 1797(Methods)S 2052(for)S 2153(Solving)S 2382(Finite)S 2559(Element)S 2801(Systems,'')S 2 F 3106(SIAM)S 1 F 448 2286(I)U 2 F 448 2094(SISSC)U 1 F (,)R 652(vol.)S 775(6,)S 847(1981.)S 470 2286(BM,)U 612(``Engineering)S 999(and)S 1116(Scienti\256c)S 1386(Subroutine)S 1697(Library,'')S 2 F 1977(IBM)S 1 F (,)R 2133(vol.)S 2256(Program)S 2505(Number:)S 2761(5668-863.)S 3223 2478(n)U 448 2574(U)U 448 2478(C.)U 539(Lawson,)S 795(R.)S 886(Hanson,)S 1135(D.)S 1230(Kincaid,)S 1485(and)S 1610(F.)S 1694(Krogh,)S 1910(``Basic)S 2130(Linear)S 2331(Algebra)S 2573(Subprograms)S 2955(for)S 3062(Fortra)S 496 2574(sage,'')U 2 F 696(ACM)S 857(Transactions)S 1224(on)S 1312(Mathematical)S 1699(Software)S 1 F (,)R 1972(vol.)S 2095(5,)S 2167(1979.)S 3234 2766(-)U 448 2862(g)U 448 2766(C.)U 541(Lawson,)S 799(R.)S 892(Hanson,)S 1143(D.)S 1240(Kincaid,)S 1496(and)S 1622(F.)S 1707(Krogh,)S 1924(``Algorithm)S 2273(539:)S 2421(Basic)S 2598(Linear)S 2800(Algebra)S 3043(Subpro)S 481 2862(rams)U 631(for)S 730(Fortran)S 946(Usage,'')S 2 F 1194(ACM)S 1355(Transactions)S 1722(on)S 1810(Mathematical)S 2197(Software)S 1 F (,)R 2470(vol.)S 2593(5,)S 2665(1979.)S 3223 3054(y)U 448 3150(S)U 448 3054(A.C.)U 597(McKellar)S 872(and)S 990(E.G.)S 1135(Coffman)S 1392(Jr.,)S 1497(``Organizing)S 1860(Matrices)S 2113(and)S 2231(Matrix)S 2433(Operations)S 2745(for)S 2845(Paged)S 3029(Memor)S 485 3150(ystems,'')U 2 F 751(CACM)S 1 F (,)R 973(vol.)S 1096(12,3,)S 1251(1969.)S 448 3342(R)U (.)R 537(Schreiber,)S 834(``Module)S 1111(Design)S 1326(Speci\256cation)S 1697(\(Version)S 1956(1.0\),'')S 2 F 2149(SAXPY)S 2366(Computer)S 2657(Corporation,)S 3031(255)S 3157(San)S 1 F 448 3630(Y)U 2 F 448 3438(Geronimo)U 738(Way,)S 894(Sunnyvale,)S 1203(CA)S 1309(94086.)S 1 F (,)R 1530(1986.)S 496 3630(.)U 536(Robert)S 738(and)S 856(P.)S 933(Sguazzero,)S 1247(``The)S 1416(LU)S 1527(Decomposition)S 1952(Algorithm)S 2249(and)S 2367(Its)S 2456(Ef\256cient)S 2705(Fortran)S 2921(Implementa)S 3234(-)S 60 Z 16 4710(-)U 66 Z 448 3726(tion)U 572(on)S 660(the)S 762(IBM)S 909(3090)S 1063(Vector)S 1264(Multiprocessor,'')S 1746(IBM)S 1893(ECSEC)S 2120(Report)S 2321(ICE-0006,)S 2620(1987.)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 22 BP 1 F 66 Z 448 786(D)U 3 F 448 462(References)U 1 F 1775 270(-)U 1819(20)S 1907(-)S 496 786(.W.)U 632(Barron)S 855(and)S 990(H.P.F)S 1146(.)S 1203(Swinnerton-Dyer,)S 1718(``Solution)S 2025(of)S 2120(Simultaneous)S 2518(Linear)S 2728(Equations)S 3030(Using)S 3227(a)S 448 1074(M)U 448 882(Magnetic-Tape)U 871(Store,'')S 2 F 1093(Computer)S 1379(J.)S 1 F (,)R 1464(vol.)S 1587(3,)S 1659(1960.)S 507 1074(.)U 551(Berry,)S 745(K.)S 837(Gallivan,)S 1107(W.)S 1213(Harrod,)S 1444(W.)S 1550(Jalby,)S 1733(S.)S 1814(Lo,)S 1931(U.)S 2023(Meier,)S 2224(B.)S 2312(Philippe,)S 2575(and)S 2697(A.)S 2789(Sameh,)S 3012(``Parallel)S 448 1362(C)U 448 1170(Algorithms)U 770(on)S 858(the)S 960(CEDAR)S 1206(System,'')S 2 F 1483(CSRD)S 1670(Report)S 1871(No.)S 1987(581)S 1 F (.)R 492 1362(.)U 533(Bischof)S 762(and)S 881(C.)S 966(Van)S 1100(Loan,)S 1276(``The)S 1446(WY)S 1580(Representation)S 1998(for)S 2099(Products)S 2354(of)S 2432(Householder)S 2792(Matrices,'')S 2 F 3106(SIAM)S 1 F 448 1650(I)U 2 F 448 1458(SISSC)U 1 F (,)R 652(vol.)S 775(8,)S 847(2,)S 919(March,)S 1130(1987.)S 470 1650(.)U 514(Bucher)S 731(and)S 853(T.)S 937(Jordan,)S 1157(``Linear)S 1399(Algebra)S 1638(Programs)S 1918(for)S 2022(use)S 2137(on)S 2230(a)S 2286(Vector)S 2491(Computer)S 2780(with)S 2923(a)S 2978(Secondary)S 3239 1746(,)U 448 1842(e)U 448 1746(Solid)U 618(State)S 780(Storage)S 1011(Device,'')S 1288(in)S 2 F 1369(Advances)S 1651(in)S 1732(Computer)S 2026(Methods)S 2283(for)S 2390(Partical)S 2635(Differential)S 2972(Equations)S 1 F 477 1842(d.)U 549(R.)S 632(Vichnevetsky)S 1016(and)S 1133(R)S 1199(Stepleman,)S 1515(pp.)S 1620(546-550,)S 1879(IMACS.)S 3238 2034(:)U 448 2130(U)U 448 2034(D.A.)U 620(Calahan,)S 894(``Block-Oriented)S 1389(Local-Memory-Based)S 2012(Linear)S 2225(Equation)S 2504(Solution)S 2769(on)S 2877(the)S 2999(CRAY-2)S 496 2130(niprocessor)U 826(Algorithms,'')S 2 F 1213(Proceedings)S 1568(International)S 1941(Conference)S 2270(on)S 2362(Parallel)S 2603(Processing)S 1 F (,)R 2939(IEEE)S 3106(Com-)S 448 2418(B)U 448 2226(puter)U 605(Society)S 824(Press,)S 1003(August)S 1216(1986.)S 492 2418(.)U 532(Chartres,)S 795(``Adaption)S 1107(of)S 1185(the)S 1288(Jacobi)S 1479(and)S 1596(Givens)S 1805(Methods)S 2058(for)S 2157(a)S 2208(Computer)S 2493(with)S 2632(Magnetic)S 2902(Tape)S 3055(Backup)S 448 2706(A)U 448 2514(Store,'')U 2 F 670(University)S 966(of)S 1039(Sydney)S 1247(Technical)S 1528(Report)S 1729(No.)S 1845(8)S 1 F (.)R 496 2706(.K.)U 604(Dave)S 769(and)S 890(I.S.)S 1009(Duff,)S 1177(``Sparse)S 1422(Matrix)S 1626(Calculati)S 1862(ons)S 1979(on)S 2070(the)S 2175(CRAY-2,'')S 2500(AERE)S 2697(Harwell)S 2934(Report)S 3138(CSS)S 448 2994(D)U 448 2802(197)U 569(\(to)S 664(appear)S 861(Parallel)S 1083(Computing\).)S 496 2994(.)U 545(Dodson)S 782(and)S 908(J.)S 982(Lewis,)S 1191(``Issues)S 1428(relating)S 1659(to)S 1741(extension)S 2024(of)S 2110(the)S 2221(Basic)S 2398(Linear)S 2600(Algebra)S 2843(Subprograms,'')S 448 3282(J)U 2 F 448 3090(ACM)U 609(SIGNUM)S 881(Newsletter)S 1 F (,)R 1201(vol.)S 1324(20,1,)S 1479(pp.)S 1584(2-18,)S 1744(1985.)S 474 3282(.J.)U 556(Dongarra,)S 844(J.)S 909(Bunch,)S 1120(C.)S 1203(Moler,)S 1403(and)S 1520(G.)S 1607(Stewart,)S 2 F 1847(LINPACK)S 2140(Users')S 2339(Guide,)S 1 F 2539(SIAM)S 2727(Pub.,)S 2886(1979.)S 3227 3474(e)U 448 3570(M)U 448 3474(J.J.)U 573(Dongarra,)S 878(F.)S 971(Gustavson,)S 1306(and)S 1439(A.)S 1542(Karp,)S 1729(``Implementi)S 2075(ng)S 2179(Linear)S 2388(Algebra)S 2638(Algorithms)S 2976(for)S 3091(Dens)S 507 3570(atrices)U 700(on)S 788(a)S 839(Vector)S 1040(Pipeline)S 1277(Machine,'')S 2 F 1590(SIAM)S 1762(Review)S 1 F (,)R 1990(vol.)S 2113(26,)S 2218(1,)S 2290(pp.)S 2395(91-112.)S 3223 3762(n)U 448 3858(t)U 448 3762(J.J.)U 560(Dongarra)S 835(and)S 956(T.)S 1039(Hewitt,)S 1261(``Implementi)S 1607(ng)S 1698(Dense)S 1888(Linear)S 2084(Algebra)S 2321(Algorithms)S 2646(Using)S 2829(Multitasking)S 3190(o)S 466 3858(he)U 550(CRAY)S 756(X-MP-4,'')S 2 F 1060(SIAM)S 1232(J.)S 1300(Sci)S 1402(Stat.)S 1543(Comp.)S 1 F (,)R 1757(vol.)S 1880(7,)S 1952(1,)S 2024(pp.)S 2129(347-350,)S 2388(January,)S 2632(1986.)S 2 F 3230 4050(s)U 448 4146(P)U 1 F 448 4050(J.J.)U 560(Dongarra)S 835(and)S 956(D.C.)S 1108(Sorensen,)S 1393(``Linear)S 1633(Algebra)S 1870(on)S 1961(High-Performance)S 2476(Computers,'')S 2851(in)S 2 F 2927(Proceeding)S 488 4146(arallel)U 685(Computing)S 1000(85)S 1 F (,)R 1105(ed.)S 1206(U.)S 1293(Schendel,)S 1573(pp.)S 1678(113-136,)S 1937(North)S 2113(Holland.)S 3234 4338(r)U 60 Z 16 4710(-)U 66 Z 448 4338(J.J.)U 560(Dongarra,)S 852(J.)S 921(DuCroz,)S 1173(S.)S 1253(Hammarling,)S 1628(and)S 1749(R.)S 1836(Hanson,)S 2081(``An)S 2232(Extended)S 2506(Set)S 2615(of)S 2695(Fortran)S 2914(Basic)S 3085(Linea)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 21 BP 1 F 66 Z 1775 270(-)U 1819(19)S 1907(-)S 598 462(I)U (n)R 680(Appendix)S 967(A)S 1042(we)S 1146(present)S 1363(a)S 1419(code)S 1570(DPLUB)S 1813(for)S 1916(block)S 2 F 2088(LU)S 1 F (-factorization,)R 2569(and)S 2690(a)S 2745(lower-level)S 3070(routine)S 3230 558(s)U 448 654(L)U 448 558(DPLU,)U 666(which)S 855(is)S 927(called)S 1111(by)S 1205(DPLUB)S 1449(and)S 1571(has)S 1686(been)S 1837(designed)S 2098(to)S 2176(have)S 2327(the)S 2434(same)S 2596(structure.)S 2892(DPLUB)S 3136(call)S 488 654(evel)U 620(3)S 676(BLAS)S 868(whereas)S 1107(DPLU)S 1302(calls)S 1444(Level)S 1615(2)S 1670(and)S 1787(Level)S 1958(1)S 2013(BLAS.)S 2243(Because)S 2484(DPLU)S 2679(needs)S 2851(to)S 2924(operate)S 3139(on)S 3227(a)S 448 846(w)U 448 750(rectangular)U 771(matrix)S 970(\(so)S 1079(that)S 1205(the)S 1313(factor)S 2 F 1494(L)S 1 F 1559(is)S 1631(lower)S 1809(trapezoidal)S 2128(rather)S 2309(than)S 2450(triangular\),)S 2772(DPLUB)S 3017(has)S 3132(been)S 496 846(ritten)U 656(with)S 795(the)S 897(same)S 1054(speci\256cation.)S 3 F 448 1038(8.4)U 2 F 553(QR)S 3 F (-Factorization)R 1 F 598 1170(We)U 719(do)S 815(not)S 929(discuss)S 1150(this)S 1275(in)S 1356(detail)S 1531(here,)S 1691(but)S 1804(draw)S 1965(attention)S 2223(to)S 2303(the)S 2412(method)S 2638(of)S 2766(\(Bischof)S 3022(and)S 3146(Van)S 60 Z 16 4710(-)U 66 Z 448 1266(Loan,)U 622(1987\),)S 815(which)S 998(was)S 1123(designed)S 1379(speci\256cally)S 1700(to)S 1773(exploit)S 1977(ef\256cient)S 2214(matrix-mat)S 2505(rix)S 2600(multiplic)S 2836(ation.)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 20 BP 1 F 66 Z 448 462(hence:)U 1775 270(-)U 1819(18)S 1907(-)S 2 F 703 594(U)U 4 F 821(=)S 2 F 880(L)S 968(A)S 1 F 48 Z 1032 609(2)U 4 F 917 567(-)U 1 F (1)R 1008 609(1)U 941(1)S 740 744(2)U 751 609(12)U 917(1)S 764 744(2)U 836(22)S 1005(22)S 1171(21)S 1267(12)S 2 F 66 Z 703 864(L)U 703 729(L)U 788(U)S 4 F 906(=)S 2 F 965(A)S 4 F 1075(-)S 2 F 1134(L)S 1219(U)S 4 F 810 864(=)U 1 F 869(\()S 2 F (A)R 4 F 1001(-)S 2 F 1060(L)S 1145(U)S 1 F 1241(\))S 2 F (U)R 1 F 48 Z 1338 837(1)U 4 F 1311(-)S 1 F 1335 879(2)U 66 Z 598 1035(T)U 48 Z 740 879(32)U 931(32)S 1097(31)S 1193(12)S 1311(2)S 66 Z 638 1035(he)U 722(computation)S 1072(of)S 1149(one)S 1266(block-column)S 1653(requires)S 1887(the)S 1989(following)S 2267(operations:)S 448 1227(1.)U 520(compute)S 768(the)S 870(superdiagonal)S 1261(block)S 1429(of)S 2 F 1506(U)S 1 F (:)R 2 F 703 1359(U)U 4 F 821(\254)S 2 F 909(L)S 997(A)S 1 F 48 Z 1061 1374(2)U 4 F 946 1332(-)U 1 F (1)R 1037 1374(1)U 970(1)S 66 Z 448 1590(2)U 48 Z 751 1374(12)U 946(1)S 66 Z 481 1590(.)U 520(update)S 717(the)S 819(diagonal)S 1067(and)S 1184(subdiagonal)S 1524(blocks:)S 0 F 703 1707 M 11 73 3 0 20 0 0 15 PS20 703 1773 M 11 74 3 0 5 0 0 15 PS5 703 1839 M 11 74 3 0 5 0 0 15 PS5 703 1905 M 11 76 3 0 21 0 0 15 PS21 2 F 753 1866(A)U 753 1749(A)U 0 F 841 1707 M 11 77 3 0 22 0 0 15 PS22 841 1773 M 11 74 3 0 5 0 0 15 PS5 841 1839 M 11 74 3 0 5 0 0 15 PS5 841 1905 M 11 79 3 0 23 0 0 15 PS23 4 F 891 1824(\254)U 0 F 979 1740 M 11 73 3 0 20 0 0 15 PS20 979 1806 M 11 74 3 0 5 0 0 15 PS5 979 1872 M 11 76 3 0 21 0 0 15 PS21 2 F 1029 1857(A)U 1029 1749(A)U 0 F 1117 1740 M 11 77 3 0 22 0 0 15 PS22 1117 1806 M 11 74 3 0 5 0 0 15 PS5 1117 1872 M 11 79 3 0 23 0 0 15 PS23 4 F 1167 1824(-)U 0 F 1226 1740 M 11 73 3 0 20 0 0 15 PS20 1226 1806 M 11 74 3 0 5 0 0 15 PS5 1226 1872 M 11 76 3 0 21 0 0 15 PS21 2 F 1276 1857(L)U 1276 1749(L)U 0 F 1361 1740 M 11 77 3 0 22 0 0 15 PS22 1361 1806 M 11 74 3 0 5 0 0 15 PS5 1361 1872 M 11 79 3 0 23 0 0 15 PS23 2 F 1389 1824(U)U 1 F 48 Z 793 1881(32)U 793 1764(2)U 4 F 793 1839(\242)U 1 F 817 1764(2)U 4 F 793 1722(\242)U 1 F 1069 1872(32)U 1069 1764(2)U (2)R 1313 1872(31)U 1313 1764(2)U (1)R 1437 1839(12)U 66 Z 1903 2124(:)U 448(3.)S 520(compute)S 768(the)S 2 F 870(LU)S 1 F 977(factorizat)S 1224(ion)S 1330(of)S 1407(the)S 1509(diagonal)S 1757(block)S 2 F 703 2256(A)U 4 F 813(\256)S 2 F 901(L)S 986(U)S 1 F 48 Z 1058 2271(2)U 4 F 743 2229(\242)U 1 F 938 2271(22)U 1034(2)S 767(2)S 66 Z 448 2487(4)U 48 Z 743 2271(2)U 66 Z 481 2487(.)U 520(compute)S 768(the)S 870(subdiagonal)S 1210(block)S 1378(of)S 2 F 1455(L)S 1 F (:)R 2 F 703 2619(L)U 4 F 810(\254)S 2 F 898(A)S 986(U)S 1 F 48 Z 740 2634(32)U 938(32)S 4 F 938 2592(\242)U 1 F 1034 2634(22)U 4 F 1034 2592(-)U 1 F (1)R 66 Z 598 2886(O)U (f)R 693(course)S 890(we)S 992(want)S 1145(to)S 1221(introduce)S 1494(partial)S 1685(pivoting)S 1928(for)S 2029(numerical)S 2315(stability.)S 2567(This)S 2708(means)S 2900(that)S 3022(at)S 3093(step)S 3223(3)S 3223 2982(k)U 448(it)S 510(is)S 580(not)S 690(suf\256cient)S 961(to)S 1038(consider)S 1287(only)S 1430(the)S 1536(diagonal)S 1788(block)S 2 F 1960(A)S 1 F 2048(:)S 2091(we)S 2193(need)S 2342(to)S 2418(involve)S 2640(the)S 2745(sub-diagonal)S 3110(bloc)S 48 Z 2000 2997(22)U 4 F 488 3054(\242)U 2000 2955(\242)U 1 F 512 3096(2)U 2 F 66 Z 448 3081(A)U 1 F 48 Z 488 3096(3)U 66 Z 558 3081(as)U 635(well.)S 787(Steps)S 952(3)S 1007(and)S 1124(4)S 1179(must)S 1329(be)S 1413(combined)S 1694(into)S 1818(the)S 1920(following:)S 2891 3276(:)U 448(3)S 493(.)S 532(compute)S 780(the)S 2 F 882(LU)S 1 F 989(factorizat)S 1236(ion)S 1342(with)S 1481(interchanges)S 1835(of)S 1912(the)S 2014(diagonal)S 2262(and)S 2379(subdiagonal)S 2719(blocks)S 4 F 48 Z 481 3249(\242)U 1 F 793 3567(32)U 793 3450(2)U 4 F 793 3525(\242)U 1 F 817 3450(2)U 4 F 793 3408(\242)U 1 F 1106 3558(32)U 1106 3450(2)U (2)R 1230 3525(22)U 66 Z 598 3750(H)U 0 F 703 3393 M 11 73 3 0 20 0 0 15 PS20 703 3459 M 11 74 3 0 5 0 0 15 PS5 703 3525 M 11 74 3 0 5 0 0 15 PS5 703 3591 M 11 76 3 0 21 0 0 15 PS21 2 F 753 