Documents Found!
As seen in
Less Work, Better Grades
Join
Course Hero
Access
best resources
Ace
your classes
Ace your courses with Course Hero!

Submit your homework question or assignment here:
352 Tutors are online
 
We are so confident that you will love our service, we will answer your first homework question for FREE!
*  Attach Assignment (optional):
 
Our current time in (CST) is
 
Study Smarter, Score Higher
View This Document For FREE!
 
Document Content (unformatted)
Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.
%%Creator: %!PS-Adobe-2.0 dvips(k) 5.92a Copyright 2002 Radical Eye Software %%Title: Trsm_llnn.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -f Trsm_llnn.dvi -t letter %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.11.25:1004 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 40258431 52099146 1000 600 600 (Trsm_llnn.dvi) @start %DVIPSBitmapFont: Fa cmsy6 6 1 /Fa 1 1 df<B712C0A322037A8D30>0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsy8 8 7 /Fb 7 95 df<B812C0A32A037A9137>0 D<006015C000E01401007014030078EC07806C EC0F006C141E6C5C6C6C5B6C6C5B6C6C485A6C6C485A90387807806D48C7FCEB1E1E6D5A EB07F86D5A6D5A497E497EEB0F3CEB1E1E497E496C7E496C7E48486C7E48486C7E484813 7848C77E001E80488048EC07800070EC03C048140100601400222376A137>2 D<EB01C0A3495AA349CBFCA2130E131E131C5B5B485A485AEA0F80007FB912F8BA12FC6C 18F8D80F80CBFCEA03C06C7EEA00707F7F131E130E7FA26D7EA36D7EA33E237CA147>32 D<170EA383A3717EA2717E841700187084181E84F007C0007FB912F8BA12FC6C18F8CBEA 07C0F00F00181E183860601701604D5AA24DC7FCA3170EA33E237CA147>I<ED018016C0 ED0380A2ED0700A2150EA25DA25DA25DA25D14015D4A5AA24AC7FCA2140EA25CA25CA25C A25C13015C495AA249C8FCA2130EA25BA25BA25BA25B12015B485AA248C9FCA2120EA25A A25AA25AA25A1260223D76AE00>54 D<B712FCA3C9121CAF160826137C972F>58 D<14301478A314FCA2EB01CEA2EB0387A29038070380A290380E01C0A3496C7EA2491370 A2497FA2497FA248487FA248487FA348C7EA0380A2000EEC01C0A248EC00E0A2481570A2 481538A248151CA248150C26297CA72F>94 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmbx8 8 23 /Fc 23 123 df<912603FF8013C0023FEBF80349B5EAFE070107ECFF0F011F9038803FDF 903A3FF80007FFD9FFE013014801807F4890C8127F4848153F485A171F485A170F485A17 07127F5BA200FF93C7FCAA127F6DED03C0A2123FA26C7EEF07806C7EEF0F006C7E6C6C15 1E6C6D5C6C01E014F8D93FF8EB03F0903A1FFF801FE0010790B55A010192C7FCD9003F13 F8020313C032307CAE3B>67 D<B712E016FE707E17E00001D9C00013F8EE1FFCEE07FE70 7E7013807013C0EF7FE0173F18F0A2171F18F8A318FCAA18F8A4EF3FF0A218E0177FEFFF