3552(A)U 753 3435(A)U 0 F 841 3393 M 11 77 3 0 22 0 0 15 PS22 841 3459 M 11 74 3 0 5 0 0 15 PS5 841 3525 M 11 74 3 0 5 0 0 15 PS5 841 3591 M 11 79 3 0 23 0 0 15 PS23 4 F 891 3510(\256)U 2 F 979(P)S 0 F 1019 3426 M 11 73 3 0 20 0 0 15 PS20 1019 3492 M 11 74 3 0 5 0 0 15 PS5 1019 3558 M 11 76 3 0 21 0 0 15 PS21 2 F 1069 3543(L)U 1069 3435(L)U 0 F 1154 3426 M 11 77 3 0 22 0 0 15 PS22 1154 3492 M 11 74 3 0 5 0 0 15 PS5 1154 3558 M 11 79 3 0 23 0 0 15 PS23 2 F 1182 3510(U)U 1 F 646 3750(ere)U 2 F 752(P)S 1 F 818(is)S 888(a)S 943(permutation)S 1286(matrix)S 1483(which)S 1670(de\256nes)S 1883(the)S 1989(row)S 2118(interchanges.)S 2515(The)S 2643(row)S 2771(interchanges)S 3128(must)S 448 3942(a)U 448 3846(also)U 582(be)S 672(applied)S 892(to)S 970(the)S 1077(other)S 1239(columns)S 1489(of)S 2 F 1571(A)S 1 F (.)R 1677(In)S 1759(our)S 1874(example)S 2123(code,)S 2291(we)S 2395(apply)S 2568(the)S 2675(interchanges)S 3034(immedi-)S 477 3942(tely)U 602(to)S 680(all)S 772(preceding)S 1058(block-columns,)S 1493(but)S 1603(delay)S 1771(applying)S 2027(them)S 2184(to)S 2261(subsequent)S 2580(block-columns)S 2997(until)S 3143(they)S 3227 4038(e)U 448 4134(d)U 448 4038(come)U 620(to)S 701(be)S 793(processed.)S 1100(This)S 1246(is)S 1319(similar)S 1530(to)S 1610(the)S 1719(block-column)S 2113(algorithm)S 2397(of)S 2525(\(DuCroz)S 2 F 2785(et)S 2861(al.)S 1 F (,)R 2975(1981\):)S 3176(th)S 481 4134(ifference)U 736(is)S 802(that)S 922(their)S 1064(algorithm)S 1341(combined)S 1622(steps)S 1776(1)S 1831(and)S 1948(2,)S 2020(and)S 2137(did)S 2243(not)S 2349(use)S 2459(any)S 2576(BLAS)S 2767(routines.)S 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 19 BP 1 F 66 Z 448 462(4.)U 520(compute)S 768(the)S 870(subdiagonal)S 1210(block)S 1378(of)S 2 F 1455(L)S 1 F (:)R 1775 270(-)U 1819(17)S 1907(-)S 2 F 703 594(L)U 4 F 810(\254)S 2 F 898(A)S 1 F 986(\()S 2 F (L)R 1 F 1093(\))S 48 Z 1142 567(1)U 2 F 1045(T)S 4 F 1115(-)S 1 F 1069 609(2)U 4 F 938 567(\242)U 1 F 1045 609(2)U 962(2)S 66 Z 448 825(O)U 48 Z 740 609(32)U 938(3)S 66 Z 496 825(perations)U 763(1,)S 838(3,)S 913(and)S 1033(4)S 1091(can)S 1207(be)S 1294(performed)S 1593(by)S 1684(calls)S 1829(to)S 1905(DSYRK,)S 2172(DGEMM,)S 2468(and)S 2588(DTRSM)S 2841(respectively)S 3183(\(in)S 3227 921(e)U 448 1017(c)U 448 921(double)U 662(precision\).)S 976(Operation)S 1273(2)S 1340(requires)S 1586(a)S 1649(separate)S 1898(lower-level)S 2231(Cholesky)S 2514(factorizat)S 2761(ion)S 2879(routine)S 3099(\(sinc)S 477 1017(urrent)U 656(standard)S 901(Fortran)S 1117(forbids)S 1326(recursion\).)S 598 1149(I)U (n)R 678(Appendix)S 963(A)S 1036(we)S 1138(give)S 1276(code)S 1425(for)S 1527(a)S 1581(block)S 1752(Cholesky)S 2026(factorizat)S 2273(ion)S 2382(routine)S 2593(DLLTB,)S 2847(calling)S 3050(Level)S 3223(3)S 3223 1245(d)U 448 1341(2)U 448 1245(BLAS)U 640(routines;)S 893(and)S 1011(also)S 1140(a)S 1191(lower-level)S 1512(routine)S 1720(DLLT)S 1910(which)S 2093(is)S 2159(called)S 2337(by)S 2425(DLLTB,)S 2676(and)S 2793(calls)S 2935(Level)S 3106(1)S 3161(an)S 505 1341(BLAS)U 698(routines.)S 973(The)S 1099(structure)S 1353(of)S 1432(DLLTB)S 1668(has)S 1780(been)S 1928(kept)S 2065(as)S 2144(similar)S 2350(as)S 2429(possible)S 2669(to)S 2744(that)S 2866(of)S 2945(DLLT.)S 3154(The)S 448 1437(call-tree)U 684(is:)S 1830 1773(DLLTB)U 2666 2061(M)U 879(DSYRK)S 1523(DLLT)S 2144(DGEMM)S 2497(DTRS)S 1203 2349(DDOT)U 1486(DGEMV)S 1828(DSCAL)S 3 F 448 2577(8.3)U 553(LU-Factorization)S 1 F 598 2709(Again,)U 800(the)S 904(details)S 1099(of)S 1178(the)S 1282(algorithm)S 1561(may)S 1698(be)S 1784(derived)S 2005(by)S 2095(partitioning)S 2425(the)S 2529(matrices.)S 2792(If)S 2860(we)S 2961(ignore)S 3153(row)S 448 2805(or)U 525(column)S 744(interchanges)S 1098(we)S 1197(have)S 1343(simply:)S 0 F 703 2922 M 11 73 3 0 20 0 0 15 PS20 703 2988 M 11 74 3 0 5 0 0 15 PS5 703 3054 M 11 74 3 0 5 0 0 15 PS5 703 3120 M 11 74 3 0 5 0 0 15 PS5 703 3186 M 11 76 3 0 21 0 0 15 PS21 2 F 753 3159(A)U 753 3051(A)U 753 2943(A)U 885 3159(A)U 885 3051(A)U 885 2943(A)U 1017 3159(A)U 1017 3051(A)U 1017 2943(A)U 0 F 1105 2922 M 11 77 3 0 22 0 0 15 PS22 1105 2988 M 11 74 3 0 5 0 0 15 PS5 1105 3054 M 11 74 3 0 5 0 0 15 PS5 1105 3120 M 11 74 3 0 5 0 0 15 PS5 1105 3186 M 11 79 3 0 23 0 0 15 PS23 4 F 1155 3051(=)U 0 F 1214 2922 M 11 73 3 0 20 0 0 15 PS20 1214 2988 M 11 74 3 0 5 0 0 15 PS5 1214 3054 M 11 74 3 0 5 0 0 15 PS5 1214 3120 M 11 74 3 0 5 0 0 15 PS5 1214 3186 M 11 76 3 0 21 0 0 15 PS21 2 F 1264 3159(L)U 1264 3051(L)U 1264 2943(L)U 1393 3159(L)U 1393 3051(L)U 1522 3159(L)U 0 F 1607 2922 M 11 77 3 0 22 0 0 15 PS22 1607 2988 M 11 74 3 0 5 0 0 15 PS5 1607 3054 M 11 74 3 0 5 0 0 15 PS5 1607 3120 M 11 74 3 0 5 0 0 15 PS5 1607 3186 M 11 79 3 0 23 0 0 15 PS23 1657 2922 M 11 73 3 0 20 0 0 15 PS20 1657 2988 M 11 74 3 0 5 0 0 15 PS5 1657 3054 M 11 74 3 0 5 0 0 15 PS5 1657 3120 M 11 74 3 0 5 0 0 15 PS5 1657 3186 M 11 76 3 0 21 0 0 15 PS21 2 F 1707 2943(U)U 1847 3051(U)U 1847 2943(U)U 1987 3159(U)U 1987 3051(U)U 1987 2943(U)U 0 F 2083 2922 M 11 77 3 0 22 0 0 15 PS22 2083 2988 M 11 74 3 0 5 0 0 15 PS5 2083 3054 M 11 74 3 0 5 0 0 15 PS5 2083 3120 M 11 74 3 0 5 0 0 15 PS5 2083 3186 M 11 79 3 0 23 0 0 15 PS23 4 F 2133 3051(=)U 1 F 48 Z 2035 2958(13)U 2059 3066(3)U 2035(2)S 3174(3)T 1895 2958(12)U 2035 3174(3)U 1919 3066(2)U 1430(22)S 1559 3174(33)U 1755 2958(11)U 1895 3066(2)U 1454 3174(2)U 1301 2958(11)U 1430 3174(3)U 1325 3066(1)U 1301(2)S 3174(1)T 1057 2958(13)U 1301 3174(3)U 1081 3066(3)U 1057(2)S 3174(3)T 925 2958(12)U 1057 3174(3)U 949 3066(2)U 925(2)S 3174(2)T 793 2958(11)U 925 3174(3)U 817 3066(1)U 793(2)S 3174(1)T 790 3678(3)U 793 3174(3)U 814 3678(1)U 886(11)S 790 3570(2)U (1)R 886(11)S 790 3462(1)U (1)R 886(11)S 1015 3678(31)U 1111(12)S 1277(32)S 1373(22)S 1015 3570(2)U (1)R 1111(12)S 1277(22)S 1373(22)S 1146 3462(1)U (1)R 1242(12)S 1502 3678(31)U 1598(13)S 1764(32)S 1860(23)S 2026(33)S 2122(33)S 1633 3570(2)U (1)R 1729(13)S 1895(22)S 1991(23)S 1764 3462(1)U (1)R 1860(13)S 0 F 66 Z 2170 3690 M 11 79 3 0 23 0 0 15 PS23 2170 3624 M 11 74 3 0 5 0 0 15 PS5 2170 3558 M 11 74 3 0 5 0 0 15 PS5 2170 3492 M 11 74 3 0 5 0 0 15 PS5 2170 3426 M 11 77 3 0 22 0 0 15 PS22 2 F 1727 3447(L)U 1812(U)S 1943 3555(U)U 1596(L)S 1681(U)S 4 F 1799(+)S 2 F 1858(L)S 2074 3663(U)U 1109 3447(L)U 1194(U)S 1465 3663(L)U 1550(U)S 4 F 1668(+)S 2 F 1727(L)S 1812(U)S 4 F 1930(+)S 2 F 1989(L)S 1325 3555(U)U 978(L)S 1063(U)S 4 F 1181(+)S 2 F 1240(L)S 1325 3663(U)U 753 3447(L)U 838(U)S 978 3663(L)U 1063(U)S 4 F 1181(+)S 2 F 1240(L)S 838 3555(U)U 753(L)S 838 3663(U)U 1 F 448 3813(This)U 587(gives)S 0 F 703 3426 M 11 73 3 0 20 0 0 15 PS20 703 3492 M 11 74 3 0 5 0 0 15 PS5 703 3558 M 11 74 3 0 5 0 0 15 PS5 703 3624 M 11 74 3 0 5 0 0 15 PS5 703 3690 M 11 76 3 0 21 0 0 15 PS21 2 F 753 3663(L)U 703 3945(A)U 4 F 813(=)S 2 F 872(L)S 957(U)S 1 F 48 Z 1029 3960(2)U 743 4095(2)U 743 3960(12)U 909(11)S 1005(1)S 767 4095(2)U 909(21)S 1005(12)S 1171(22)S 1267(22)S 2 F 66 Z 703 4215(A)U 703 4080(A)U 4 F 813(=)S 2 F 872(L)S 957(U)S 4 F 1075(+)S 2 F 1134(L)S 1219(U)S 4 F 813 4215(=)U 2 F 872(L)S 957(U)S 4 F 1075(+)S 2 F 1134(L)S 1219(U)S 1 F 48 Z 1291 4230(2)U 60 Z 16 4710(-)U 48 Z 743 4230(32)U 909(31)S 1005(12)S 1171(32)S 1267(2)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 18 BP 1 F 66 Z 1775 270(-)U 1819(16)S 1907(-)S 3 F 448 462(8.2)U 553(Cholesky)S 836(Factorization)S 1 F 598 594(We)U 723(assume)S 951(that)S 1083(we)S 1193(are)S 1306(given)S 1485(a)S 1547(positive-de\256nite)S 2007(symmetric)S 2317(matrix)S 2 F 2521(A)S 1 F 2594(whose)S 2796(lower)S 2979(triangle)S 3212(is)S 448 786(e)U 448 690(stored)U 634(in)S 710(the)S 814(lower)S 988(triangle)S 1212(of)S 1291(a)S 1344(2-dimensional)S 1744(array,)S 1920(and)S 2039(we)S 2140(wish)S 2289(to)S 2364(compute)S 2 F 2614(L)S 1 F 2675(overwriting)S 3006(the)S 3110(given)S 477 786(lements)U 703(of)S 2 F 780(A)S 1 F (.)R 598 918(W)U (e)R 711(can)S 824(partition)S 1068(the)S 1170(matrices)S 1414(so)S 1495(that)S 4 F 2100 1173(=)U 0 F 703 1035 M 11 73 3 0 20 0 0 15 PS20 703 1101 M 11 74 3 0 5 0 0 15 PS5 703 1167 M 11 74 3 0 5 0 0 15 PS5 703 1233 M 11 74 3 0 5 0 0 15 PS5 703 1299 M 11 76 3 0 21 0 0 15 PS21 2 F 753 1281(A)U 753 1173(A)U 753 1056(A)U 885 1281(A)U 885 1173(A)U 885 1056(A)U 1017 1281(A)U 1017 1173(A)U 1017 1056(A)U 0 F 1105 1035 M 11 77 3 0 22 0 0 15 PS22 1105 1101 M 11 74 3 0 5 0 0 15 PS5 1105 1167 M 11 74 3 0 5 0 0 15 PS5 1105 1233 M 11 74 3 0 5 0 0 15 PS5 1105 1299 M 11 79 3 0 23 0 0 15 PS23 4 F 1155 1173(=)U 0 F 1214 1044 M 11 73 3 0 20 0 0 15 PS20 1214 1110 M 11 74 3 0 5 0 0 15 PS5 1214 1176 M 11 74 3 0 5 0 0 15 PS5 1214 1242 M 11 74 3 0 5 0 0 15 PS5 1214 1308 M 11 76 3 0 21 0 0 15 PS21 2 F 1264 1281(L)U 1264 1173(L)U 1264 1065(L)U 1393 1281(L)U 1393 1173(L)U 1522 1281(L)U 0 F 1607 1044 M 11 77 3 0 22 0 0 15 PS22 1607 1110 M 11 74 3 0 5 0 0 15 PS5 1607 1176 M 11 74 3 0 5 0 0 15 PS5 1607 1242 M 11 74 3 0 5 0 0 15 PS5 1607 1308 M 11 79 3 0 23 0 0 15 PS23 1657 1038 M 11 73 3 0 20 0 0 15 PS20 1657 1104 M 11 74 3 0 5 0 0 15 PS5 1657 1170 M 11 74 3 0 5 0 0 15 PS5 1657 1236 M 11 74 3 0 5 0 0 15 PS5 1657 1302 M 11 76 3 0 21 0 0 15 PS21 2 F 1707 1056(L)U 1836 1173(L)U 1836 1056(L)U 1965 1290(L)U 1965 1173(L)U 1965 1056(L)U 0 F 2050 1038 M 11 77 3 0 22 0 0 15 PS22 2050 1104 M 11 74 3 0 5 0 0 15 PS5 2050 1170 M 11 74 3 0 5 0 0 15 PS5 2050 1236 M 11 74 3 0 5 0 0 15 PS5 2050 1302 M 11 79 3 0 23 0 0 15 PS23 1 F 48 Z 793 1296(31)U 793 1188(2)U (1)R 793 1071(1)U (1)R 925 1296(32)U 925 1188(2)U (2)R 925 1071(2)U (1)R 2 F 925 1029(T)U 1 F 1057 1296(33)U 1057 1188(3)U (2)R 1057 1071(3)U 2 F 1057 1146(T)U 1 F 1081 1071(1)U 2 F 1057 1029(T)U 1 F 1301 1296(31)U 1301 1188(2)U (1)R 1301 1080(1)U (1)R 1430 1296(32)U 1430 1188(2)U (2)R 1559 1296(33)U 1744 1071(11)U 2 F 1744 1029(T)U 1 F 1873 1188(22)U 1873 1071(2)U 2 F 1873 1146(T)U 1 F 1897 1071(1)U 2 F 1873 1029(T)U 1 F 2002 1305(33)U 2002 1188(3)U 2 F 2002 1263(T)U 1 F 2026 1188(2)U 2002 1071(3)U 2 F 2002 1146(T)U 1 F 2026 1071(1)U 2 F 1805 1632(T)U 2002 1029(T)U 1 F 1829 1674(1)U 1720(11)S 1805(3)S 2 F 1930 1749(T)U 1 F 1954 1791(2)U 2 F 1679 1749(T)U 1 F 1845 1791(22)U 1930(3)S 1703(1)S 1594(21)S 1679(3)S 2 F 2056 1866(T)U 1 F 2080 1908(3)U 2 F 1805 1866(T)U 1 F 1971 1908(33)U 2056(3)S 1829(2)S 2 F 1554 1866(T)U 1 F 1720 1908(32)U 1805(3)S 1578(1)S 2 F 1214 1632(T)U 1 F 1469 1908(31)U 1554(3)S 1238 1674(1)U 1129(11)S 1214(2)S 2 F 1340 1749(T)U 1 F 1364 1791(2)U 2 F 1089 1749(T)U 1 F 1255 1791(22)U 1340(2)S 1113(1)S 1004(21)S 1089(2)S 2 F 1340 1866(T)U 1 F 1364 1908(2)U 2 F 1089 1866(T)U 1 F 1255 1908(32)U 1340(2)S 1113(1)S 2 F 875 1632(T)U 1 F 1004 1908(31)U 1089(2)S 899 1674(1)U 790(11)S 875(1)S 2 F 875 1749(T)U 1 F 899 1791(1)U 790(21)S 875(1)S 2 F 875 1866(T)U 1 F 899 1908(1)U 0 F 66 Z 703 1905 M 11 76 3 0 21 0 0 15 PS21 703 1839 M 11 74 3 0 5 0 0 15 PS5 703 1773 M 11 74 3 0 5 0 0 15 PS5 703 1707 M 11 74 3 0 5 0 0 15 PS5 703 1641 M 11 73 3 0 20 0 0 15 PS20 1 F 48 Z 790 1908(31)U 875(1)S 2 F 66 Z 753 1893(L)U 838(L)S 753 1776(L)U 838(L)S 753 1659(L)U 838(L)S 967 1893(L)U 1052(L)S 4 F 1159(+)S 2 F 1218(L)S 1303(L)S 967 1776(L)U 1052(L)S 4 F 1159(+)S 2 F 1218(L)S 1303(L)S 1092 1659(L)U 1177(L)S 1432 1893(L)U 1517(L)S 4 F 1624(+)S 2 F 1683(L)S 1768(L)S 4 F 1875(+)S 2 F 1934(L)S 2019(L)S 1557 1776(L)U 1642(L)S 4 F 1749(+)S 2 F 1808(L)S 1893(L)S 1683 1659(L)U 1768(L)S 0 F 2104 1641 M 11 77 3 0 22 0 0 15 PS22 2104 1707 M 11 74 3 0 5 0 0 15 PS5 2104 1773 M 11 74 3 0 5 0 0 15 PS5 2104 1839 M 11 74 3 0 5 0 0 15 PS5 2104 1905 M 11 79 3 0 23 0 0 15 PS23 1 F 3239 2031(,)U 448(Here)S 2 F 599(L)S 1 F 707(and)S 2 F 825(L)S 1 F 933(constitute)S 1211(the)S 1313(current)S 1521(block)S 1689(of)S 1766(columns)S 2011(of)S 2 F 2088(L)S 1 F 2147(to)S 2220(be)S 2304(computed,)S 2602(and)S 2719(we)S 2818(assume)S 3034(that)S 2 F 3154(L)S 1 F 48 Z 636 2046(22)U 862(32)S 3191(11)S 2 F 66 Z 448 2130(L)U 1 F 48 Z 485 2145(21)U 735(31)S 66 Z 533 2130(,)U 577(and)S 2 F 698(L)S 1 F 809(constitute)S 1090(the)S 1196(blocks,)S 1411(if)S 1477(any,)S 1615(which)S 1802(have)S 1952(already)S 2171(been)S 2321(computed.)S 2623(Note)S 2777(that)S 2901(the)S 3007(blocks)S 3205(in)S 448 2229(the)U 550(above)S 729(partitioning)S 1057(are)S 1159(not)S 1265(all)S 1352(of)S 1429(equal)S 1593(size:)S 1735(the)S 1837(off-diagonal)S 2184(blocks)S 2378(are)S 2480(in)S 2553(general)S 2768(rectangular.)S 598 2361(Equating)U 857(blocks,)S 1068(we)S 1167(have:)S 2 F 703 2493(A)U 4 F 813(=)S 2 F 872(L)S 957(L)S 4 F 1064(+)S 2 F 1123(L)S 1208(L)S 1 F 48 Z 743 2508(22)U 909(21)S 994(21)S 2 F 994 2466(T)U 1 F 1160 2508(22)U 1245(22)S 2 F 1245 2601(T)U 1245 2466(T)U 1 F 1269 2643(2)U 