C018805E04071300EE1FFCEEFFF8B85A17C04CC7FC16E0362E7DAD3E>I<B712C016FC16 FF17C00001D9C00113E09238007FF0EE3FF8161FEE0FFCA217FEA717FCA2EE1FF8163FEE 7FF0923801FFE091B612C0170016FC16C002C0C8FCAFB67EA42F2E7DAD37>80 D<B7FC16F816FE707E0001D9C00313E09238007FF0707E161F707EA283A55FA24C5A163F 4C5A923803FFE091B612804CC8FC5E16FFDAC00F7F03037F6F7F81707EA9F00780A2EE3F F8180FB63A801FFC1F0070B5FC705B040113F8CAEA3FF0392F7DAD3D>82 D<EBFFF8000713FF4814C04814F0393FF01FF8EC07FCA26E7EA2EA1FE0EA0FC0C7FCEB03 FF137F0003B5FC000F13E3381FFC03EA3FF0EA7FC0EAFF80A5387FC007383FF03F3A1FFF FDFFF86C13F86C5BC6EB803F251E7E9D28>97 D<EA03F812FFA4120FABEC1FF0ECFFFC01 FB13FF90B6128002C013C09039FE003FE001F8EB1FF0A2ED0FF8A316FCA816F8A2151F16 F06D14E06D133F9039FF80FFC001E7B5128001C3EBFE00018013F89038003FE0262E7DAD 2D>I<EB0FFE90387FFFC048B512E04814F03907FC1FF8EA0FF0EA1FE0123F127F9038C0 0FF0EC07E000FF90C7FCA8127F7FA2003F143C6C7E6C6C13783907FE01F06CB512E06C14 C06C6C138090380FFC001E1E7D9D24>I<ED1FC0EC07FFA4EC007FABEB1FF0EBFFFE4890 B5FC1207380FFC07381FF0019038E0007F123F127F5BA212FFA8127FA36C7E15FF381FF0 03380FFC0F6CB612FC6C147FC613FCEB1FE0262E7DAD2D>I<EB0FFC90387FFF8048B512 C04814F0390FFC0FF8391FF003FCEBE001003F14FE1400485A15FF12FF90B6FCA401C0C7 FCA3127F7F123F150F6C7ED80FFC131F6CB4137E6CEBFFFCC614F8013F13E00107130020 1E7E9D25>I<EA03F812FFA4120FABEC0FF0EC7FFC4A7E01F97FD9FBE01380EBFF0049EB 7FC05BA35BB0B53887FFFCA4262E7CAD2D>104 D<EA0380EA0FE0487EA2487EA36C5AA2 6C5AEA0380C8FCA6EA03F812FFA4120FB3A3B51280A4112F7DAE16>I<EA03F812FFA412 0FAB913803FFF0A4913800FC004A5AEC07E04A5A4A5A4AC7FC14FC13F913FBEBFFFE8081 497F5B9038F07FE06E7E6E7E6E7EA26E7E6E7E6E1380B5000713FCA4262E7DAD2B>107 D<EA03F812FFA4120FB3B3B51280A4112E7DAD16>I<2703F00FF8EB3FE000FFD93FFEEB FFF891B500037F01F102877F903BF3E07FCF81FF260FF780EBDE019028FE003FF8001380 A2495CA2495CB0B52783FFFE0F13F8A43D1E7C9D44>I<3903F00FF000FFEB7FFC4A7E01 F17FD9F3E01380380FF70001FEEB7FC05BA35BB0B53887FFFCA4261E7C9D2D>I<EB07FE 90387FFFE048B512F848803907FC03FE390FF000FF4848EB7F80003F15C049133F007F15 E0A200FF15F0A8007F15E0A26D137F003F15C0001F15806D13FF260FFC0313000003B512 FC6C5C6C6C13E0D907FEC7FC241E7E9D29>I<3903F81FF000FFEBFFFC01FB13FF90B612 8002C013C03A0FFE007FE001F8EB3FF0151F16F8A2150F16FCA816F8151FA2ED3FF06D14 E06D137F9039FF81FFC091B5128001FBEBFE0001F813F8EC3FE091C8FCA9B57EA4262B7D 9D2D>I<3803F0FC39FFF1FF8001F313C001F713E09038FFBFF0380FFE3FA213FCA2EC1F E09038F80FC091C7FCAFB512C0A41C1E7D9D22>114 D<3801FF8E000713FE121F5AEA7F 00007C133E00FC131EA3B46CC7FC13FCEBFFC06C13F06C7F6C7F6C7F6C7FC6FC010F1380 EB007F0070133F00F0131F7E15006C5B38FF807EEBFFFC5C00FB5B00E01380191E7D9D20 >I<131EA4133EA2137EA213FE12011207121FB6FCA4D803FEC7FCAEEC03C0A6EC078038 01FF0F6CEBFF006D5A6D5AEB0FF01A2A7FA920>I<D803F8EB1FC000FFEB07FFA4000FEB 007FB115FFA25C0007130790B612FC6CEBFE7FC613FCEB3FE0261E7C9D2D>I<B53A0FFF E03FFCA4260FF80090380003C06D90387F8007000702FF14806DECC00F000317005CD801 FFECE01E5C913983CFF03E6C163C9139C787F87C017F157814CF9139EF03FCF8D93FFF5C 9138FE01FF011F5DA26D486C5BA24A137F01075D4A133F010392C7FCA26D48131E361E7E 9D3B>119 D<003FB512FCA49038C03FF89038007FF0003E14E048EBFFC05BD878031380 491300495A5CEA001F495A90387FF03CEBFFE014C05A48EB807C481300485A4913F8485A 003F1301387FF00FB6FCA41E1E7E9D24>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmmi6 6 4 /Fd 4 85 df<90B6128016F016F8903907E001FC90390FC0007EA2163EA2495A167EA216 FC90393F0001F8ED03F0ED0FC091B512005B90397E001FC0ED03F015014914F8150016FC A2484814F81501A2ED03F04848EB07E0ED1FC0ED7F80B6EAFE005D15C027227DA12E>66 D<90B512C05DA2D907E0C7FC495AA4495AA449C8FCA4137EA449EB01C0A215031680485A ED07005D5D4848133E157EEC03FC007FB5FCB6FC5D22227DA12A>76 D<90B512FEEDFF8016E0903907E007F090390FC001F8ED00FCA3495AA3ED01F8D93F0013 F0ED03E0ED07C0ED3F0090387FFFFC5D90387E00FE153F5BED1F80A34848EB3F00A44848 137E1607A23A7FFF803E0EB538003F0CED1FF8C8EA03F028237DA12E>82 D<000FB612FEA33A1FC01F807E3A1E003F001E121C003C151C1238147E5AA348491338C7 1400A3495AA4495AA4495AA4495AA3000FB57EA292C7FC27227EA124>84 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmbx12 12 9 /Fe 9 117 df<EA07C0EA1FF0EA3FF8EA7FFCA2EAFFFEA5EA7FFCA2EA3FF8EA1FF0EA07 C00F0F788E1F>46 D<EC01E01407140F143FEB01FF131FB6FCA413E3EA0003B3B3AD007F B612FEA5274178C038>49 D<ECFFE0010F13FE017F6D7E90B612E0000315F82607F8037F 260FE0007FD81F80EB3FFF48C76C138001E06D13C0486C6D13E07F00FF6E13F07F8117F8 A36C5A6C5A6C5A6C5AC8FC17F05D17E0A24B13C0A24B138017004B5A5E4B5AED7FE04B5A 4A5B93C7FCEC03FC4A5AEC0FE04A5A4A4813F84AC7FC14FC495A4948EB01F0495AEB0F80 49C7FC013E140349140790B7FC4816E05A5A5A5A5A5AB8FC17C0A42D417BC038>I<B76C 91B6FCA5D8003F0180C9EA7C001AFC6D606F15016D608119036D6D5E19076D606F150F6D 6081191F6D6D93C7FC61027F163E6F157E6E167C8219FC6E6D5C18016E5E7013036E5E82 18076E6D5C180F6E6D5C181F6E93C8FC705B037F143E82187E6FEB807C18FC6F5C17C16F 5C17E117E36FEBF3E017F76F5C17FF6F5CA36F91C9FCA2705AA2705AA3705AA2705AA270 5AA250457EC355>86 D<903801FFF0011F13FF017F14C048B612F04848C66C7ED807F8EB 