2 F 994 2601(T)U 1 F 1160 2643(32)U 1245(2)S 1018(1)S 2 F 66 Z 703 2628(A)U 1 F 48 Z 743 2643(32)U 909(31)S 994(2)S 4 F 66 Z 813 2628(=)U 2 F 872(L)S 957(L)S 4 F 1064(+)S 2 F 1123(L)S 1208(L)S 1 F 448 2763(so)U 529(that)S 2 F 703 2895(L)U 788(L)S 4 F 895(=)S 2 F 954(A)S 4 F 1064(-)S 2 F 1123(L)S 1208(L)S 48 Z 1245 2868(T)U 1 F 1269 2910(1)U 2 F 825 2868(T)U 1 F 994 2910(22)U 1160(21)S 1245(2)S 849(2)S 740 3045(3)U 740 2910(22)U 825(2)S 764 3045(2)U 931(32)S 1097(31)S 1182(21)S 2 F 1182 3003(T)U 1 F 1311 3045(22)U 2 F 1311 3003(T)U 4 F 1381(-)S 1 F (1)R 66 Z 448 3165(T)U 2 F 703 3030(L)U 4 F 810(=)S 1 F 869(\()S 2 F (A)R 4 F 1001(-)S 2 F 1060(L)S 1145(L)S 1 F 1230(\)\()S 2 F (L)R 1 F 1359(\))S 488 3165(hus)U 602(the)S 704(computation)S 1054(of)S 1131(one)S 1248(block-column)S 1635(of)S 1712(the)S 1814(result)S 1982(involves)S 2227(the)S 2329(following)S 2607(operations:)S 448 3357(1.)U 520(update)S 717(the)S 819(diagonal)S 1067(block:)S 2 F 703 3489(A)U 4 F 813(\254)S 2 F 901(A)S 4 F 1011(-)S 2 F 1070(L)S 1155(L)S 48 Z 1192 3462(T)U 1 F 1216 3504(1)U 4 F 743 3462(\242)U 1 F 941 3504(22)U 1107(21)S 1192(2)S 767(2)S 66 Z 448 3720(2)U 48 Z 743 3504(2)U 66 Z 481 3720(.)U 520(compute)S 768(the)S 870(Cholesky)S 1141(factorizat)S 1388(ion)S 1494(of)S 1571(the)S 1673(diagonal)S 1921(block:)S 2 F 703 3852(A)U 4 F 813(\256)S 2 F 901(L)S 986(L)S 1 F 48 Z 743 3867(22)U 4 F 743 3825(\242)U 1 F 938 3867(22)U 1023(22)S 2 F 1023 3825(T)U 1 F 66 Z 1305 4083(:)U 448(3.)S 520(update)S 717(the)S 819(subdiagonal)S 1159(block)S 2 F 703 4215(A)U 4 F 813(\254)S 2 F 901(A)S 4 F 1011(-)S 2 F 1070(L)S 1155(L)S 48 Z 1192 4188(T)U 1 F 1216 4230(1)U 4 F 743 4188(\242)U 1 F 941 4230(32)U 1107(31)S 1192(2)S 767(2)S 60 Z 16 4710(-)U 48 Z 743 4230(3)U 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 17 BP 1 F 66 Z 448 462(w)U 1775 270(-)U 1819(15)S 1907(-)S 496 462(hich)U 644(potentiall)S 891(y)S 959(high)S 1111(performance)S 1478(can)S 1604(easily)S 1791(be)S 1887(degraded)S 2162(by)S 2262(excessive)S 2548(transfer)S 2783(of)S 2872(data)S 3037(between)S 3234 558(\))U 448 654(\()U 448 558(different)U 703(levels)S 885(of)S 969(memory)S 1216(\(vector)S 1430(registers,)S 1698(cache,)S 1892(local)S 2047(memory,)S 2311(main)S 2470(memory)S 2717(or)S 2800(solid-state)S 3098(disks)S 470 654(Dongarra)U 755(and)S 886(Hewitt,)S 1118(1986\),)S 1336(\(Berry)S 2 F 1544(et)S 1627(al.)S 1 F (,)R 1748(1986\),)S 1966(\(IBM,)S 2165(1986\),)S 2382(\(Robert)S 2618(and)S 2748(Sguazzero,)S 3074(1987\),)S 3228 750(,)U 448 846(\()U 448 750(\(Bischof)U 702(and)S 824(Van)S 961(Loan,)S 1140(1987\),)S 1349(\(Bucher)S 1588(and)S 1710(Jordan,)S 1929(1984\),)S 2126(and)S 2269(\(Schreiber,)S 2586(1986\),)S 2794(\(Calahan,)S 3074(1986\))S 470 846(Dongarra)U 741(and)S 858(Sorensen,)S 1139(1986\).)S 598 978(H)U (ere)R 752(we)S 855(illustrate)S 1110(how)S 1250(the)S 1356(proposed)S 1624(Level)S 1799(3)S 1858(BLAS)S 2053(routines)S 2291(can)S 2408(be)S 2495(used)S 2641(to)S 2717(implement)S 3022(two)S 3146(fun-)S 3234 1074(r)U 448 1170(w)U 448 1074(damental)U 713(algorithms)S 1019(of)S 1099(numerical)S 1385(linear)S 1558(algebra:)S 1793(Cholesky)S 2066(factorizat)S 2313(ion,)S 2438(and)S 2557(LU)S 2669(factorizat)S 2916(ion,)S 3041(togethe)S 496 1170(ith)U 587(a)S 638(simple)S 835(illustration)S 1141(of)S 1218(use)S 1328(in)S 1401(a)S 1452(high)S 1591(level)S 1740(matrix-mult)S 2053(iply)S 2177(routine.)S 3234 1302(-)U 448 1398(s)U 598 1302(For)U 713(the)S 816(factorizat)S 1063(ion)S 1170(routines)S 1405(the)S 1508(strategy)S 1738(in)S 1811(each)S 1953(case)S 2088(is)S 2154(to)S 2227(compute)S 2475(at)S 2544(each)S 2686(stage)S 2843(a)S 2894(block)S 3062(of)S 3139(con)S 474 1398(ecutive)U 690(columns)S 940(of)S 1022(the)S 1129(result.)S 1319(The)S 1448(size)S 1577(of)S 1659(the)S 1766(block)S 1939(is)S 2010(a)S 2066(parameter,)S 2 F 2371(nb)S 1 F (,)R 2480(which)S 2667(may)S 2806(be)S 2894(varied)S 3084(to)S 3161(suit)S 448 1590(m)U 448 1494(the)U 552(size)S 678(of)S 757(the)S 861(problem)S 1104(and)S 1222(the)S 1325(architect)S 1550(ure)S 1657(of)S 1735(the)S 1838(machine.)S 2100(\(For)S 2237(transportable)S 2603(software,)S 2870(we)S 2970(need)S 3117(some)S 499 1590(eans)U 644(of)S 727(determining)S 2 F 1072(nb)S 1 F 1166(within)S 1362(the)S 1470(routine,)S 1701(rather)S 1882(than)S 2023(requiring)S 2292(it)S 2356(to)S 2435(be)S 2525(passed)S 2728(as)S 2810(a)S 2866(parameter,)S 3172(but)S 448 1686(we)U 547(set)S 642(that)S 762(issue)S 916(aside)S 1073(here.\))S 598 1818(There)U 776(are)S 881(other)S 1041(ways)S 1202(to)S 1278(organize)S 1529(the)S 1634(computation:)S 2005(for)S 2107(example,)S 2370(it)S 2430(is)S 2498(equally)S 2715(possible)S 2955(to)S 3030(compute)S 3227 1914(e)U 448 2010(e)U 448 1914(a)U 509(block)S 687(of)S 774(consecutive)S 1116(rows)S 1277(at)S 1356(each)S 1508(stage.)S 1692(The)S 1826(analysis)S 2070(of)S 2201(\(Dongarra)S 2 F 2504(et)S 2583(al.)S 1 F (,)R 2700(1984\))S 2886(can)S 3009(easily)S 3194(b)S 477 2010(xtended)U 722(to)S 810(algorithms)S 1128(which)S 1326(work)S 1499(by)S 1602(blocks.)S 1827(We)S 1954(have)S 2114(chosen)S 2333(an)S 2431(organization)S 2795(which)S 2992(works)S 3190(by)S 448 2106(columns)U 693(rather)S 868(than)S 1003(rows,)S 1171(and)S 1288(which)S 1471(involves)S 1716(fewest)S 1910(memory)S 2151(references.)S 598 2238(Also,)U 765(we)S 867(have)S 1016(implement)S 1296(ed)S 1383(the)S 1488(algorithms)S 1794(in)S 1870(such)S 2016(a)S 2070(way)S 2205(that)S 2328(submatrices)S 2667(passed)S 2868(to)S 2944(the)S 3049(Level)S 3223(3)S 3238 2334(t)U 448 2430(s)U 448 2334(BLAS)U 641(routines)S 877(are)S 981(kept)S 1118(as)S 1197(large)S 1352(as)S 1431(possible)S 1671(\(once)S 1841(the)S 1945(parameter)S 2 F 2230(nb)S 1 F 2319(has)S 2430(been)S 2577(\256xed\):)S 2772(this)S 2890(gives)S 3052(greates)S 474 2430(cope)U 627(for)S 733(achieving)S 1016(ef\256ciency)S 1303(within)S 1499(the)S 1607(Level)S 1784(3)S 1845(BLAS.)S 2081(Alternativel)S 2394(y)S 2455(one)S 2578(might)S 2759(explicitl)S 2973(y)S 3034(partition)S 3223 2526(3)U 448 2622(B)U 448 2526(the)U 555(matrix)S 753(into,)S 899(say,)S 1031(square)S 1230(blocks)S 1429(of)S 1511(size)S 2 F 1640(nb)S 4 F (\264)R 2 F (nb)R 1 F (:)R 1854(this)S 1976(would)S 2168(require)S 2381(many)S 2554(more)S 2716(calls)S 2863(to)S 2941(the)S 3048(Level)S 492 2622(LAS)U 639(routines,)S 890(but)S 996(might)S 1171(allow)S 1339(a)S 1390(more)S 1547(precise)S 1755(control)S 1963(of)S 2040(memory)S 2281(or)S 2358(of)S 2435(paralleli)S 2649(sm.)S 2880 2754(.)U 3 F 448 2946(8)U 1 F 598 2754(A)U 668(similar)S 872(strategy)S 1102(is)S 1168(used)S 1311(for)S 1410(the)S 1512(matrix)S 1705(multiplic)S 1941(ation)S 2094(and)S 2211(we)S 2310(present)S 2522(that)S 2642(case)S 2777(\256rst)S 3 F 481 2946(.1)U 553(Matrix)S 772(Multiply)S 1 F 598 3078(G)U (iven)R 786(matrices)S 2 F 1035(A)S 1 F 1102(and)S 2 F 1224(B)S 1 F 1291(we)S 1395(wish)S 1547(to)S 1625(compute)S 1878(the)S 1985(product)S 2 F 2213(C)S 4 F 2279(\254)S 2 F 2367(A*B)S 1 F (.)R 2546(The)S 2675(matrices)S 2 F 2924(A)S 1 F (,)R 2 F 3008(B)S 1 F (,)R 3091(and)S 2 F 3212(C)S 1 F 448 3174(are)U 550(of)S 627(dimension)S 2 F 923(m)S 4 F (\264)R 2 F (k)R 1 F (,)R 2 F 1076(k)S 4 F (\264)R 2 F (n)R 1 F 1197(and)S 2 F 1314(m)S 4 F (\264)R 2 F (n)R 1 F 1454(respectively.)S 1832(We)S 1945(can)S 2058(partition)S 2302(the)S 2404(matrices)S 2648(so)S 2729(that)S 0 F 703 3288 M 11 73 3 0 20 0 0 15 PS20 703 3354 M 11 74 3 0 5 0 0 15 PS5 703 3420 M 11 74 3 0 5 0 0 15 PS5 703 3486 M 11 74 3 0 5 0 0 15 PS5 703 3552 M 11 74 3 0 5 0 0 15 PS5 703 3618 M 11 76 3 0 21 0 0 15 PS21 2 F 929 3606(C)U 929 3513(.)U 929 3405(C)U 929 3297(C)U 0 F 1173 3288 M 11 77 3 0 22 0 0 15 PS22 1173 3354 M 11 74 3 0 5 0 0 15 PS5 1173 3420 M 11 74 3 0 5 0 0 15 PS5 1173 3486 M 11 74 3 0 5 0 0 15 PS5 1173 3552 M 11 74 3 0 5 0 0 15 PS5 1173 3618 M 11 79 3 0 23 0 0 15 PS23 4 F 1223 3480(=)U 0 F 1282 3288 M 11 73 3 0 20 0 0 15 PS20 1282 3354 M 11 74 3 0 5 0 0 15 PS5 1282 3420 M 11 74 3 0 5 0 0 15 PS5 1282 3486 M 11 74 3 0 5 0 0 15 PS5 1282 3552 M 11 74 3 0 5 0 0 15 PS5 1282 3618 M 11 76 3 0 21 0 0 15 PS21 2 F 1508 3480(A)U 0 F 1750 3288 M 11 77 3 0 22 0 0 15 PS22 1750 3354 M 11 74 3 0 5 0 0 15 PS5 1750 3420 M 11 74 3 0 5 0 0 15 PS5 1750 3486 M 11 74 3 0 5 0 0 15 PS5 1750 3552 M 11 74 3 0 5 0 0 15 PS5 1750 3618 M 11 79 3 0 23 0 0 15 PS23 1800 3288 M 11 73 3 0 20 0 0 15 PS20 1800 3354 M 11 74 3 0 5 0 0 15 PS5 1800 3420 M 11 74 3 0 5 0 0 15 PS5 1800 3486 M 11 74 3 0 5 0 0 15 PS5 1800 3552 M 11 74 3 0 5 0 0 15 PS5 1800 3618 M 11 76 3 0 21 0 0 15 PS21 2 F 2026 3606(B)U 2026 3513(.)U 2026 3405(B)U 2026 3297(B)U 0 F 2266 3288 M 11 77 3 0 22 0 0 15 PS22 2266 3354 M 11 74 3 0 5 0 0 15 PS5 2266 3420 M 11 74 3 0 5 0 0 15 PS5 2266 3486 M 11 74 3 0 5 0 0 15 PS5 2266 3552 M 11 74 3 0 5 0 0 15 PS5 2266 3618 M 11 79 3 0 23 0 0 15 PS23 2 F 2294 3480(.)U 48 Z 973 3621(p)U 1 F 973 3420(2)U 973 3312(1)U 2 F 1548 3495(ij)U 2066 3621(p)U 1 F 2066 3420(2)U 2066 3312(1)U 2 F 1329 3759(ij)U 1434(j)S 1647(i)S 1 F 66 Z 448 3744(T)U (he)R 573(partitions)S 844(for)S 943(the)S 1045(matrices)S 2 F 1289(A)S 1 F 1355(,)S 2 F 1394(B)S 1 F 1447(,)S 1486(and)S 2 F 1603(C)S 1 F 1682(are)S 1784(of)S 1861(dimension)S 2 F 2157(nr)S 4 F (\264)R 2 F (nc)R 1 F (,)R 2 F 2354(nc)S 4 F (\264)R 2 F (n)R 1 F 2508(and)S 2 F 2625(nr)S 4 F (\264)R 2 F (n)R 1 F 2776(respectively.)S 3154(The)S 448 3843(update)U 645(of)S 722(one)S 839(block)S 1007(row)S 1132(of)S 1209(the)S 1311(matrix)S 2 F 1504(C)S 1 F 1570(follows)S 1790(by)S 1878(computing)S 2 F 703 3975(C)U 4 F 782(\254)S 2 F 870(C)S 4 F 949(+)S 2 F 1008(A)S 1074(B)S 1127(.)S 1 F 448 4110(T)U 2 F 48 Z 747 3990(i)U 914(i)S 1048(ij)S 1114(j)S 1 F 66 Z 488 4110(his)U 587(algorithm)S 864(is)S 930(implement)S 1210(ed)S 1294(in)S 1367(the)S 1469(subroutine)S 1769(DMM)S 1957(in)S 2030(Appendix)S 2312(A.)S 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 16 BP 1 F 66 Z 1775 270(-)U 1819(14)S 1907(-)S 3223 462(h)U 448 558(l)U 448 462(likely)U 623(to)S 700(be)S 788(so)S 873(useful)S 1060(in)S 1137(the)S 1243(type)S 1382(of)S 1463(applicati)S 1688(on)S 1780(we)S 1883(are)S 1989(aiming)S 2197(at.)S 2286(Also)S 2436(such)S 2582(storage)S 2797(is)S 2866(required)S 3110(muc)S 466 558(ess)U 571(with)S 712(large)S 866(memory)S 1108(machines)S 1379(available)S 1638(today)S 1807(and,)S 1942(in)S 2016(addition,)S 2271(we)S 2371(also)S 2500(wish)S 2648(to)S 2722(keep)S 2869(the)S 2972(set)S 3068(of)S 3146(rou-)S 448 654(tines)U 594(as)S 671(small)S 835(as)S 912(possible.)S 598 786(Again)U 783(in)S 858(this)S 977(type)S 1114(of)S 1193(applicati)S 1418(on,)S 1524(we)S 1624(are)S 1727(not)S 1834(aware)S 2014(of)S 2092(a)S 2144(need)S 2291(for)S 2391(simple)S 2589(transposition)S 2952(of)S 3030(complex)S 3234 882(-)U 448(matrices;)S 711(this)S 828(contrasts)S 1084(with)S 1223(the)S 1325(Level)S 1496(2)S 1551(BLAS)S 1742(which)S 1925(provide)S 2148(for)S 2247(computing)S 2 F 2550(A)S 2617(x)S 1 F 2668(as)S 2745(a)S 2796(means)S 2986(of)S 3063(achiev)S 2 F 48 Z 2590 855(T)U 1 F 66 Z 448 978(i)U 2 F 48 Z 583 951(T)U 1 F 66 Z 466 978(ng)U 2 F 554(x)S 610(A)S 1 F (,)R 689(with)S 828(complex)S 2 F 1076(x)S 1 F 1127(and)S 2 F 1244(A)S 1 F (.)R 598 1110(W)U (e)R 713(also)S 843(have)S 991(not)S 1099(proposed)S 1365(a)S 1418(set)S 1515(of)S 1594(extended)S 1855(precision)S 2119(routines)S 2354(analogous)S 2644(to)S 2718(the)S 2821(ES)S 2921(and)S 3039(EC)S 3146(rou-)S 448 1206(tines)U 594(in)S 667(the)S 769(Level)S 940(2)S 995(BLAS,)S 1203(since)S 1360(this)S 1477(would)S 1664(require)S 1872(a)S 1923(2-dimensional)S 2321(array)S 2478(in)S 2551(extended)S 2810(precision.)S 598 1338(As)U 696(with)S 837(the)S 941(Level)S 1114(2)S 1171(BLAS)S 1364(no)S 1454(check)S 1631(has)S 1743(been)S 1891(included)S 2141(for)S 2242(singularity,)S 2564(or)S 2643(near)S 2780(singularity,)S 3102(in)S 3176(the)S 448 1530(w)U 448 1434(triangular)U 726(equation)S 975(solving)S 1192(routines.)S 1466(The)S 1591(requirements)S 1957(for)S 2057(such)S 2201(a)S 2252(test)S 2365(depend)S 2577(on)S 2665(the)S 2767(applicati)S 2992(on)S 3080(and)S 3197(so)S 496 1530(e)U 547(felt)S 656(that)S 776(this)S 893(should)S 1091(not)S 1197(be)S 1281(included,)S 1546(but)S 1652(should)S 1850(instead)S 2058(be)S 2142(performed)S 2438(outside)S 2650(the)S 2752(triangular)S 3029(solver.)S 3238 1662(;)U 448 1758(h)U 598 1662(We)U 717(have)S 869(tried)S 1017(to)S 1096(adhere)S 1299(to)S 1378(the)S 1486(conventions,)S 1849(and)S 1972(maintain)S 2229(consistency)S 2564(with,)S 2726(the)S 2833(Level)S 3009(2)S 3069(BLAS)S 481 1758(owever,)U 724(we)S 832(have)S 987(deliberate)S 1245(ly)S 1327(departed)S 1584(from)S 1743(this)S 1869(in)S 1951(a)S 2011(few)S 2141(cases.)S 2328(The)S 2461(input-output)S 2817(matrix)S 2 F 3019(C)S 1 F 3094(in)S 3176(the)S 3227 1854(e)U 2 F 448 1950(C)U 1 F 448 1854(matrix)U 644(multiply)S 891(routines)S 1128(is)S 1197(the)S 1301(analogue)S 1562(of)S 1641(the)S 1745(vector)S 2 F 1933(y)S 1 F 1986(in)S 2061(the)S 2165(matrix-vect)S 2467(or)S 2546(product)S 2771(routines.)S 3024(But)S 3143(her)S 519 1950(always)U 729(has)S 844(the)S 951(same)S 1113(dimensions,)S 1457(whereas)S 2 F 1700(y)S 1 F 1756(was)S 1886(either)S 2062(of)S 2144(length)S 2 F 2335(m)S 1 F 2410(or)S 2 F 2492(n)S 1 F 2551(depending)S 2851(on)S 2943(context.)S 3201(In)S 3223 2046(k)U 448 2142(u)U 448 2046(the)U 553(rank-)S 2 F (k)R 1 F 746(update)S 946(routines)S 1183(we)S 1285(have)S 1434(included)S 1685(a)S 1739(parameter)S 4 F 2026(b)S 1 F 2088(which)S 2274(was)S 2402(not)S 2511(present)S 2726(in)S 2802(the)S 2907(Level)S 3081(2)S 3139(ran)S 481 2142(pdate)U 646(routines.)S 920(Here)S 1071(we)S 1171(felt)S 1281(that)S 1402(the)S 1505(parameter)S 4 F 1790(b)S 1 F 1849(is)S 1915(useful)S 2098(in)S 2171(applicati)S 2396(ons,)S 2527(and)S 2644(since)S 2801(the)S 2903(matrix)S 3096(multi-)S 448 2334(r)U 448 2238(ply)U 555(routines)S 789(can)S 902(also)S 1030(be)S 1114(viewed)S 1326(as)S 1403(rank)S 1542(update)S 1739(routines,)S 1990(we)S 2089(have)S 2235(consistency)S 2564(between)S 2805(the)S 2907(MM)S 3047(and)S 3164(RK)S 470 2334(outines.)U 725(Finally,)S 954(as)S 1035(mentioned)S 1338(above,)S 1538(we)S 1641(have)S 1791(not)S 1901(allowed)S 2135(transposition)S 2501(in)S 2578(the)S 2684(complex)S 2936(case)S 3074(for)S 3176(the)S 448 2430(Level)U 619(3)S 674(BLAS.)S 598 2562(Finally)U 808(we)S 909(would)S 1098(welcome)S 1359(suggestions)S 1690(for)S 1790(a)S 1842(better)S 2014(way)S 2147(of)S 2225(distinguishing)S 2621(between)S 2863(the)S 2966(two)S 3088(opera-)S 3 F 448 2850(8)U 1 F 448 2658(tions)U 598(performed)S 894(by)S 982(SYRK/HERK)S 1379(than)S 1514(our)S 1624(current)S 1832(use)S 1942(of)S 2019(TRANSA.)S 3 F 481 2850(.)U 542(Applications)S 1 F 598 2982(T)U (he)R 725(primary)S 958(intended)S 1209(applicati)S 1434(on)S 1525(of)S 1605(the)S 1710(Level)S 1883(3)S 1940(BLAS)S 2133(is)S 2201(in)S 2276(implement)S 2556(ing)S 2664(algorithms)S 2969(of)S 3048(numeri-)S 448 3174(a)U 448 3078(cal)U 552(linear)S 729(algebra)S 950(in)S 1029(terms)S 1203(of)S 1286(operations)S 1588(on)S 1682(submatrices)S 2024(\(or)S 2129(blocks\).)S 2367(There)S 2547(is)S 2618(a)S 2674(long)S 2818(history)S 3028(of)S 3110(block)S 477 3174(lgorithms:)U 776(a)S 834(few)S 962(references)S 1261(are)S 1413(\(Barron)S 1646(and)S 1769(Swinnerton-Dyer,)S 2272(1960\),)S 2482(\(Chartres,)S 2772(1960\),)S 2982(\(McKellar)S 448 3366(N)U 448 3270(and)U 570(Coffman,)S 848(1969\),)S 1057(\(DuCroz)S 2 F 1315(et)S 1388(al.)S 1 F (,)R 1499(1981\),)S 1707(\(Calahan,)S 1987(1986\),)S 2184(and)S 2327(\(Dave)S 2514(and)S 2635(Duff,)S 2803(1986\).)S 3022(Both)S 3176(the)S 496 3366(AG)U 625(and)S 753(the)S 866(IMSL)S 1057(libraries)S 1305(include)S 1530(such)S 1683(algorithms)S 1996(\(F01BTF)S 2274(and)S 2401(F01BXF)S 2665(in)S 2748(NAG;)S 2942(LQEIF)S 3161(and)S 3223 3462(d)U 448 3558(b)U 448 3462(LQEOF)U 703(in)S 796(IMSL\).)S 1057(The)S 1200(earlier)S 1408(work)S 1585(was)S 1729(usually)S 1960(concerned)S 2271(with)S 2429(submatrices)S 2784(being)S 2971(transferre)S 481 3558(etween)U 691(the)S 794(main)S 948(memory)S 1190(and)S 1308(disk)S 1441(or)S 1519(tape.)S 1668(Similar)S 1884(concerns)S 2141(motivated)S 2426(work)S 2585(designed)S 2842(to)S 2916(exploit)S 3121(com-)S 3234 3654(-)U 448 3750(e)U 448 3654(mon)U 595(page-swapping)S 1023(algorithms)S 1333(in)S 1413(virtual)S 1613(memory)S 1861(machines.)S 2155(Indeed)S 2363(the)S 2472(techniques)S 2782(are)S 2891(similar)S 3102(wher)S 477 3750(ver)U 584(there)S 738(exists)S 911(any)S 1029(hierarchy)S 1300(of)S 1378(data)S 1510(storage)S 1723(\(in)S 1819(terms)S 1988(of)S 2065(access)S 2255(speed\).)S 2466(Additionally,)S 2837(full)S 2950(blocks,)S 3161(and)S 3227 3846(e)U 448 3942(s)U 448 3846(hence)U 632(the)S 743(multiplic)S 979(ation)S 1141(of)S 1227(full)S 1349(matrices,)S 1619(might)S 1803(appear)S 2009(as)S 2095(a)S 2155(subproblem)S 2497(when)S 2670(handling)S 2930(large)S 3091(spars)S 474 3942(ystems)U 689(of)S 776(equations)S 1060(\(for)S 1191(example,)S 1506(\(Duff,)S 1702(1981\),)S 1916(\(George)S 2164(and)S 2291(Rashwan,)S 2582(1985\),)S 2796(\(Dave)S 2988(and)S 3114(Duff,)S 448 4038(1986\)\).)U 598 4170(More)U 777(recently)S 1024(several)S 1246(workers)S 1495(have)S 1655(demonstrated)S 2045(the)S 2161(effectivene)S 2452(ss)S 2539(of)S 2629(block)S 2810(algorithms)S 3126(on)S 3227(a)S 60 Z 16 4710(-)U 66 Z 448 4266(variety)U 655(of)S 735(modern)S 961(computer)S 1234(architect)S 1459(ures,)S 1611(with)S 1753(vector-processing)S 2246(or)S 2325(parallel)S 2545(processing)S 2851(capabilit)S 3076(ies,)S 3190(on)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 15 BP 1 F 66 Z 1775 270(-)U 1819(13)S 1907(-)S 3 F 448 462(7.)U 542(Rationale)S 1 F 598 594(In)U 689(the)S 805(design)S 1013(of)S 1104(all)S 1205(levels)S 1394(of)S 1485(BLAS,)S 1707(one)S 1838(of)S 1929(the)S 2045(main)S 2212(concerns)S 2482(is)S 2562(to)S 2649(keep)S 2809(both)S 2962(the)S 3078(calling)S 3234 690(-)U 448 786(t)U 448 690(sequences)U 740(simple)S 940(and)S 1060(the)S 1165(range)S 1336(of)S 1416(options)S 1635(limited,)S 1862(while)S 2033(at)S 2105(the)S 2210(same)S 2370(time)S 2511(maintaini)S 2758(ng)S 2848(suf\256cient)S 3117(func)S 466 786(ionality.)U 719(This)S 872(clearly)S 1085(implies)S 1313(a)S 1377(compromise,)S 1754(and)S 1884(a)S 1948(good)S 2115(decision)S 2369(is)S 2448(vital)S 2599(if)S 2674(the)S 2789(BLAS)S 2993(are)S 3108(to)S 3194(be)S 448 978(w)U 448 882(accepted)U 700(as)S 778(a)S 830(useful)S 1014(standard.)S 1277(In)S 1354(this)S 1471(proposal,)S 1737(we)S 1836(have)S 1982(had)S 2099(to)S 2172(make)S 2336(several)S 2544(decisions)S 2811(of)S 2888(this)S 3005(kind,)S 3161(and)S 496 978(e)U 548(are)S 651(anxious)S 879(that)S 1000(our)S 1111(reasoning)S 1390(is)S 1457(questioned)S 1765(by)S 1854(as)S 1932(wide)S 2083(a)S 2135(community)S 2457(as)S 2535(possible.)S 2790(Therefore,)S 3088(in)S 3161(this)S 448 1170(d)U 448 1074(Section,)U 685(we)S 785(discuss)S 999(some)S 1161(of)S 1239(the)S 1342(functionalit)S 1644(y)S 1700(which)S 1884(we)S 1984(have)S 2131(omitted,)S 2371(and)S 2489(some)S 2651(which)S 2835(we)S 2935(were)S 3086(a)S 3137(little)S 481 1170(oubtful)U 698(about)S 871(including)S 1145(\(for)S 1270(example,)S 1535(features)S 1769(which)S 1956(were)S 2110(included)S 2362(without)S 2589(unanimous)S 2904(support)S 3128(from)S 448 1266(all)U 535(authors!\).)S 598 1398(As)U 700(with)S 845(the)S 953(previous)S 1208(BLAS,)S 1422(the)S 1530(three)S 1689(basic)S 1851(matrix-mat)S 2142(rix)S 2242(operations)S 2543(were)S 2698(chosen)S 2908(because)S 3143(they)S 3227 1494(e)U 448 1590(b)U 448 1494(occur)U 617(in)S 691(a)S 743(wide)S 894(range)S 1062(of)S 1139(linear)S 1310(algebra)S 1525(applicati)S 1750(ons.)S 1903(We)S 2016(have)S 2162(again)S 2326(aimed)S 2508(at)S 2577(a)S 2628(reasonable)S 2931(compromis)S 481 1590(etween)U 690(a)S 742(much)S 911(larger)S 1087(number)S 1311(of)S 1389(routines)S 1624(each)S 1767(performing)S 2085(only)S 2224(one)S 2341(type)S 2476(of)S 2553(operation)S 2823(\(e.g.)S 2 F 2963(B)S 4 F 3025(\254)S 2 F 3113(L)S 3177(B)S 1 F (\),)R 2 F 48 Z 3150 1563(T)U 1 F 66 Z 3223 1686(h)U 448 1782(d)U 448 1686(and)U 572(a)S 630(smaller)S 852(number)S 1082(of)S 1166(routines)S 1407(with)S 1553(a)S 1611(more)S 1775(complicat)S 2033(ed)S 2123(set)S 2224(of)S 2307(options.)S 2546(There)S 2727(are)S 2835(in)S 2914(fact,)S 3057(in)S 3136(eac)S 481 1782(ata)U 586(type,)S 745(5)S 807(routines)S 1048(performing)S 1373(altogether)S 1664(48)S 1759(different)S 2014(operations.)S 2356(The)S 2486(number)S 2715(of)S 2798(routines)S 3038(is)S 3110(much)S 448 1878(smaller)U 663(than)S 798(in)S 871(the)S 973(Level)S 1144(2)S 1199(BLAS.)S 598 2010(The)U 730(routines)S 972(that)S 1099(we)S 1205(have)S 1358(proposed)S 1629(are)S 1738(not)S 1851(intended)S 2106(as)S 2190(high)S 2336(level)S 2492(matrix)S 2692(algebra)S 2914(routines,)S 3172(but)S 448 2202(e)U 448 2106(rather)U 635(as)S 724(tools)S 886(for)S 996(the)S 1109(construction)S 1467(of)S 1555(such)S 1709(routines.)S 1971(To)S 2077(emphasize)S 2387(the)S 2500(point)S 2668(we)S 2778(have)S 2935(included)S 3194(an)S 477 2202(xample)U 693(of)S 771(a)S 823(high)S 963(level)S 1113(matrix)S 1307(multiply)S 1552(routine,)S 1778(in)S 1852(Appendix)S 2135(A,)S 2223(based)S 2396(upon)S 2551(the)S 2653(Level)S 2824(3)S 2879(BLAS)S 3070(routine)S 448 2298(DGEMM.)U 598 2430(In)U 680(each)S 827(case,)S 983(where)S 1170(appropriate,)S 1512(we)S 1615(include)S 1834(operations)S 2134(involving)S 2412(a)S 2467(matrix)S 2664(and)S 2785(its)S 2873(transpose.)S 3165(We)S 448 2622(i)U 448 2526(could)U 620(ask)S 734(the)S 840(user)S 976(to)S 1053(transpose)S 1328(the)S 1434(input)S 1595(matrix)S 1792(but)S 1901(feel)S 2024(that)S 2147(this)S 2267(would)S 2457(be)S 2544(an)S 2631(imposition,)S 2954(particularl)S 3223(y)S 466 2622(f)U 523(the)S 638(BLAS)S 842(routine)S 1063(is)S 1141(being)S 1321(called)S 1511(from)S 1673(deep)S 1831(within)S 2033(the)S 2147(user's)S 2339(code.)S 2514(It)S 2588(would)S 2787(also)S 2927(increase)S 3176(the)S 3234 2718(f)U 448 2814(s)U 448 2718(amount)U 675(of)S 760(data)S 899(movement,)S 1223(whereas)S 1469(one)S 1594(of)S 1679(the)S 1789(aims)S 1942(of)S 2026(our)S 2143(proposal)S 2399(is)S 2472(to)S 2552(assist)S 2724(the)S 2833(development)S 3201(o)S 474 2814(oftware)U 697(that)S 817(minimize)S 1064(s)S 1112(data)S 1243(movement.)S 598 2946(I)U (t)R 668(could)S 844(also)S 980(be)S 1072(argued)S 1281(that)S 1409(algorithms)S 1720(can)S 1841(be)S 1933(rewritten)S 2200(to)S 2281(require)S 2497(only)S 2644(one)S 2769(of)S 2854(the)S 2964(patterns)S 3201(of)S 3238 3042(l)U 448 3138(t)U 448 3042(access)U 639(for)S 739(symmetric,)S 1056(Hermitian)S 1345(or)S 1423(triangular)S 1701(matrices)S 1946(\(i.e.)S 2072(upper)S 2245(or)S 2323(lower)S 2496(triangle\),)S 2758(but)S 2865(we)S 2964(do)S 3052(not)S 3158(fee)S 466 3138(hat)U 568(the)S 670(BLAS)S 861(should)S 1059(be)S 1143(dictating)S 1394(this)S 1511(to)S 1584(the)S 1686(user.)S 598 3270(W)U (e)R 712(allow)S 881(pre-)S 1010(or)S 1088(post-multiplic)S 1456(ation)S 1610(of)S 1688(a)S 1740(general)S 1955(matrix)S 2148(by)S 2236(a)S 2287(symmetric)S 2586(or)S 2663(Hermitian)S 2951(matrix,)S 3161(and)S 3234 3366(-)U 448 3462(c)U 448 3366(can)U 568(devise)S 765(instances)S 1035(when)S 1207(it)S 1272(is)S 1345(required.)S 1610(However)S 1881(they)S 2023(do)S 2117(not)S 2229(occur)S 2403(frequently,)S 2718(and)S 2841(we)S 2946(would)S 3139(wel)S 477 3462(ome)U 612(comments)S 904(on)S 992(the)S 1094(utility)S 1272(of)S 1349(this)S 1466(operation.)S 598 3594(W)U (e)R 728(do)S 833(not)S 956(provide)S 1196(routines)S 1447(for)S 1563(operations)S 1875(involving)S 2165(trapezoidal)S 2494(matrices;)S 2772(all)S 2875(our)S 3001(triangular)S 3234 3690(-)U 448 3786(t)U 448 3690(matrices)U 694(are)S 798(square.)S 1011(This)S 1152(is)S 1220(consistent)S 1507(with)S 1648(the)S 1752(Level)S 1925(2)S 1982(BLAS.)S 2192(It)S 2256(would)S 2445(be)S 2531(possible)S 2771(to)S 2845(extend)S 3043(the)S 3146(rou)S 466 3786(ines)U 599(for)S 703(triangular)S 985(matrices,)S 1251(so)S 1337(that)S 1462(they)S 1602(could)S 1775(handle)S 1977(trapezoidal)S 2295(matrices,)S 2561(at)S 2634(the)S 2740(cost)S 2872(of)S 2953(introducing)S 3234 3882(r)U 448 3978(m)U 448 3882(extra)U 609(arguments.)S 930(On)S 1041(the)S 1151(other)S 1316(hand,)S 1491(a)S 1550(trapezoidal)S 1871(matrix)S 2072(can)S 2193(always)S 2406(be)S 2498(partitioned)S 2812(into)S 2943(a)S 3001(triangula)S 499 3978(atrix)U 641(and)S 758(a)S 809(rectangular)S 1126(matrix.)S 598 4110(W)U (e)R 713(have)S 861(not)S 969(proposed)S 1234(specialize)S 1492(d)S 1548(routines)S 1783(to)S 1857(take)S 1989(advantage)S 2278(of)S 2356(packed)S 2565(storage)S 2778(schemes)S 3024(for)S 3124(sym-)S 448 4302(s)U 448 4206(metric,)U 656(Hermitian,)S 962(or)S 1040(triangular)S 1318(matrices,)S 1580(nor)S 1691(of)S 1769(compact)S 2014(storage)S 2227(schemes)S 2473(for)S 2573(banded)S 2786(matrices,)S 3048(because)S 474 4302(uch)U 600(storage)S 821(schemes)S 1075(do)S 1172(not)S 1287(seem)S 1453(to)S 1535(lend)S 1679(themselves)S 2002(to)S 2084(partitioning)S 2420(into)S 2552(blocks)S 2754(and)S 2879(hence)S 3062(are)S 3172(not)S 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 14 BP 