1FFC486C6D7E6D6D7E486C81818381836C5A6C5A6C5A6C5AC8FCA30203B5FC91B6FC1307 013F13F19038FFFE01000313F0481380481300485A485A485AA2485AA45DA26C6C5BA26C 6C010E13F86C6C013CEBFFC03A0FFF80F87F6CEBFFF06CECE01FC66CEB8007D90FFCC9FC 322F7DAD36>97 D<137C48B4FC487F487FA2487FA56C5BA26C5B6C90C7FCEA007C90C8FC AAEB3FE0EA7FFFA512037EB3AFB61280A519467DC51F>105 D<90397FC003FFB5011F13 E04B13F892B57E9139C1F81FFE9139C3C00FFF00039038C780076CD9CE008002DE7F02DC 8114F8A25CA35CB3A7B600C1B61280A5392D7DAC3E>110 D<90397FC03F80B5EBFFE002 C113F802C313FC9138C7C7FEECCF0700039038CE0FFF6C13DE14FC14F8ED07FEA29138F0 03FCED00F01600A25CB3A6B612F0A5282D7EAC2E>114 D<EB03E0A61307A3130FA3131F A2133F137F13FF5A5A001F90B51280B7FCA4000101E0C7FCB3A3ED03E0AA6CEC07C014F0 017F1480ECF80F90393FFC1F0090381FFFFE6D5B010313F09038007FC023407EBE2C> 116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi8 8 10 /Ff 10 111 df<13FE6D7E80EB1FE0130F130780A26D7EA2130180A2130080A2147FA280 81A2141F81A26E7E141F143F4A7E14F7EB01E3903803C3F8EB0781010F7FEB1F01EB3E00 497F485A4848137F485A48487F001F1580485A48C7121F00FE15C048140F4815E0007014 03232F7DAD29>21 D<123C127E12FFA4127E123C08087A8714>58 D<15C01401A2EC0380A3EC0700A2140EA35CA35CA35CA35CA2495AA3495AA349C7FCA313 0EA25BA35BA35BA35BA3485AA2485AA348C8FCA3120EA35AA35AA25AA35AA25A1A437CB1 23>61 D<011FB6FC4915E06D15F8903A00FE0003FCEE00FE177FA24948801880A3494815 005FA217FE49485C4C5AEE07F04C5A4948EB7F8091B6C7FC16FC16FF903A1FC0003FC0EE 0FE0707E1603494880A21601A249C7FC1603A25F01FE14074C5AA24C5A4848EC7F804CC7 FCED07FCB75A16C04BC8FC312D7DAC35>66 D<011FB5FC5B6D5B010090C8FC5CA3495AA4 495AA4495AA4495AA4495AA44948EB01801603A2EE070049C7FC5E160E161E01FE141C16 3C167C5E484813011507ED3FF0B7FC5EA2292D7DAC30>76 D<90260FFFFC90B5FC814B14 FE9026003FE0EB1FE04B14006E6C131C5F5F6E6C5B4C5A913807F8034C5A6E6C48C7FC16 1E6E6C5A5E5E6E6C5A5E6F5AA26F7EA24B7E15FF5CEDCFF0EC038F91380707F8140E4A6C 7E14384A6C7E14E0130149486C7E495A49C76C7E131E496E7E13FC00074B7ED87FFF0103 B5FCB56C5A6C497E382D7EAC3A>88 D<13FC121FA3EA01F8A4485AA4485AA4380FC3E0EB CFF8EBFFFCEBFC3E381FF01F9038E00F8013C0018013C0EA3F00A4007E131FA448EB3F80 A315005C147E485B7E387C01F85C383C03E0381E0FC06CB4C7FC6C5AEA03F01A2F7DAD1E >98 D<137EEA0FFEA3EA00FCA4EA01F8A4EA03F0A4EA07E0A4EA0FC0A4EA1F80A4EA3F00 A4127EA31318EAFC38A4EAF870A3EAFCE0127FEA3FC0EA0F000F2F7DAD15>108 D<2707C007F013FE3C0FF01FFC03FF80281FF87FFE0F7F3C3CFCF83F1F07E03C387DE01F BC03F026707F8013F0A2020013E0D8E0FE14C0491480A21240D801F84948485AA34D5A48 