1 F 66 Z 448 462(For)U 562(complex)S 810(matrices)S 1775 270(-)U 1819(12)S 1907(-)S 1522 606(TRANSA)U 1809(=`N')S 2161(TRANSA)S 2448(=`C')S 1068 798(SIDE)U 1237(=`L')S 2 F 1461(B)S 4 F 1523(\254)S 2 F 1611(AB)S 2098(B)S 4 F 2160(\254)S 2 F 2248(A)S 4 F 2225 804(`)U 2226(`)S 2 F 2315 798(B)U 48 Z 2288 768(T)U 66 Z 2588 894(m)U 1 F 1068 1086(S)U 2 F 1461 894(A)U 1 F 1523(is)S 1589(triangular)S 2 F 1866(m)S 4 F (\264)R 2 F (m)R 2098(A)S 1 F 2160(is)S 2226(triangular)S 2 F 2503(m)S 4 F (\264)R 1 F 1105 1086(IDE)U 1237(=`R')S 2 F 1461(B)S 4 F 1523(\254)S 2 F 1611(BA)S 2098(B)S 4 F 2160(\254)S 2 F 2248(BA)S 4 F 2265 1092(`)U 2266(`)S 2 F 48 Z 2328 1056(T)U 66 Z 2573 1182(n)U 1461(A)S 1 F 1523(is)S 1589(triangular)S 2 F 1866(n)S 4 F (\264)R 2 F (n)R 2098(A)S 1 F 2160(is)S 2226(triangular)S 2 F 2503(n)S 4 F (\264)R 1 F 1702 1506(:)U 470(e\))S 543(Solution)S 788(of)S 865(triangular)S 1142(systems)S 1373(of)S 1450(equations)S 5 F 514 1698(_)U 1 F (TRSM)R 749(\(SIDE,)S 957(UPLO,)S 1169(TRANSA,)S 1473(DIAG,)S 1678(M,)S 1776(N,)S 1863(A,)S 1950(LDA,)S 2125(B,)S 2208(LDB\))S 448 2082(F)U 514 1890(Operation)U 799(\()S 2 F (B)R 1 F 883(is)S 949(always)S 2 F 1154(m)S 4 F (\264)R 2 F (n)R 1 F (,)R 2 F 1311(A)S 1 F 1373(is)S 1439(triangular\):)S 485 2082(or)U 562(real)S 682(matrices)S 1493 2226(TRANSA)U 1780(=`N')S 2069(TRANSA)S 2356(=`T')S 2499(or)S 2576(`C')S 1039 2418(SIDE)U 1208(=`L')S 2 F 1432(B)S 4 F 1494(\254)S 2 F 1582(A)S 1673(B)S 2069(B)S 4 F 2131(\254)S 2 F 2219(A)S 2313(B)S 4 F 48 Z 1622 2391(-)U 1 F (1)R 4 F 2259(-)S 2 F (T)R 66 Z 2559 2610(m)U 1 F 1039 2802(S)U 2 F 1432 2610(A)U 1 F 1494(is)S 1560(triangular)S 2 F 1837(m)S 4 F (\264)R 2 F (m)R 2069(A)S 1 F 2131(is)S 2197(triangular)S 2 F 2474(m)S 4 F (\264)R 1 F 1076 2802(IDE)U 1208(=`R')S 2 F 1432(B)S 4 F 1494(\254)S 2 F 1582(BA)S 2069(B)S 4 F 2131(\254)S 2 F 2219(BA)S 48 Z 2326 2775(T)U 66 Z 1432 2898(A)U 4 F 48 Z 1662 2775(-)U 1 F (1)R 4 F 2299(-)S 1 F 66 Z 1494 2898(is)U 1560(triangular)S 2 F 1837(n)S 4 F (\264)R 2 F (n)R 2069(A)S 1 F 2131(is)S 2197(triangular)S 2 F 2474(n)S 4 F (\264)R 2 F (n)R 1 F 448 3126(For)U 562(complex)S 810(matrices)S 1522 3270(TRANSA)U 1809(=`N')S 2161(TRANSA)S 2448(=`C')S 1068 3462(SIDE)U 1237(=`L')S 2 F 1461(B)S 4 F 1523(\254)S 2 F 1611(A)S 1702(B)S 2098(B)S 4 F 2160(\254)S 2 F 2248(A)S 4 F 2225 3468(`)U 2226(`)S 2 F 2342 3462(B)U 4 F 48 Z 1651 3435(-)U 1 F (1)R 4 F 2288 3432(-)U 2 F (T)R 66 Z 2588 3558(m)U 1 F 1068 3750(S)U 2 F 1461 3558(A)U 1 F 1523(is)S 1589(triangular)S 2 F 1866(m)S 4 F (\264)R 2 F (m)R 2098(A)S 1 F 2160(is)S 2226(triangular)S 2 F 2503(m)S 4 F (\264)R 1 F 1105 3750(IDE)U 1237(=`R')S 2 F 1461(B)S 4 F 1523(\254)S 2 F 1611(BA)S 2098(B)S 4 F 2160(\254)S 2 F 2248(BA)S 4 F 2265 3756(`)U 2266(`)S 2 F 48 Z 2355 3720(T)U 66 Z 1461 3846(A)U 4 F 48 Z 1691 3723(-)U 1 F (1)R 4 F 2328 3720(-)U 1 F 66 Z 1523 3846(is)U 1589(triangular)S 2 F 1866(n)S 4 F (\264)R 2 F (n)R 2098(A)S 1 F 2160(is)S 2226(triangular)S 2 F 2503(n)S 4 F (\264)R 2 F (n)R 1 F 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 13 BP 1 F 66 Z 1775 270(-)U 1819(11)S 1907(-)S 2309 462(:)U 470(c\))S 543(Rank-)S 2 F (k)R 1 F 755(updates)S 978(of)S 1055(a)S 1106(real)S 1226(symmetric)S 1525(or)S 1602(complex)S 1850(Hermitian)S 2138(matrix)S 5 F 514 654(_)U 1 F (SYRK)R 746(\(UPLO,)S 980(TRANSA,)S 1284(N,)S 1371(K,)S 1458(ALPHA,)S 1718(A,)S 1805(LDA,)S 1980(BETA,)S 2191(C,)S 2274(LDC\))S 2409 750(\))U 514 942(O)U 5 F 514 750(_)U 1 F (HERK)R 749(\(UPLO,)S 983(TRANSA,)S 1287(N,)S 1374(K,)S 1461(ALPHA,)S 1721(A,)S 1808(LDA,)S 1983(BETA,)S 2194(C,)S 2277(LDC)S 562 942(peration)U 799(\()S 2 F (C)R 1 F 887(is)S 953(always)S 2 F 1158(n)S 4 F (\264)R 2 F (n)R 1 F 1283(symmetric)S 1582(or)S 1659(Hermitian\):)S 448 1134(For)U 562(real)S 682(matrices)S 1312 1278(TRANSA)U 1599(=`N')S 1880(TRANSA)S 2167(=`T')S 2 F 2273 1374(C)U 1288(C)S 4 F 1354(\254)S 1442(a)S 2 F (AA)R 4 F 1613(+)S 1672(b)S 2 F (C)R 1852(C)S 4 F 1918(\254)S 2006(a)S 2 F (A)R 2115(A)S 4 F 2177(+)S 2236(b)S 2 F 48 Z 1564 1347(T)U 2088(T)S 1 F 66 Z 448 1698(For)U 562(complex)S 810(matrices)S 2 F 1288 1470(A)U 1 F 1350(is)S 2 F 1416(n)S 4 F (\264)R 2 F (k)R 1852(A)S 1 F 1914(is)S 2 F 1980(k)S 4 F (\264)R 2 F (n)R 1 F 1312 1842(TRANSA)U 1599(=`N')S 1878(TRANSA)S 2165(=`C')S 2 F 2273 1938(C)U 1288(C)S 4 F 1354(\254)S 1442(a)S 2 F (AA)R 4 F 1501 1944(`)U 1502(`)S 1613 1938(+)U 1672(b)S 2 F (C)R 1852(C)S 4 F 1918(\254)S 2006(a)S 2 F (A)R 4 F 2025 1944(`)U 2026(`)S 2 F 2115 1938(A)U 4 F 2177(+)S 2236(b)S 2 F 48 Z 1564 1908(T)U 2088(T)S 66 Z 1288 2034(A)U 1 F 1350(is)S 2 F 1416(n)S 4 F (\264)R 2 F (k)R 1852(A)S 1 F 1914(is)S 2 F 1980(k)S 4 F (\264)R 2 F (n)R 1 F 1459 2454(:)U 470(d\))S 547(Triangular)S 846(matrix-mat)S 1137(rix)S 1232(products)S 5 F 514 2646(_)U 1 F (TRMM)R 771(\(SIDE,)S 979(UPLO,)S 1191(TRANSA,)S 1495(DIAG,)S 1700(M,)S 1798(N,)S 1885(A,)S 1972(LDA,)S 2147(B,)S 2230(LDB\))S 448 3030(F)U 514 2838(Operation)U 799(\()S 2 F (B)R 1 F 883(is)S 949(always)S 2 F 1154(m)S 4 F (\264)R 2 F (n)R 1 F (,)R 2 F 1311(A)S 1 F 1373(is)S 1439(triangular\):)S 485 3030(or)U 562(real)S 682(matrices)S 1493 3174(TRANSA)U 1780(=`N')S 2069(TRANSA)S 2356(=`T')S 2499(or)S 2576(`C')S 1039 3366(SIDE)U 1208(=`L')S 2 F 1432(B)S 4 F 1494(\254)S 2 F 1582(AB)S 2069(B)S 4 F 2131(\254)S 2 F 2219(A)S 2286(B)S 48 Z 2259 3339(T)U 66 Z 2559 3462(m)U 1 F 1039 3654(S)U 2 F 1432 3462(A)U 1 F 1494(is)S 1560(triangular)S 2 F 1837(m)S 4 F (\264)R 2 F (m)R 2069(A)S 1 F 2131(is)S 2197(triangular)S 2 F 2474(m)S 4 F (\264)R 1 F 1076 3654(IDE)U 1208(=`R')S 2 F 1432(B)S 4 F 1494(\254)S 2 F 1582(BA)S 2069(B)S 4 F 2131(\254)S 2 F 2219(BA)S 48 Z 2299 3627(T)U 66 Z 2544 3750(n)U 1 F 60 Z 16 4710(-)U 2 F 66 Z 1432 3750(A)U 1 F 1494(is)S 1560(triangular)S 2 F 1837(n)S 4 F (\264)R 2 F (n)R 2069(A)S 1 F 2131(is)S 2197(triangular)S 2 F 2474(n)S 4 F (\264)R 1 F 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 12 BP 1 F 66 Z 1775 270(-)U 1819(10)S 1907(-)S 2759 462(:)U 470(b\))S 547(Matrix-matrix)S 941(products)S 1190(where)S 1373(one)S 1490(matrix)S 1683(is)S 1749(real)S 1869(symmetric)S 2168(or)S 2245(complex)S 2493(Hermitian)S 5 F 514 654(_)U 1 F (SYMM)R 772(\(SIDE,)S 980(UPLO,)S 1192(TRANSB,)S 1492(M,)S 1590(N,)S 1677(ALPHA,)S 1937(A,)S 2024(LDA,)S 2199(B,)S 2282(LDB,)S 2453(BETA,)S 2664(C,)S 2747(LDC\))S 2882 750(\))U 514 942(O)U 5 F 514 750(_)U 1 F (HEMM)R 775(\(SIDE,)S 983(UPLO,)S 1195(TRANSB,)S 1495(M,)S 1593(N,)S 1680(ALPHA,)S 1940(A,)S 2027(LDA,)S 2202(B,)S 2285(LDB,)S 2456(BETA,)S 2667(C,)S 2750(LDC)S 562 942(peration)U 799(\()S 2 F (C)R 1 F 887(is)S 953(always)S 2 F 1158(m)S 4 F (\264)R 2 F (n)R 1 F (\):)R 448 1134(For)U 562(real)S 682(matrices)S 1497 1278(TRANSB)U 1780(=`N')S 2082(TRANSB)S 2365(=`T')S 2508(or)S 2585(`C')S 1030 1470(SIDE)U 1199(=`L')S 2 F 1423(C)S 4 F 1489(\254)S 1577(a)S 2 F (AB)R 4 F 1721(+)S 1780(b)S 2 F (C)R 2082(C)S 4 F 2148(\254)S 2236(a)S 2 F (AB)R 4 F 2407(+)S 2466(b)S 2 F (C)R 48 Z 2358 1443(T)U 66 Z 2594 1566(m)U 1423 1662(B)U 1423 1566(A)U 1 F 1485(is)S 1551(symmetric)S 2 F 1850(m)S 4 F (\264)R 2 F (m)R 2082(A)S 1 F 2144(is)S 2210(symmetric)S 2 F 2509(m)S 4 F (\264)R 1 F 1485 1662(is)U 2 F 1551(m)S 4 F (\264)R 2 F (n)R 2082(B)S 1 F 2144(is)S 2 F 2210(n)S 4 F (\264)R 2 F (m)R 2503 1854(C)U 1 F 1030(SIDE)S 1199(=`R')S 2 F 1423(C)S 4 F 1489(\254)S 1577(a)S 2 F (BA)R 4 F 1721(+)S 1780(b)S 2 F (C)R 2082(C)S 4 F 2148(\254)S 2236(a)S 2 F (B)R 2345(A)S 4 F 2407(+)S 2466(b)S 2 F 48 Z 2318 1827(T)U 66 Z 1423 2046(A)U 1423 1950(B)U 1 F 1485(is)S 2 F 1551(m)S 4 F (\264)R 2 F (n)R 2082(B)S 1 F 2144(is)S 2 F 2210(n)S 4 F (\264)R 2 F (m)R 1 F 1485 2046(is)U 1551(symmetric)S 2 F 1850(n)S 4 F (\264)R 2 F (n)R 2082(A)S 1 F 2144(is)S 2210(symmetric)S 2 F 2509(n)S 4 F (\264)R 2 F (n)R 1 F 448 2274(For)U 562(complex)S 810(matrices)S 1518 2418(TRANSB)U 1801(=`N')S 2168(TRANSB)S 2451(=`C')S 1057 2610(SIDE)U 1226(=`L')S 2 F 1450(C)S 4 F 1516(\254)S 1604(a)S 2 F (AB)R 4 F 1748(+)S 1807(b)S 2 F (C)R 2098(C)S 4 F 2164(\254)S 2252(a)S 2 F (AB)R 4 F 2311 2616(`)U 2312(`)S 2423 2610(+)U 2482(b)S 2 F (C)R 48 Z 2374 2580(T)U 66 Z 2599 2706(m)U 1450 2802(B)U 1450 2706(A)U 1 F 1512(is)S 1578(Hermitian)S 2 F 1866(m)S 4 F (\264)R 2 F (m)R 2098(A)S 1 F 2160(is)S 2226(Hermitian)S 2 F 2514(m)S 4 F (\264)R 1 F 1512 2802(is)U 2 F 1578(m)S 4 F (\264)R 2 F (n)R 2098(B)S 1 F 2160(is)S 2 F 2226(n)S 4 F (\264)R 2 F (m)R 2519 2994(C)U 1 F 1057(SIDE)S 1226(=`R')S 2 F 1450(C)S 4 F 1516(\254)S 1604(a)S 2 F (BA)R 4 F 1748(+)S 1807(b)S 2 F (C)R 2098(C)S 4 F 2164(\254)S 2252(a)S 2 F (B)R 4 F 2271 3000(`)U 2272(`)S 2 F 2361 2994(A)U 4 F 2423(+)S 2482(b)S 2 F 48 Z 2334 2964(T)U 66 Z 1450 3186(A)U 1450 3090(B)U 1 F 1512(is)S 2 F 1578(m)S 4 F (\264)R 2 F (n)R 2098(B)S 1 F 2160(is)S 2 F 2226(n)S 4 F (\264)R 2 F (m)R 1 F 1512 3186(is)U 1578(Hermitian)S 2 F 1866(n)S 4 F (\264)R 2 F (n)R 2098(A)S 1 F 2160(is)S 2226(Hermitian)S 2 F 2514(n)S 4 F (\264)R 2 F (n)R 1 F 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 11 BP 1 F 66 Z 470 462(a\))U 543(General)S 773(matrix-mat)S 1064(rix)S 1159(products:)S 1791 270(-)U 1835(9)S 1890(-)S 5 F 514 654(_)U 1 F (GEMM)R 775(\(TRANSA,)S 1101(TRANSB,)S 1401(M,)S 1499(N,)S 1586(K,)S 1673(ALPHA,)S 1933(A,)S 2020(LDA,)S 2195(B,)S 2278(LDB,)S 2449(BETA,)S 2660(C,)S 2743(LDC\))S 514 1038(F)U 514 846(Operation)U 799(\()S 2 F (C)R 1 F 887(is)S 953(always)S 2 F 1158(m)S 4 F (\264)R 2 F (n)R 1 F (\):)R 551 1038(or)U 628(real)S 748(matrices)S 1642 1182(TRANSA)U 1929(=`N')S 2203(TRANSA)S 2490(=`T')S 2633(or)S 2710(`C')S 906 1374(TRANSB)U 1189(=`N')S 2 F 1596(C)S 4 F 1662(\254)S 1750(a)S 2 F (AB)R 4 F 1894(+)S 1953(b)S 2 F (C)R 2203(C)S 4 F 2269(\254)S 2357(a)S 2 F (A)R 2466(B)S 4 F 2528(+)S 2587(b)S 2 F (C)R 48 Z 2439 1347(T)U 66 Z 2678 1470(n)U 1 F 906 1662(T)U 2 F 1596 1470(A)U 1 F 1658(is)S 2 F 1724(m)S 4 F (\264)R 2 F (k)R 1 F (,)R 2 F 1877(B)S 1 F 1939(is)S 2 F 2005(k)S 4 F (\264)R 2 F (n)R 2203(A)S 1 F 2265(is)S 2 F 2331(k)S 4 F (\264)R 2 F (m)R 1 F (,)R 2 F 2484(B)S 1 F 2546(is)S 2 F 2612(k)S 4 F (\264)R 1 F 946 1662(RANSB)U 1189(=`T')S 1332(or)S 1409(`C')S 2 F 1596(C)S 4 F 1662(\254)S 1750(a)S 2 F (AB)R 4 F 1921(+)S 1980(b)S 2 F (C)R 2203(C)S 4 F 2269(\254)S 2357(a)S 2 F (A)R 2466(B)S 4 F 2555(+)S 2614(b)S 2 F (C)R 1596 1758(A)U 48 Z 1872 1635(T)U 2439(T)S 2506(T)S 1 F 66 Z 1658 1758(is)U 2 F 1724(m)S 4 F (\264)R 2 F (k)R 1 F (,)R 2 F 1877(B)S 1 F 1939(is)S 2 F 2005(n)S 4 F (\264)R 2 F (k)R 2203(A)S 1 F 2265(is)S 2 F 2331(k)S 4 F (\264)R 2 F (m)R 1 F (,)R 2 F 2484(B)S 1 F 2546(is)S 2 F 2612(n)S 4 F (\264)R 2 F (k)R 1 F 514 1986(For)U 628(complex)S 876(matrices)S 1605 2130(TRANSA)U 1892(=`N')S 2203(TRANSA)S 2490(=)S 2549(`C')S 1030 2322(TRANSB)U 1313(=`N')S 2 F 1559(C)S 4 F 1625(\254)S 1713(a)S 2 F (AB)R 4 F 1857(+)S 1916(b)S 2 F (C)R 2166(C)S 4 F 2232(\254)S 2320(a)S 2 F (A)R 4 F 2339 2328(`)U 2340(`)S 2 F 2429 2322(B)U 4 F 2491(+)S 2550(b)S 2 F (C)R 48 Z 2402 2292(T)U 66 Z 2641 2418(n)U 1 F 1030 2610(T)U 2 F 1559 2418(A)U 1 F 1621(is)S 2 F 1687(m)S 4 F (\264)R 2 F (k)R 1 F (,)R 2 F 1840(B)S 1 F 1902(is)S 2 F 1968(k)S 4 F (\264)R 2 F (n)R 2166(A)S 1 F 2228(is)S 2 F 2294(k)S 4 F (\264)R 2 F (m)R 1 F (,)R 2 F 2447(B)S 1 F 2509(is)S 2 F 2575(k)S 4 F (\264)R 1 F 1070 2610(RANSB)U 1313(=)S 1372(`C')S 2 F 1559(C)S 4 F 1625(\254)S 1713(a)S 2 F (AB)R 4 F 1772 2616(`)U 1773(`)S 1884 2610(+)U 1943(b)S 2 F (C)R 2166(C)S 4 F 2232(\254)S 2320(a)S 2 F (A)R 4 F 2339 2616(`)U 2340(`)S 2 F 2429 2610(B)U 4 F 2406 2616(`)U 2407(`)S 2518 2610(+)U 2577(b)S 2 F (C)R 1559 2706(A)U 48 Z 1835 2580(T)U 2402(T)S 2469(T)S 1 F 66 Z 1621 2706(is)U 2 F 1687(m)S 4 F (\264)R 2 F (k)R 1 F (,)R 2 F 1840(B)S 1 F 1902(is)S 2 F 1968(n)S 4 F (\264)R 2 F (k)R 2166(A)S 1 F 2228(is)S 2 F 2294(k)S 4 F (\264)R 2 F (m)R 1 F (,)R 2 F 2447(B)S 1 F 2509(is)S 2 F 2575(n)S 4 F (\264)R 2 F (k)R 1 F 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 10 BP 3 F 66 Z 448 462(6.)U 520(Speci\256cation)S 901(of)S 978(the)S 1088(Level)S 1263(3)S 1318(BLAS)S 1 F 1791 270(-)U 1835(8)S 1890(-)S 598 594(Type)U 755(and)S 872(dimension)S 1168(for)S 1267(variables)S 1526(occurring)S 1800(in)S 1873(the)S 1975(subroutine)S 2275(speci\256cations)S 2655(are)S 2757(as)S 2834(follows:)S 668 882(C)U 668 786(INTEGER)U 1060(M,)S 1158(N,)S 1245(K,)S 1332(LDA,)S 1507(LDB,)S 1678(LDC)S 712 882(HARACTER*1)U 1156(SIDE,)S 1342(UPLO,)S 1554(TRANSA,)S 1858(TRANSB,)S 2158(DIAG)S 448 1074(For)U 562(routines)S 796(whose)S 987(\256rst)S 1112(letter)S 1268(is)S 1334(an)S 1418(S:)S 668 1266(REAL)U 884(ALPHA,)S 1144(BETA)S 1758 1362(\))U 448 1554(F)U 668 1362(REAL)U 884(A\(LDA,*\),)S 1201(B\(LDB,*\),)S 1510(C\(LDC,*)S 485 1554(or)U 562(routines)S 796(whose)S 987(\256rst)S 1112(letter)S 1268(is)S 1334(a)S 1385(D)S 1728 1746(A)U 668 1842(D)U 668 1746(DOUBLE)U 958(PRECISION)S 1344(ALPHA,)S 1604(BET)S 716 1842(OUBLE)U 958(PRECISION)S 1344(A\(LDA,*\),)S 1661(B\(LDB,*\),)S 1970(C\(LDC,*\))S 448 2034(For)U 562(routines)S 796(whose)S 987(\256rst)S 1112(letter)S 1268(is)S 1334(a)S 1385(C:)S 668 2226(COMPLEX)U 1028(ALPHA,)S 1288(BETA)S 1902 2322(\))U 448 2514(e)U 668 2322(COMPLEX)U 1028(A\(LDA,*\),)S 1345(B\(LDB,*\),)S 1654(C\(LDC,*)S 477 2514(xcept)U 641(that)S 761(for)S 860(CHERK)S 1106(the)S 1208(scalars)S 4 F 1409(a)S 1 F 1473(and)S 4 F 1590(b)S 1 F 1649(are)S 1751(real)S 1871(so)S 1952(that)S 2072(the)S 2174(\256rst)S 2299(declarati)S 2524(on)S 2612(above)S 2791(is)S 2857(replaced)S 3101(by:)S 448 2994(F)U 668 2706(REAL)U 884(ALPHA,)S 1144(BETA)S 485 2994(or)U 562(routines)S 796(whose)S 987(\256rst)S 1112(letter)S 1268(is)S 1334(Z:)S 536 3186(C)U (OMPLEX*16)R 995(ALPHA,)S 1255(BETA)S 1438(DOUBLE)S 1728(COMPLEX)S 2088(ALPHA,)S 2348(BETA)S 2509 3282(\))U 536 3378(C)U 536 3282(COMPLEX*16)U 995(A\(LDA,*\))S 1603(DOUBLE)S 1893(COMPLEX)S 2253(A\(LDA,*)S 580 3378(OMPLEX*16)U 995(B\(LDB,*\))S 1603(DOUBLE)S 1893(COMPLEX)S 2253(B\(LDB,*\))S 448 3666(e)U 536 3474(COMPLEX*16)U 995(C\(LDC,*\))S 1438(DOUBLE)S 1728(COMPLEX)S 2088(C\(LDC,*\))S 477 3666(xcept)U 641(that)S 761(for)S 860(ZHERK)S 1102(the)S 1204(scalars)S 4 F 1405(a)S 1 F 1469(and)S 4 F 1586(b)S 1 F 1645(are)S 1747(real)S 1867(so)S 1948(that)S 2068(the)S 2170(\256rst)S 2295(declarati)S 2520(on)S 2608(above)S 2787(is)S 2853(replaced)S 3097(by:)S 60 Z 16 4710(-)U 66 Z 668 3858(DOUBLE)U 958(PRECISION)S 1344(ALPHA,)S 1604(BETA)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 9 BP 1 F 66 Z 448 462(m)U 1791 270(-)U 1835(7)S 1890(-)S 499 462(atrix)U 647(arguments.)S 966(If)S 1038(M)S 1125(and)S 1248(N)S 4 F 1323(>)S 1 F 1387(0,)S 1464(but)S 1575(K)S 1650(=)S 1714(0,)S 1791(the)S 1898(operation)S 2173(reduces)S 2401(to)S 2 F 2479(C)S 4 F 2545(\254)S 2633(b)S 2 F (C)R 1 F 2741(\(this)S 2885(applies)S 3098(to)S 3176(the)S 3230 558(s)U 448 654(a)U 448 558(GEMM,)U 696(SYRK)S 898(and)S 1017(HERK)S 1221(routines\).)S 1496(The)S 1622(input-output)S 1971(matrix)S 2166(\()S 2 F 2212(B)S 1 F 2276(for)S 2377(the)S 2481(TR)S 2589(routines,)S 2 F 2842(C)S 1 F 2910(otherwise\))S 3212(i)S 477 654(lways)U 2 F 653(m)S 4 F (\264)R 2 F (n)R 1 F 815(if)S 877(rectangular,)S 1211(and)S 2 F 1328(n)S 4 F (\264)R 2 F (n)R 1 F 1453(if)S 1515(square.)S 598 786(T)U (he)R 731(description)S 1054(of)S 1139(the)S 1249(matrix)S 1450(consists)S 1689(of)S 1774(the)S 1884(array)S 2049(name)S 2221(\(A,)S 2338(B)S 2412(or)S 2497(C\))S 2593(followed)S 2857(by)S 2953(the)S 3063(leading)S 448 882(dimension)U 744(of)S 821(the)S 923(array)S 1080(as)S 1157(declared)S 1401(in)S 1474(the)S 1576(calling)S 1776(\(sub\)program)S 2157(\(LDA,)S 2354(LDB)S 2508(or)S 2585(LDC\).)S 598 1014(The)U 722(scalars)S 923(always)S 1128(have)S 1274(the)S 1376(dummy)S 1599(argument)S 1869(names)S 2059(ALPHA)S 2302(and)S 2419(BETA.)S 580 1338(A)U 598 1146(The)U 722(following)S 1000(values)S 1190(of)S 1267(arguments)S 1563(are)S 1665(invalid:)S 628 1338(ny)U 716(value)S 880(of)S 957(the)S 1059(character)S 1321(arguments)S 1617(SIDE,)S 1803(TRANSA,)S 2107(TRANSB,)S 580 1530(M)U 580 1434(UPLO,)U 792(or)S 869(DIAG,)S 1074(whose)S 1265(meaning)S 1513(is)S 1579(not)S 1685(speci\256ed.)S 4 F 661 1530(<)U 1 F 720(0)S 580 1722(K)U 580 1626(N)U 4 F 650(<)S 1 F 709(0)S 4 F 650 1722(<)U 1 F 709(0)S 580 1818(L)U (DA)R 4 F 738(<)S 1 F 797(the)S 899(number)S 1122(of)S 1199(rows)S 1350(in)S 1423(the)S 1525(matrix)S 2 F 1718(A)S 1 F (.)R 580 2010(L)U 580 1914(LDB)U 4 F 734(<)S 1 F 793(the)S 895(number)S 1118(of)S 1195(rows)S 1346(in)S 1419(the)S 1521(matrix)S 2 F 1714(B)S 1 F (.)R 620 2010(DC)U 4 F 734(<)S 1 F 793(the)S 895(number)S 1118(of)S 1195(rows)S 1346(in)S 1419(the)S 1521(matrix)S 2 F 1714(C)S 1 F (.)R 448 2202(I)U (f)R 515(a)S 567(routine)S 776(is)S 843(called)S 1022(with)S 1162(an)S 1246(invalid)S 1450(value)S 1614(for)S 1713(any)S 1830(of)S 1907(its)S 1991(arguments,)S 2304(then)S 2439(it)S 2497(must)S 2647(report)S 2826(the)S 2928(fact)S 3048(and)S 3165(ter-)S 448 2394(d)U 448 2298(minate)U 660(execution)S 949(of)S 1038(the)S 1152(routine.)S 1411(In)S 1499(the)S 1612(model)S 1809(implement)S 2089(ation)S 2253(\(see)S 2392(Appendix)S 2685(B\),)S 2801(each)S 2954(routine,)S 3190(on)S 481 2394(etecting)U 712(an)S 798(error,)S 967(calls)S 1111(a)S 1164(common)S 1418(error)S 1570(handling)S 1824(routine)S 2034(XERBLA,)S 2339(passing)S 2561(to)S 2636(it)S 2696(the)S 2800(name)S 2965(of)S 3043(the)S 3146(rou-)S 3238 2490(l)U 448 2586(s)U 448 2490(tine)U 578(and)S 705(the)S 817(number)S 1050(of)S 1137(the)S 1249(\256rst)S 1384(argument)S 1664(which)S 1857(is)S 1933(in)S 2016(error.)S 2215(Specialize)S 2484(d)S 2549(implement)S 2829(ations)S 3018(may)S 3162(cal)S 474 2586(ystem-speci\256c)U 876(exception-handl)S 1299(ing)S 1405(and)S 1522(diagnostic)S 1814(faciliti)S 1984(es.)S 3 F 448 2778(5.)U 520(Storage)S 758(Conventions)S 1 F 598 2910(All)U 708(matrices)S 956(are)S 1062(stored)S 1249(conventionall)S 1606(y)S 1665(in)S 1741(a)S 1795(2-dimensional)S 2196(array)S 2356(with)S 2498(matrix-ele)S 2767(ment)S 2 F 2923(a)S 1 F 3019(stored)S 3205(in)S 2 F 48 Z 2956 2925(i)U 1 F (,)R 2 F (j)R 1 F 66 Z 3234 3009(r)U 448 3105(m)U 448 3009(array-eleme)U 761(nt)S 842(A\(I,J\);)S 1047(there)S 1207(is)S 1280(no)S 1375(provision)S 1653(for)S 1759(packed)S 1974(storage)S 2193(for)S 2299(symmetric,)S 2622(Hermitian)S 2917(or)S 3001(triangula)S 499 3105(atrices.)U 598 3237(F)U (or)R 714(symmetric)S 1015(and)S 1134(Hermitian)S 1424(matrices,)S 1687(only)S 1828(the)S 1932(upper)S 2106(triangle)S 2330(\(UPLO=`U'\))S 2700(or)S 2779(the)S 2883(lower)S 3056(triangle)S 448 3333(\(UPLO=`L'\))U 808(is)S 874(stored.)S 598 3465(For)U 732(triangular)S 1029(matrices,)S 1310(the)S 1432(argument)S 1721(UPLO)S 1935(serves)S 2141(to)S 2233(de\256ne)S 2435(whether)S 2688(the)S 2809(matrix)S 3021(is)S 3106(upper)S 448 3561(\(UPLO=`U'\))U 816(or)S 893(lower)S 1065(\(UPLO=`L'\))S 1425(triangular.)S 598 3693(For)U 716(a)S 771(Hermitian)S 1063(matrix)S 1260(the)S 1366(imaginary)S 1658(parts)S 1812(of)S 1892(the)S 1997(diagonal)S 2248(elements)S 2506(are)S 2611(of)S 2691(course)S 2888(zero)S 3026(and)S 3146(thus)S 448 3885(z)U 448 3789(the)U 552(imaginary)S 842(parts)S 994(of)S 1073(the)S 1177(corresponding)S 1577(Fortran)S 1794(array)S 1952(elements)S 2208(need)S 2355(not)S 2462(be)S 2547(set,)S 2660(but)S 2767(are)S 2870(assumed)S 3120(to)S 3194(be)S 477 3885(ero.)U 603(In)S 683(the)S 788(RK)S 905(routines)S 1142(these)S 1302(imaginary)S 1593(parts)S 1746(will)S 1873(be)S 1960(set)S 2058(to)S 2133(zero)S 2270(on)S 2360(return,)S 2558(except)S 2753(when)S 4 F 2920(b)S 2979(=)S 1 F 3038(1)S 3095(and)S 4 F 3214(a)S 1 F 60 Z 16 4710(-)U 66 Z 448 3981(or)U 2 F 525(k)S 4 F 576(=)S 1 F 635(0,)S 707(in)S 780(which)S 963(case)S 1098(the)S 1200(routines)S 1434(exit)S 1554(immediat)S 1801(ely.)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 8 BP 1 F 66 Z 598 498(SIDE)U 767(is)S 833(used)S 976(by)S 1064(the)S 1166(routines)S 1400(as)S 1477(follows:)S 1791 270(-)U 1835(6)S 1890(-)S 938 738(Value)U 1194(Meaning)S 5 F 2733 768(_)U 1 F 938 864(`)U 5 F 938 768(_)U 951(______________________________________________________)S 1 F 960 864(L')U 1194(Multiply)S 1446(general)S 1661(matrix)S 1854(by)S 1942(symmetric)S 2241(or)S 2318(triangular)S 2595(matrix)S 938 1056(`)U 1194 960(on)U 1282(the)S 1384(left.)S 960 1056(R')U 1194(Multiply)S 1446(general)S 1661(matrix)S 1854(by)S 1942(symmetric)S 2241(or)S 2318(triangular)S 2595(matrix)S 598 1416(T)U 1194 1152(on)U 1282(the)S 1384(right.)S 638 1416(RANSA,)U 902(and)S 1019(TRANSB)S 1302(are)S 1404(used)S 1547(by)S 1635(the)S 1737(routines)S 1971(as)S 2048(follows:)S 5 F 1031 1590(_)U 1 F 1031 1560(Value)U 1287(Meaning)S 5 F 1056 1590(_________________________________________________)U 1 F 1031 1782(`)U 1031 1686(`N')U 1287(Operate)S 1517(with)S 1656(the)S 1758(matrix.)S 1053 1782(T')U 1287(Operate)S 1517(with)S 1656(the)S 1758(transpose)S 2029(of)S 2106(the)S 2208(matrix.)S 2656 1878(.)U 448 2010(I)U 1031 1878(`C')U 1287(Operate)S 1517(with)S 1656(the)S 1758(conjugate)S 2035(transpose)S 2306(of)S 2383(the)S 2485(matrix)S 470 2010(n)U 526(the)S 629(real)S 750(case)S 886(the)S 989(values)S 1179(`T')S 1285(and)S 1402(`C')S 1512(have)S 1658(the)S 1760(same)S 1917(meaning)S 2165(and)S 2282(in)S 2355(the)S 2457(complex)S 2705(case)S 2840(the)S 2942(value)S 3106(`T')S 3212(is)S 448 2106(not)U 554(allowed.)S 598 2238(UPLO)U 795(is)S 863(used)S 1008(by)S 1098(the)S 1202(Hermitian,)S 1509(symmetric,)S 1827(and)S 1946(triangular)S 2225(matrix)S 2419(routines)S 2654(to)S 2728(specify)S 2941(whether)S 3176(the)S 448 2334(upper)U 620(or)S 697(lower)S 869(triangle)S 1091(is)S 1157(being)S 1325(referenced)S 1624(as)S 1701(follows:)S 1527 2478(Value)U 1783(Meaning)S 5 F 2144 2508(_)U 1 F 1527 2604(`)U 5 F 1527 2508(_)U 1550(__________________)S 1 F 1549 2604(U')U 1783(Upper)S 1970(triangle)S 2148 2700(e)U 598 2868(D)U 1527 2700(`L')U 1783(Lower)S 1977(triangl)S 646 2868(IAG)U 792(is)S 864(used)S 1013(by)S 1107(the)S 1215(triangular)S 1498(matrix)S 1697(routines)S 1937(to)S 2016(specify)S 2234(whether)S 2474(or)S 2557(not)S 2669(the)S 2777(matrix)S 2976(is)S 3047(unit)S 3176(tri-)S 448 2964(angular,)U 684(as)S 761(follows:)S 1466 3108(Value)U 1722(Meaning)S 5 F 2204 3138(_)U 1 F 1466 3234(`)U 5 F 1466 3138(_)U 1478(______________________)S 1 F 1488 3234(U')U 1722(Unit)S 1861(triangular)S 2215 3330(r)U 1466(`N')S 1722(Non-unit)S 1982(triangula)S 2407 3462(.)U 448(When)S 627(DIAG)S 815(is)S 881(supplied)S 1126(as)S 1203(`U')S 1317(the)S 1419(diagonal)S 1667(elements)S 1922(are)S 2024(not)S 2130(referenced)S 598 3594(Thus,)U 771(UPLO)S 968(and)S 1087(DIAG)S 1277(have)S 1425(the)S 1529(same)S 1688(values)S 1879(and)S 1997(meanings)S 2272(as)S 2350(for)S 2450(the)S 2553(Level)S 2725(2)S 2781(BLAS;)S 2991(TRANSA)S 3223 3690(o)U 448 3786(t)U 448 3690(and)U 565(TRANSB)S 848(have)S 994(the)S 1096(same)S 1253(values)S 1443(and)S 1560(meanings)S 1834(as)S 1911(TRANS,)S 2167(where)S 2350(TRANSA)S 2637(and)S 2754(TRANSB)S 3037(apply)S 3205(t)S 466 3786(he)U 550(matrices)S 794(A)S 864(or)S 941(B,)S 1024(respectively,)S 1380(except)S 1573(that)S 1693(in)S 1766(the)S 1868(complex)S 2116(routines)S 2350(transposition)S 2712(is)S 2778(not)S 2884(allowed.)S 3239 3918(,)U 448 4014(a)U 598 3918(We)U 721(recommend)S 1063(that)S 1193(the)S 1305(equivalent)S 1610(lower)S 1792(case)S 1937(characters)S 2235(be)S 2328(accepted)S 2588(with)S 2736(the)S 2847(same)S 3013(meaning)S 477 4014(lthough,)U 718(because)S 949(they)S 1085(are)S 1188(not)S 1295(included)S 1544(in)S 1618(the)S 1721(standard)S 1967(Fortran)S 2184(character)S 2447(set,)S 2560(their)S 2703(use)S 2814(may)S 2950(not)S 3057(be)S 3142(sup-)S 448 4110(ported)U 638(on)S 726(all)S 813(systems.)S 1061(See)S 1178(Section)S 1397(7)S 1452(of)S 1595(\(Dongarra)S 2 F 1888(et)S 1957(al.)S 1 F (,)R 2064(1986a\))S 2269(for)S 2368(further)S 2569(discussion.)S 598 4242(The)U 725(sizes)S 878(of)S 958(the)S 1063(matrices)S 1310(are)S 1415(determined)S 1735(by)S 1826(the)S 1931(arguments)S 2230(M,)S 2331(N,)S 2420(and)S 2539(K.)S 2628(It)S 2692(is)S 2760(permissible)S 3087(to)S 3162(call)S 60 Z 16 4710(-)U 66 Z 448 4338(the)U 557(routines)S 798(with)S 944(M)S 1032(or)S 1116(N)S 1193(=)S 1259(0,)S 1338(in)S 1418(which)S 1608(case)S 1750(the)S 1859(routines)S 2099(exit)S 2225(immediat)S 2472(ely)S 2580(without)S 2809(referencing)S 3136(their)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 7 BP 1 F 66 Z 1791 270(-)U 1835(5)S 1890(-)S 3239 462(.)U 448 558(S)U 448 462(initial)U 625(C)S 694(may)S 832(be)S 919(replaced)S 1166(by)S 1257(Z.)S 1339(In)S 1419(the)S 1524(second)S 1732(column,)S 1970(under)S 2 F 2144(real)S 1 F (,)R 2291(the)S 2395(initial)S 2571(S)S 2632(may)S 2769(be)S 2855(replaced)S 3101(by)S 3191(D)S 485 558(ee)U 565(Appendix)S 847(C)S 913(for)S 1012(the)S 1114(full)S 1227(subroutine)S 1527(calling)S 1727(sequences.)S 2 F 3234 690(-)U 448 786(b)U 1 F 598 690(The)U 726(collecti)S 918(on)S 1010(of)S 1091(routines)S 1329(can)S 1446(be)S 1534(thought)S 1761(of)S 1842(as)S 1922(being)S 2093(divided)S 2315(into)S 2442(four)S 2577(separate)S 2817(parts,)S 2 F 2987(real)S 1 F (,)R 2 F 3135(dou)S 481 786(le)U 558(precision)S 1 F (,)R 2 F 850(complex)S 1 F (,)R 1116(and)S 2 F 1241(complex*16)S 1 F (.)R 1628(The)S 1760(routines)S 2002(can)S 2123(be)S 2215(written)S 2431(in)S 2512(ANSI)S 2697(standard)S 2950(Fortran)S 