48137EA24D5AF081804848491483EF3F03A2F0070048484848137E180E173E183C484848 48EB1FF8000F01016E5A6CC748EB07C0391F7E9D3E>I<39078007F0390FE03FFC486C48 7E3938F8F83F9038F9C01F2670FF80138014005BEAE1FCA25B1241D803F0EB3F00A3157E 485AA25D161848481438EC01F8A21670391F8003F0A2020113E0EDF1C048C7EAFF806CEC 7F00000E143E251F7E9D2B>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr8 8 45 /Fg 45 122 df<1307130F131E133C1378137013F0EA01E0120313C012071380120F1300 5AA2121E123EA35AA612FC5AAD7E127CA67EA3121E121FA27E1380120713C0120313E012 01EA00F013701378133C131E130F130710437AB11B>40 D<126012F012787E7E120E120F EA078013C0120313E0120113F0120013F8A21378137CA3133EA6133F131FAD133F133EA6 137CA3137813F8A213F0120113E0120313C012071380EA0F00120E121E5A5A5A12601043 7CB11B>I<123C127EB4FCA21380A2127F123F1203A312071300A2120EA25A123C12385A 122009157A8714>44 D<B512C0A512057F9018>I<123C127E12FFA4127E123C08087A87 14>I<EB3FC0EBFFF0000313FC3807F0FE380FC03F497E391F000F80A24814C0003E1307 007E14E0A500FE14F0B0007E14E0A46CEB0FC0A36C1480390F801F006D5A3807F0FE6CB4 5AC613F0EB3FC01C2D7DAB23>48 D<130E131E137EEA01FE12FFA2EAFE7E1200B3AF13FF 007F13FFA3182C7BAB23>I<EB7F803801FFF0000713FC380F81FE381E007F0038EB3F80 48EB1FC0007814E000FC130F7E6C14F01407A2127E123CC7120FA215E0A2EC1FC0158014 3F1500147E5C495A495A14C0495A49C7FC131E5B4913705B485A484813E048C7FC120E00 1FB5FC5A5AB612C0A31C2C7DAB23>I<EB3F803801FFF04813FC3807C0FE380E007F001E EB3F80123F9038801FC0A4381F003F000E1480C7FCA2EC7F00147E5CEB03F8EBFFF014C0 14F0EB00FC147E80EC1F8015C0EC0FE0A215F0A2123C127EB4FCA215E0A248EB1FC0127C 0078EB3F806C1400381F80FE3807FFFC6C13F038007F801C2D7DAB23>I<140F5CA25C5C A25C5BA25BEB07BF143F130F131E131C1338137813F013E01201EA03C013801207EA0F00 120E121E5A123812785AB612FCA3C7EA3F00A8EC7F8090381FFFFCA31E2D7EAC23>I<00 0CEB0180381F801F90B5FC15005C14F85C14C0D81C7CC7FC90C8FCA7EB1FC0EB7FF0381D FFFC381FE07EEB801F01001380001EEB0FC0120CC7EA07E0A315F0A3123C127E12FEA215 E0A248130F007014C00078EB1F8000381400001E133E380F80FC6CB45A000113E038007F 801C2D7DAB23>I<EB03F8EB0FFEEB3FFF90387E0780EBF8033901E00FC03803C01F1207 EA0F80121F9038000F804890C7FCA3127EA2EB1FF038FE3FF8EB7FFEEBE03F6C487E9038 800F80010013C0EC07E0A25A15F0A4127EA4003E14E0123FA26CEB0FC01580EA0F803907 C01F003803F07E6CB45A38007FF0EB1FC01C2D7DAB23>I<1238123E003FB512F8A34814 F015E015C038700001EC0380EC070048130EA25CC75A5CA25C495AA21303495AA2130F91 C7FCA25BA25B133EA2137EA513FEA8137C13381D2E7CAC23>I<EB1FC0EBFFF04813FC38 