3173(77,)S 448 978(c)U 448 882(with)U 593(the)S 701(exception)S 983(of)S 1065(the)S 1172(routines)S 1411(that)S 1536(use)S 1651(COMPLEX*16)S 2093(variables.)S 2396(These)S 2580(routines)S 2819(are)S 2926(included)S 3179(for)S 477 978(ompleteness)U 827(and)S 947(for)S 1049(their)S 1194(usefulness)S 1494(on)S 1585(those)S 1749(systems)S 1983(which)S 2169(support)S 2392(this)S 2512(data)S 2646(type;)S 2802(but)S 2911(because)S 3143(they)S 448 1266(T)U 448 1074(do)U 536(not)S 642(conform)S 887(to)S 960(the)S 1062(Fortran)S 1278(standard,)S 1540(they)S 1675(may)S 1810(not)S 1916(be)S 2000(available)S 2258(on)S 2346(all)S 2433(machines.)S 488 1266(he)U 572(combinations)S 948(provided)S 1204(are:)S 1725 1362(Table)U 1896(3.1)S 2327 1506(M)U 1318(Complex)S 1660(Real)S 1882(MM)S 2099(RK)S 2290(S)S 1372 1698(CGE)U 1658(SGE)S 1924(*)S 2128 1890(*)U 1372(CHE)S 1659(SSY)S 1924(*)S 1374 2082(CTR)U 1660(STR)S 1924(*)S 2321(*)S 2921 2310(.)U 3 F 448 2502(4)U 1 F 448 2310(However,)U 729(note)S 864(that)S 984(rank-)S 2 F (k)R 1 F 1174(updates)S 1397(of)S 1474(general)S 1689(matrices)S 1933(are)S 2035(provided)S 2291(by)S 2379(the)S 2481(GEMM)S 2709(routines)S 3 F 481 2502(.)U 520(Argument)S 832(Conventions)S 1 F 598 2634(W)U (e)R 719(follow)S 921(a)S 980(similar)S 1192(convention)S 1514(for)S 1621(the)S 1731(argument)S 2009(lists)S 2145(to)S 2226(that)S 2354(for)S 2461(the)S 2571(Level)S 2750(2)S 2813(BLAS,)S 3029(with)S 3176(the)S 448 2730(necessary)U 726(adaptations.)S 1064(The)S 1188(order)S 1349(of)S 1426(arguments)S 1722(is)S 1788(as)S 1865(follows:)S 580 2922(a\))U 653(Arguments)S 968(specifying)S 1264(options)S 1837 3018(s)U 580 3114(c)U 580 3018(b\))U 657(Arguments)S 972(de\256ning)S 1210(the)S 1312(sizes)S 1462(of)S 1539(the)S 1641(matrice)S 609 3114(\))U 653(Input)S 814(scalar)S 580 3210(d)U (\))R 657(Description)S 986(of)S 1063(input)S 1220(matrices)S 1960 3306(\))U 580 3402(f)U 580 3306(e\))U 653(Input)S 814(scalar)S 989(\(associated)S 1303(with)S 1442(input-output)S 1789(matrix)S 602 3402(\))U 646(Description)S 975(of)S 1052(the)S 1154(input-output)S 1501(matrix)S 2124 3594(.)U 448(Note)S 598(that)S 718(not)S 824(every)S 992(category)S 1240(is)S 1306(present)S 1518(in)S 1591(each)S 1733(of)S 1810(the)S 1912(routines)S 598 3726(The)U 733(arguments)S 1040(that)S 1171(specify)S 1394(options)S 1621(are)S 1734(character)S 2007(arguments)S 2314(with)S 2464(the)S 2577(names)S 2778(SIDE,)S 2974(TRANSA,)S 60 Z 16 4710(-)U 66 Z 448 3822(TRANSB,)U 748(UPLO)S 943(and)S 1060(DIAG.)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 6 BP 1 F 66 Z 1791 270(-)U 1835(4)S 1890(-)S 2418 462(:)U 448(d\))S 525(Solving)S 752(triangular)S 1029(systems)S 1260(of)S 1337(equations)S 1611(with)S 1750(multiple)S 1990(right-hand)S 2286(sides)S 2 F 703 594(B)U 4 F 765(\254)S 2 F 853(T)S 941(B)S 4 F 48 Z 890 567(-)U 1 F (1)R 2 F 917 699(T)U 4 F 890(-)S 2 F 66 Z 944 726(B)U 703 858(B)U 703 726(B)U 4 F 765(\254)S 2 F 853(T)S 4 F 765 858(\254)U 2 F 853(BT)S 1 F 48 Z 957 831(1)U 4 F 930(-)S 2 F 957 963(T)U 1 F 66 Z 598 1158(H)U 2 F 703 990(B)U 4 F 765(\254)S 2 F 853(BT)S 4 F 48 Z 963(-)T 1 F 66 Z 646 1158(ere)U 4 F 754(a)S 1 F 824(and)S 4 F 947(b)S 1 F 1012(are)S 1120(scalars,)S 2 F 1344(A)S 1 F (,)R 2 F 1429(B)S 1 F 1497(and)S 2 F 1620(C)S 1 F 1692(are)S 1800(rectangular)S 2122(matrices)S 2371(\(in)S 2471(some)S 2637(cases)S 2803(square)S 3002(and)S 3124(sym-)S 448 1254(metric\),)U 676(and)S 2 F 793(T)S 1 F 852(is)S 918(an)S 1002(upper)S 1174(or)S 1251(lower)S 1423(triangular)S 1700(matrix)S 1893(\(and)S 2032(non-singular)S 2387(in)S 2460(\(d\)\).)S 598 1386(Analogous)U 919(operations)S 1228(are)S 1343(proposed)S 1619(in)S 1704(complex)S 1964(arithmeti)S 2200(c:)S 2281(conjugate)S 2570(transposition)S 2944(is)S 3022(speci\256ed)S 3 F 448 1674(3)U 1 F 448 1482(instead)U 656(of)S 733(simple)S 930(transposition)S 1292(and)S 1409(in)S 1482(\(b\))S 2 F 1581(C)S 1 F 1647(is)S 1713(Hermitian)S 2001(and)S 4 F 2118(a)S 1 F 2182(and)S 4 F 2299(b)S 1 F 2358(are)S 2460(real.)S 3 F 481 1674(.)U 520(Naming)S 766(conventions)S 1 F 598 1806(T)U (he)R 729(name)S 900(of)S 984(a)S 1042(Level)S 1220(3)S 1282(BLAS)S 1480(routine)S 1695(follows)S 1922(the)S 2031(conventions)S 2378(of)S 2462(the)S 2571(Level)S 2748(2)S 2809(BLAS.)S 3023(The)S 3153(\256rst)S 448 1902(character)U 710(in)S 783(the)S 885(name)S 1049(denotes)S 1272(the)S 1374(Fortran)S 1590(data)S 1721(type)S 1856(of)S 1933(the)S 2035(matrix,)S 2245(as)S 2322(follows:)S 1036 2142(S)U 1183(REAL)S 1036 2238(D)U 1183(DOUBLE)S 1473(PRECISION)S 1036 2334(C)U 1183(COMPLEX)S 1036 2430(Z)U 1183(COMPLEX*16)S 1620(or)S 1697(DOUBLE)S 1987(COMPLEX)S 2325(\(if)S 2409(available)S 2645(\))S 2755 2658(:)U 448(Characters)S 751(two)S 872(and)S 989(three)S 1142(in)S 1215(the)S 1317(name)S 1481(refer)S 1627(to)S 1700(the)S 1802(kind)S 1941(of)S 2018(matrix)S 2211(involved,)S 2480(as)S 2557(follows)S 1277 2898(GE)U 1464(All)S 1570(matrices)S 1814(are)S 1916(general)S 2131(rectangular)S 1277 3090(S)U 1277 2994(HE)U 1464(One)S 1596(of)S 1673(the)S 1775(matrices)S 2019(is)S 2085(Hermitian)S 1314 3090(Y)U 1464(One)S 1596(of)S 1673(the)S 1775(matrices)S 2019(is)S 2085(symmetric)S 448 3414(T)U 1277 3186(TR)U 1464(One)S 1596(of)S 1673(the)S 1775(matrices)S 2019(is)S 2085(triangular)S 488 3414(he)U 572(fourth)S 755(and)S 872(\256fth)S 1004(characters)S 1292(in)S 1365(the)S 1467(name)S 1631(denote)S 1828(the)S 1930(type)S 2065(of)S 2142(operation,)S 2429(as)S 2506(follows:)S 921 3750(R)U 921 3558(MM)U 1138(Matrix-matrix)S 1532(product)S 965 3750(K)U 1138(Rank-)S 2 F (k)R 1 F 1350(update)S 1547(of)S 1624(a)S 1675(symmetric)S 1974(or)S 2051(Hermitian)S 2339(matrix)S 2749 3942(d)U 921(SM)S 1138(Solve)S 1310(a)S 1361(system)S 1566(of)S 1643(linear)S 1814(equations)S 2088(for)S 2187(a)S 2238(matrix)S 2431(of)S 2508(right-han)S 1138 4038(sides)U 448 4266(T)U (he)R 575(suggested)S 860(combinations)S 1239(are)S 1344(indicated)S 1609(in)S 1685(Table)S 1859(3.1)S 1967(below.)S 2192(In)S 2272(the)S 2377(\256rst)S 2504(column,)S 2742(under)S 2 F 2916(complex)S 1 F (,)R 3176(the)S 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 5 BP 1 F 66 Z 1791 270(-)U 1835(3)S 1890(-)S 448 462(extent)U 630(for)S 729(experienced)S 1068(applicati)S 1293(ons)S 1407(programmers.)S 598 594(The)U 727(details)S 925(of)S 1007(this)S 1129(proposal)S 1383(are)S 1490(concerned)S 1787(speci\256cally)S 2113(with)S 2257(de\256ning)S 2500(a)S 2555(set)S 2654(of)S 2735(subroutines)S 3065(for)S 3168(use)S 448 786(l)U 448 690(in)U 538(Fortran)S 771(77)S 876(programs.)S 1181(However,)S 1479(the)S 1598(essential)S 1863(features)S 2110(could)S 2294(be)S 2394(adapted)S 2636(to)S 2725(other)S 2898(programming)S 466 786(anguages.)U 3 F 448 978(2)U (.)R 520(Scope)S 707(of)S 784(the)S 894(Level)S 1069(3)S 1124(BLAS)S 1 F 598 1110(T)U (he)R 723(routines)S 958(proposed)S 1223(here)S 1359(have)S 1506(been)S 1653(derived)S 1873(in)S 1947(a)S 1999(fairly)S 2164(obvious)S 2396(manner)S 2616(from)S 2767(some)S 2928(of)S 3005(the)S 3107(Level)S 448 1302(o)U 448 1206(2)U 505(BLAS,)S 715(by)S 805(replacing)S 1073(the)S 1177(vectors)S 2 F 1390(x)S 1 F 1442(and)S 2 F 1560(y)S 1 F 1612(with)S 1752(matrices)S 2 F 1997(B)S 1 F 2060(and)S 2 F 2178(C)S 1 F (.)R 2262(The)S 2387(advantage)S 2676(in)S 2750(keeping)S 2981(the)S 3084(design)S 481 1302(f)U 533(the)S 643(software)S 900(as)S 985(consistent)S 1277(as)S 1361(possible)S 1606(with)S 1752(that)S 1879(of)S 1963(the)S 2072(Level)S 2250(2)S 2312(BLAS)S 2510(is)S 2583(that)S 2710(it)S 2775(will)S 2906(be)S 2997(easier)S 3179(for)S 448 1398(users)U 606(to)S 679(remember)S 967(the)S 1069(calling)S 1269(sequences)S 1558(and)S 1675(parameter)S 1959(conventions.)S 598 1530(In)U 675(real)S 795(arithmeti)S 1031(c)S 1082(the)S 1184(operations)S 1480(proposed)S 1744(for)S 1843(the)S 1945(Level)S 2116(3)S 2171(BLAS)S 2362(have)S 2508(the)S 2610(following)S 2888(forms.)S 448 1722(a\))U 521(Matrix-matrix)S 915(products)S 2 F 703 1950(C)U 4 F 769(\254)S 857(a)S 2 F (AB)R 4 F 1001(+)S 1060(b)S 2 F (C)R 1124 2082(C)U 703(C)S 4 F 769(\254)S 857(a)S 2 F (A)R 966(B)S 4 F 1028(+)S 1087(b)S 2 F 48 Z 939 2055(T)U 979 2187(T)U 66 Z 1124 2214(C)U 703 2346(C)U 703 2214(C)U 4 F 769(\254)S 857(a)S 2 F (AB)R 4 F 1028(+)S 1087(b)S 769 2346(\254)U 857(a)S 2 F (A)R 966(B)S 4 F 1055(+)S 1114(b)S 2 F (C)R 1 F 448 2574(N)U 2 F 48 Z 939 2319(T)U 1006(T)S 1 F 66 Z 496 2574(ote)U 601(that)S 724(these)S 884(operations)S 1183(are)S 1288(more)S 1448(accuratel)S 1684(y)S 1741(described)S 2017(as)S 2096(matrix-mat)S 2387(rix)S 2484(multiply-and-a)S 2874(dd)S 2964(operations;)S 448 2862(b)U 448 2670(they)U 583(include)S 798(rank-)S 2 F (k)R 1 F 988(updates)S 1211(of)S 1288(a)S 1339(general)S 1554(matrix.)S 481 2862(\))U 525(Rank-)S 2 F (k)R 1 F 737(updates)S 960(of)S 1037(a)S 1088(symmetric)S 1387(matrix:)S 2 F 703 2994(C)U 4 F 769(\254)S 857(a)S 2 F (AA)R 4 F 1028(+)S 1087(b)S 2 F (C)R 48 Z 979 2967(T)U 939 3099(T)U 66 Z 1124 3126(C)U 1 F 448 3354(c)U 2 F 703 3126(C)U 4 F 769(\254)S 857(a)S 2 F (A)R 966(A)S 4 F 1028(+)S 1087(b)S 1 F 477 3354(\))U 521(Multiplying)S 857(a)S 908(matrix)S 1101(by)S 1189(a)S 1240(triangular)S 1517(matrix:)S 2 F 703 3618(B)U 703 3486(B)U 4 F 765(\254)S 2 F 853(TB)S 4 F 765 3618(\254)U 2 F 853(T)S 917(B)S 48 Z 890 3591(T)U 66 Z 893 3750(T)U 703 3882(B)U 703 3750(B)U 4 F 765(\254)S 2 F 853(B)S 4 F 765 3882(\254)U 2 F 853(BT)S 48 Z 3855(T)T 1 F 60 Z 16 4710(-)U (-)R 3259(--)S EP %%Page: ? 4 BP 1 F 66 Z 598 462(A)U 1791 270(-)U 1835(2)S 1890(-)S 646 462(n)U 702(extended)S 962(set)S 1058(of)S 1136(Fortran)S 1353(BLAS)S 1545(aimed)S 1728(at)S 1798(matrix-vect)S 2100(or)S 2178(operations)S 2475(\(Level)S 2669(2)S 2724(BLAS\))S 2937(were)S 3087(subse-)S 448 654(\()U 448 558(quently)U 689(proposed)S 975(by)S 1085(Dongarra,)S 1395(Du)S 1520(Croz,)S 1709(Hammarling)S 2085(and)S 2224(Hanson)S 2513(\(Dongarra)S 2 F 2827(et)S 2917(al.)S 1 F (,)R 3045(1986a\),)S 470 654(Dongarra)U 2 F 747(et)S 822(al.)S 1 F (,)R 935(1986b\).)S 1189(The)S 1319(Level)S 1496(2)S 1557(BLAS)S 1754(were)S 1910(proposed)S 2180(in)S 2259(order)S 2425(to)S 2503(support)S 2728(the)S 2835(development)S 3201(of)S 448 846(e)U 448 750(software)U 702(that)S 827(would)S 1019(be)S 1108(both)S 1252(portable)S 1494(and)S 1616(ef\256cient)S 1858(across)S 2050(a)S 2106(wide)S 2260(range)S 2432(of)S 2513(machine)S 2761(architect)S 2986(ures,)S 3139(with)S 477 846(mphasis)U 715(on)S 803(vector-processing)S 1293(machines.)S 598 978(M)U (any)R 776(of)S 855(the)S 959(frequently)S 1253(used)S 1398(algorithms)S 1703(of)S 1781(numerical)S 2066(linear)S 2238(algebra)S 2454(can)S 2568(be)S 2653(coded)S 2833(so)S 2915(that)S 3036(the)S 3139(bulk)S 3223 1074(y)U 448 1170(u)U 448 1074(of)U 527(the)S 630(computation)S 981(is)S 1048(performed)S 1345(by)S 1434(calls)S 1577(to)S 1651(Level)S 1823(2)S 1879(BLAS)S 2071(routines;)S 2324(ef\256ciency)S 2606(can)S 2720(then)S 2856(be)S 2941(obtained)S 3190(b)S 481 1170(tilizing)U 689(tailored)S 912(implement)S 1192(ations)S 1372(of)S 1450(the)S 1553(Level)S 1724(2)S 1779(BLAS)S 1970(routines.)S 2221(On)S 2324(vector-processing)S 2814(machines)S 3084(one)S 3201(of)S 3234 1266(-)U 448 1362(r)U 448 1266(the)U 555(aims)S 705(of)S 786(such)S 933(implement)S 1213(ations)S 1396(is)S 1466(to)S 1543(keep)S 1693(the)S 1799(vector)S 1989(lengths)S 2205(as)S 2286(long)S 2429(as)S 2510(possible,)S 2769(and)S 2890(in)S 2967(most)S 3121(algo)S 470 1362(ithms)U 643(the)S 750(results)S 949(are)S 1056(computed)S 1341(one)S 1462(vector)S 1652(\(row)S 1803(or)S 1884(column\))S 2129(at)S 2202(a)S 2257(time.)S 2438(In)S 2519(addition,)S 2777(on)S 2869(vector)S 3059(register)S 3234 1458(r)U 448 1554(b)U 448 1458(machines)U 719(performance)S 1074(is)S 1141(increased)S 1412(by)S 1500(reusing)S 1716(the)S 1818(results)S 2012(of)S 2089(a)S 2140(vector)S 2326(register,)S 2562(and)S 2679(not)S 2785(storing)S 2990(the)S 3092(vecto)S 481 1554(ack)U 594(into)S 718(memory.)S 598 1686(U)U (nfortunately,)R 1008(this)S 1127(approach)S 1392(to)S 1467(software)S 1718(construction)S 2067(is)S 2135(often)S 2293(not)S 2400(well)S 2536(suited)S 2716(to)S 2790(computers)S 3087(with)S 3227(a)S 3227 1782(e)U 448 1878(p)U 448 1782(hierarchy)U 724(of)S 807(memory)S 1054(\(such)S 1225(as)S 1308(global)S 1500(memory,)S 1764(cache)S 1941(or)S 2024(local)S 2179(memory,)S 2443(and)S 2566(vector)S 2758(registers\))S 3031(and)S 3154(tru)S 481 1878(arallel-proc)U 783(essing)S 995(computers.)S 1333(\(For)S 1494(a)S 1570(description)S 1909(of)S 2011(many)S 2203(advanced)S 2497(computer)S 2791(architect)S 3016(ures)S 3172(see)S 3234 1974(r)U 448 2070(m)U 448 1974(\(Dongarra)U 752(and)S 879(Duff,)S 1053(1987\).\))S 1278(For)S 1402(those)S 1573(architect)S 1798(ures)S 1940(it)S 2008(is)S 2084(often)S 2251(preferable)S 2549(to)S 2632(partition)S 2886(the)S 2998(matrix)S 3201(o)S 499 2070(atrices)U 701(into)S 834(blocks)S 1037(and)S 1163(to)S 1245(perform)S 1488(the)S 1599(computation)S 1958(by)S 2055(matrix-mat)S 2346(rix)S 2450(operations)S 2754(on)S 2850(the)S 2960(blocks.)S 3179(By)S 3223 2166(n)U 448 2262(t)U 448 2166(organizing)U 754(the)S 859(computation)S 1212(in)S 1288(this)S 1408(fashion)S 1627(we)S 1729(provide)S 1955(for)S 2057(full)S 2173(reuse)S 2337(of)S 2417(data)S 2551(while)S 2722(the)S 2827(block)S 2998(is)S 3067(held)S 3205(i)S 466 2262(he)U 558(cache)S 737(or)S 822(local)S 979(memory.)S 1245(This)S 1392(approach)S 1663(avoids)S 1865(excessive)S 2147(movement)S 2454(of)S 2538(data)S 2676(to)S 2756(and)S 2880(from)S 3037(memory)S 3223 2358(n)U 448 2454(a)U 448 2358(and)U 576(gives)S 748(a)S 2 F 809(surface-to-volume)S 1 F 1320(effect)S 1501(for)S 1610(the)S 1722(ratio)S 1874(of)S 1961(operations)S 2267(to)S 2350(data)S 2491(movement.)S 2839(In)S 2926(addition,)S 3190(o)S 477 2454(rchitecture)U 757(s)S 810(that)S 934(provide)S 1161(for)S 1264(parallel)S 1486(processing,)S 1811(paralleli)S 2025(sm)S 2128(can)S 2245(be)S 2333(exploited)S 2603(in)S 2680(two)S 2805(ways:)S 2985(\(1\))S 3088(opera-)S 3239 2550(,)U 448 2646(s)U 448 2550(tions)U 607(on)S 704(distinct)S 928(blocks)S 1131(may)S 1275(be)S 1368(performed)S 1673(in)S 1755(parallel;)S 2000(and)S 2126(\(2\))S 2234(within)S 2433(the)S 2543(operations)S 2847(on)S 2943(each)S 3093(block)S 474 2646(calar)U 623(or)S 700(vector)S 886(operations)S 1182(may)S 1317(be)S 1401(performed)S 1697(in)S 1770(parallel.)S 3227 2778(e)U 448 2874(p)U 598 2778(The)U 726(Level)S 901(3)S 960(BLAS)S 1155(proposed)S 1423(here)S 1562(are)S 1668(targeted)S 1905(at)S 1978(the)S 2084(matrix-mat)S 2375(rix)S 2474(operations)S 2774(required)S 3019(for)S 3121(thes)S 481 2874(urposes.)U 728(If)S 800(the)S 908(vectors)S 1126(and)S 1249(matrices)S 1499(involved)S 1757(are)S 1865(of)S 1948(order)S 2 F 2115(n)S 1 F (,)R 2193(then)S 2334(the)S 2441(original)S 2672(BLAS)S 2868(include)S 3088(opera-)S 3234 2970(r)U 448(tions)S 600(that)S 722(are)S 826(of)S 905(order)S 2 F 1068(O)S 1 F (\()R 2 F (n)R 1 F (\),)R 1234(the)S 1338(extended)S 1599(or)S 1678(Level)S 1851(2)S 1908(BLAS)S 2100(provide)S 2324(operations)S 2621(of)S 2699(order)S 2 F 2861(O)S 1 F (\()R 2 F (n)R 1 F 2988(\),)S 3050(and)S 3168(ou)S 48 Z 2964 2943(2)U 66 Z 448 3066(c)U 48 Z 1797 3039(3)U 66 Z 477 3066(urrent)U 657(proposal)S 907(provides)S 1157(operations)S 1454(of)S 1532(order)S 2 F 1694(O)S 1 F (\()R 2 F (n)R 1 F 1821(\);)S 1884(hence)S 2060(our)S 2171(use)S 2282(of)S 2359(the)S 2461(term)S 2603(Level)S 2774(3)S 2829(BLAS.)S 3059(Section)S 3239 3162(.)U 448 3258(S)U 448 3162(8)U 515(illustrates)S 804(how)S 952(some)S 1125(common)S 1389(algorithms)S 1704(can)S 1829(be)S 1925(implement)S 2205(ed)S 2301(by)S 2401(calls)S 2555(to)S 2639(the)S 2752(proposed)S 3027(routines)S 485 3258(uch)U 603(implement)S 883(ations)S 1063(can,)S 1194(we)S 1294(believe,)S 1523(be)S 1608(portable)S 1846(across)S 2034(a)S 2085(wide)S 2235(variety)S 2439(of)S 2516(vector)S 2702(and)S 2819(parallel)S 3037(comput-)S 3239 3354(.)U 448 3450(T)U 448 3354(ers)U 558(and)S 686(also)S 825(ef\256cient)S 1073(\(assuming)S 1377(that)S 1507(ef\256cient)S 1754(implement)S 2034(ations)S 2223(of)S 2310(the)S 2422(Level)S 2603(3)S 2668(BLAS)S 2869(are)S 2981(available)S 3217(\))S 488 3450(here)U 631(is)S 705(certainly)S 963(considerable)S 1324(evidence)S 1586(for)S 1692(the)S 1801(ef\256ciency)S 2089(of)S 2173(such)S 2323(algorithms)S 2633(on)S 2728(particular)S 3008(machines)S 3230 3546(s)U 448 3642(s)U 448 3546(\(see,)U 598(for)S 702(example,)S 968(the)S 1075(references)S 1372(quoted)S 1578(in)S 1656(Section)S 1880(8\);)S 1980(the)S 2087(question)S 2337(of)S 2419(portability)S 2719(has)S 2834(been)S 2985(much)S 3157(les)S 474 3642(tudied)U 667(but)S 780(we)S 886(hope,)S 1060(by)S 1155(proposing)S 1448(a)S 1506(standard)S 1758(set)S 1860(of)S 1943(building)S 2190(blocks,)S 2407(to)S 2486(encourage)S 2784(research)S 3031(into)S 3161(this)S 448 3738(aspect.)U 598 3870(The)U 730(scope)S 910(of)S 995(this)S 1120(proposal)S 1377(is)S 1451(limited.)S 1682(It)S 1751(does)S 1901(not)S 2014(include)S 2236(any)S 2360(routines)S 2601(for)S 2707(matrix)S 2907(factorizat)S 3154(ion;)S 448 4062(L)U 448 3966(these)U 608(are)S 713(covered)S 946(by)S 1037(LINPACK)S 1415(\(Dongarra)S 2 F 1711(et)S 1783(al.)S 1 F (,)R 1893(1976\))S 2072(and)S 2192(we)S 2294(continue)S 2545(to)S 2621(advocate)S 2879(the)S 2984(use)S 3097(of)S 3176(the)S 488 4062(INPACK)U 759(calling)S 961(sequences)S 1252(as)S 1331(a)S 1384(standard)S 1631(interface,)S 1901(even)S 2049(if)S 2112(the)S 2215(details)S 2409(of)S 2487(the)S 2590(underlying)S 2898(algorithm)S 3176(are)S 3234 4158(f)U 448 4254(r)U 448 4158(modi\256ed)U 707(to)S 783(suit)S 903(particular)S 1179(architect)S 1404(ures.)S 1578(Nor)S 1706(does)S 1852(the)S 1956(proposal)S 2207(aim)S 2329(to)S 2404(provide)S 2629(a)S 2682(comprehensive)S 3104(set)S 3201(o)S 470 4254(outines)U 686(for)S 789(elementa)S 1025(ry)S 1106(matrix)S 1303(algebra.)S 1538(It)S 1603(is)S 1672(intended)S 1923(primarily)S 2192(for)S 2294(software)S 2546(developers)S 2856(and)S 2976(to)S 3052(a)S 3106(lesser)S 60 Z 3279 4710(-)U 16(--)S 3259(-)S EP %%Page: ? 3 BP 3 F 72 Z 1092 582(B)U 1158 438(A)U 1234(PROPOSAL)S 1650(FOR)S 1826(A)S 1902(SET)S 2062(OF)S 2186(LEVEL)S 2454(3)S 1140 582(ASIC)U 1336(LINEAR)S 1640(ALGEBRA)S 2020(SUBPROGRAMS)S 1 F 60 Z 1235 834(M)U 2 F 66 Z 1590 726(Jack)U 1732(Dongarra)S 42 Z 1997 702(\262)U 1 F 60 Z 1288 834(athemat)U 1480(ics)S 1567(and)S 1674(Computer)S 1935(Science)S 2143(Division)S 1430 978(A)U 1430 906(Argonne)U 1660(National)S 1888(Laboratory)S 1473 978(rgonne,)U 1675(Illinois)S 1866(60439-4844)S 1372 1230(N)U 2 F 66 Z 1571 1122(Jeremy)U 1783(Du)S 1886(Croz)S 1 F 60 Z 1415 1230(umerical)U 1647(Algorithms)S 1941(Group)S 2114(Ltd.)S 2227 1302(e)U 1324 1374(2)U 1332 1302(NAG)U 1481(Central)S 1679(Of\256ce,)S 1864(May\256eld)S 2101(Hous)S 1354 1374(56)U 1434(Banbury)S 1661(Road,)S 1823(Oxford)S 2016(OX2)S 2152(7DE)S 1334 1626(C)U 2 F 66 Z 1713 1518(Iain)U 1841(Duff)S 1 F 60 Z 1374 1626(omputer)U 1595(Science)S 1803(and)S 1910(Systems)S 2130(Division)S 1535 1770(O)U 1595 1698(Harwell)U 1809(Laboratory)S 1578 1770(xfordshire)U 1845(OX11)S 2011(ORA)S 1372 2022(N)U 2 F 66 Z 1551 1914(Sven)U 1697(Hammarling)S 1 F 60 Z 1415 2022(umerical)U 1647(Algorithms)S 1941(Group)S 2114(Ltd.)S 2227 2094(e)U 1324 2166(2)U 1332 2094(NAG)U 1481(Central)S 1679(Of\256ce,)S 1864(May\256eld)S 2101(Hous)S 1354 2166(56)U 1434(Banbury)S 1661(Road,)S 1823(Oxford)S 2016(OX2)S 2152(7DE)S 3102 2454(3)U 540 2526(B)U 540 2454(Abstract)U 771(\320)S 858(This)S 992(paper)S 1153(describes)S 1404(a)S 1458(proposal)S 1692(for)S 1788(Level)S 1952(3)S 2008(Basic)S 2168(Linear)S 2352(Algebra)S 2572(Subprograms)S 2918(\(Level)S 580 2526(LAS\).)U 756(The)S 878(Level)S 1044(3)S 1102(BLAS)S 1282(are)S 1383(targeted)S 1605(at)S 1676(matrix-ma)S 1928(trix)S 2039(operations)S 2317(with)S 2451(the)S 2552(aim)S 2670(of)S 2747(providing)S 3008(more)S 3105 2598(e)U 540 2670(w)U 540 2598(ef\256cient,)U 779(but)S 881(portable,)S 1119(impleme)S 1331(ntations)S 1547(of)S 1622(algorithms)S 1905(on)S 1990(high-performance)S 2447(computers,)S 2738(especial)S 2933(ly)S 3005(thos)S 583 2670(ith)U 667(hierarchic)S 909(al)S 973(memory)S 1194(and)S 1301(parallel)S 1503(processing)S 1780(capabili)S 1972(ty.)S 3 F 66 Z 448 2934(1.)U 520(Introduction)S 1 F 598 3066(In)U 676(1973)S 831(Hanson,)S 1073(Krogh,)S 1282(and)S 1400(Lawson)S 1632(wrote)S 1805(an)S 1890(article)S 2076(in)S 2150(the)S 2253(SIGNUM)S 2538(Newsletter)S 2846(\(Vol.)S 3029(8,)S 3101(no.)S 3206(4,)S 3239 3162(.)U 448 3258(T)U 448 3162(page)U 599(16\))S 714(describing)S 1015(the)S 1122(advantages)S 1441(of)S 1523(adopting)S 1780(a)S 1836(set)S 1936(of)S 2018(basic)S 2180(routines)S 2419(for)S 2523(problems)S 2794(in)S 2871(linear)S 3046(algebra)S 488 3258(he)U 585(original)S 824(basic)S 994(linear)S 1178(algebra)S 1406(subprograms,)S 1799(now)S 1948(commonly)S 2264(referred)S 2507(to)S 2593(as)S 2683(the)S 2798(BLAS)S 3002(and)S 3132(fully)S 3227 3354(e)U 448 3450(b)U 448 3354(described)U 731(in)S 857(\(Lawson)S 2 F 1119(et)S 1197(al.)S 1 F (,)R 1313(1979a\),)S 1555(\(Lawson)S 2 F 1817(et)S 1895(al.)S 1 F (,)R 2011(1979b\),)S 2246(have)S 2401(been)S 2556(very)S 2704(successful)S 3006(and)S 3132(hav)S 481 3450(een)U 596(used)S 741(in)S 815(a)S 867(wide)S 1018(range)S 1187(of)S 1265(software)S 1515(including)S 1786(LINPACK)S 2162(\(Dongarra)S 2 F 2456(et)S 2526(al.)S 1 F (,)R 2634(1976\))S 2811(and)S 2929(many)S 3098(of)S 3176(the)S 448 3642(t)U 448 3546(algorithms)U 752(published)S 1031(by)S 1120(the)S 1223(ACM)S 1397(Transactions)S 1756(on)S 1845(Mathematic)S 2158(al)S 2228(Software.)S 2505(In)S 2582(particular)S 2855(they)S 2990(are)S 3092(an)S 3176(aid)S 466 3642(o)U 526(clarity,)S 737(portability,)S 1054(modularity)S 1369(and)S 1491(maintenanc)S 1793(e)S 1849(of)S 1931(software)S 2185(and)S 2307(they)S 2447(have)S 2598(become)S 2828(a)S 2 F 2883(de)S 2971(facto)S 1 F 3128(stan-)S 448 3834(i)U 448 3738(dard)U 591(for)S 694(the)S 800(elementa)S 1036(ry)S 1117(vector)S 1307(operations.)S 1646(An)S 1753(excellent)S 2015(discussion)S 2316(of)S 2396(the)S 2 F 2501(raison)S 2695(d')S 5 F 2775(\303)S 2 F 2775(etre)S 1 F 2902(of)S 2982(the)S 3087(BLAS)S 466 3834(s)U 514(given)S 682(in)S 799(\(Dodson)S 1049(and)S 1166(Lewis,)S 1366(1986\).)S 0 F 48 Z 448 3918 M 8 22 0 0 16 0 0 18 PS16 472 3918 M 8 22 0 0 16 0 0 18 PS16 496 3918 M 8 22 0 0 16 0 0 18 PS16 520 3918 M 8 22 0 0 16 0 0 18 PS16 544 3918 M 8 22 0 0 16 0 0 18 PS16 568 3918 M 8 22 0 0 16 0 0 18 PS16 592 3918 M 8 22 0 0 16 0 0 18 PS16 616 3918 M 8 22 0 0 16 0 0 18 PS16 640 3918 M 8 22 0 0 16 0 0 18 PS16 664 3918 M 8 22 0 0 16 0 0 18 PS16 688 3918 M 8 22 0 0 16 0 0 18 PS16 712 3918 M 8 22 0 0 16 0 0 18 PS16 736 3918 M 8 22 0 0 16 0 0 18 PS16 760 3918 M 8 22 0 0 16 0 0 18 PS16 784 3918 M 8 22 0 0 16 0 0 18 PS16 808 3918 M 8 22 0 0 16 0 0 18 PS16 832 3918 M 8 22 0 0 16 0 0 18 PS16 856 3918 M 8 22 0 0 16 0 0 18 PS16 1 F 54 Z 478 3984(Work)U 630(supported)S 872(in)S 943(part)S 1056(by)S 1139(the)S 1234(Applied)S 1437(Mathematical)S 1762(Sciences)S 1979(subprogram)S 2265(of)S 2338(the)S 2432(Of\256ce)S 2595(of)S 2668(Energy)S 448 4050(R)U 2 F 42 Z 448 3960(\262)U 1 F 54 Z 484 4050(esearch,)U 678(U.)S 749(S.)S 811(Department)S 1084(of)S 1147(Energy,)S 1335(under)S 1476(Contract)S 1680(W-31-109-Eng-38.)S 60 Z 16 4710(-)U 54 Z 448 4182(Typeset)U 637(on)S 709(November)S 955(16,)S 1041(1987.)S 60 Z 36 4710(-)U 3259(--)S EP %%Page: ? 2 BP 1 F 60 Z 1838 438(.)U 3 F 66 Z 1937 1230(t)U 1 F 60 Z 432 1518(T)U 3 F 66 Z 1713 1230(Abstrac)U 1 F 60 Z 469 1518(his)U 565(paper)S 725(describes)S 974(a)S 1026(proposal)S 1258(for)S 1353(Level)S 1516(3)S 1571(Basic)S 1730(Linear)S 1913(Algebra)S 2132(Subprograms)S 2477(\(Level)S 2660(3)S 2715(BLAS\).)S 2928(The)S 3047(Level)S 3210(3)S 3220 1590(-)U 432 1662(t)U 432 1590(BLAS)U 606(are)S 701(targeted)S 917(at)S 982(matrix-ma)S 1234(trix)S 1339(operations)S 1611(with)S 1739(the)S 1834(aim)S 1946(of)S 2017(providing)S 2271(more)S 2415(ef\256cient,)S 2648(but)S 2745(portable,)S 2978(impleme)S 3190(n)S 449 1662(ations)U 623(of)S 702(algorithms)S 989(on)S 1078(high-performance)S 1539(computers,)S 1834(especial)S 2029(ly)S 2105(those)S 2261(with)S 2397(hierarchic)S 2639(al)S 2712(memory)S 2942(and)S 3058(parallel)S 16 4710(-)U 432 1734(processing)U 709(capabili)S 901(ty.)S 36 4710(-)U 3259(--)S EP %%Page: ? 1 BP 1 F 60 Z 16 -42(--)U 3259(--)S 1838 438(.)U 2318 1230(Y)U 1311(ARGONNE)S 1623(NATIONAL)S 1952(LABORATOR)S 1526 1302(9700)U 1666(South)S 1826(Cass)S 1959(Avenue)S 3 F 72 Z 1158 1704(A)U 1 F 60 Z 1533 1374(Argonne,)U 1778(Illinois)S 1989(60439)S 3 F 72 Z 1234 1704(PROPOSAL)U 1650(FOR)S 1826(A)S 1902(SET)S 2062(OF)S 2186(LEVEL)S 2454(3)S 2540 1848(S)U 66 Z 876 2136(J)U 72 Z 1092 1848(BASIC)U 1336(LINEAR)S 1640(ALGEBRA)S 2020(SUBPROGRAM)S 66 Z 909 2136(ack)U 1030(Dongarra,)S 1344(Jeremy)S 1574(Du)S 1681(Croz,)S 1859(Iain)S 1995(Duff)S 2146(and)S 2275(Sven)S 2433(Hammarling)S 1 F 60 Z 1278 2640(Mathemati)U 1540(cs)S 1610(and)S 1717(Computer)S 1978(Science)S 2186(Division)S 1442 2856(Technica)U 1664(l)S 1701(Memorandum)S 2062(No.)S 2170(88)S 1702 3375(April)U 1849(1987)S 3279 4710(-)U 16(--)S 3259(-)S EP %%Trailer pscatsave end restore %%Pages: 30 .