03E07E3807801F390F000F805A001EEB07C0123EA4123F9038800F80EA1FC09038F01F00 380FF81EEBFE7C6C6C5A6C5B7E6C6C7E3801FFFE3803EFFFD807831380D80F0113C0381E 007F003EEB3FE048130FEC07F0481303A21401A315E0127C14036C14C0EC0780391F800F 00380FE07E3803FFFC6C13F038003FC01C2D7DAB23>I<123C127E12FFA4127E123C1200 AD123C127E12FFA4127E123C081D7A9C14>58 D<B812FCA27ECBFCAD007FB712FCB8FCA2 2E137C9937>61 D<EC03C0A24A7EA34A7EA34A7EA3EC39FCA2EC79FE1470A2ECF0FF4A7E A249486C7EA349486C7EA2010780EC000FA24980010E1307A2011FB57EA2498090383800 01A201788001701300A249147FA30001ED3F801203D80FF0EC7FC0D8FFFE0107B5FCA330 2F7EAE35>65 D<B712FCA33903FC00030001EC00FE163EA2161EA2160EA316071538A316 00A21578A2EC01F890B5FCA3EBFC01EC0078A21538A592C7FCA9487EB512FCA3282D7EAC 2E>70 D<D8FFFC49B5FC7FA2D801FF9038001FF06EEB07C06EEB038013DFEBCFE08013C7 EBC3F8EBC1FC8013C0147F1580143FEC1FC0EC0FE015F01407EC03F815FCEC01FE140015 7F1683153FED1FC3ED0FE316F31507ED03FB16FF8181167FA2163F161F487ED80FF8140F B56C13071603A2302D7EAC35>78 D<90383F80303901FFF07048EBFCF0380FC07E381F00 0F001E1307481303007C1301127800F81300A21570A27EA26C1400127F7FEA3FF8EBFF80 6C13F86C13FE6C7F6C14C0C614E0130F010013F0EC0FF814031401EC00FCA20060147C12 E0A46C1478A26C14F07E6CEB01E039FF8003C039F7F00F8000F1B5120038E07FFE38C00F F01E2F7CAD27>83 D<007FB712F8A39039800FE007D87E00140000781678A20070163800 F0163CA348161CA5C71500B3A74A7E011FB512F0A32E2D7EAC33>I<136013F0487E487E 487EEA0F9F383F0FC0387E07E038F801F0387000E000201340140B79AD23>94 D<EB7F803803FFE0000F13F848C67E143E383F803FEC1F80A2391F000FC0120EC7FCA3EB 07FF137F3803FF8F3807F80FEA1FE0EA3F8013005A00FE14C7A4141F127F1477393FC1E7 FE381FFFC30007EB81FC3901FC00F020207E9E23>97 D<EA03E012FFA3120F1207AB147F 9038E3FFE001E77F9038FF01F89038FC007C497F49133F49EB1F80A2ED0FC0A216E0A816 C0A2151F1680A26DEB3F006D137E01DC5B90388F03F8903887FFF0010313C0C76CC7FC23 2F7FAD27>I<EB1FE0EB7FF83801FFFE3803F01FEA07E0390FC03F80EA1F80130048EB1F 0048130E91C7FC127E12FEA8127E127FA26CEB01C01380001FEB0380EA0FC03907E00700 3803F81E3801FFFC6C6C5AEB1FC01A207E9E1F>I<157CEC1FFCA314011400ABEB1FE0EB 7FF83801FFFE3803F81F3807E007380FC001381F800048C7FCA25A127EA212FEA7127EA3 7EA2381F8001000F13036C6C487E3A03F03EFFE03801FFFC6C13F090391FC0F800232F7E AD27>I<EB1F80EBFFF0487F3803F0FC3807C03E380F803F48487E481480140F127E15C0 A212FEB6FCA348C8FCA4127EA2127F6CEB01C0A2391F800380EA0FC03907E007003803F8 1E3801FFFC6C6C5AEB1FC01A207E9E1F>I<EB01F8EB07FEEB1FFF90383F1F80EB7C3F13 F81201EC1F003803F00E91C7FCA9B512E0A3D803F0C7FCB3A47F387FFFC0A3192F7FAE16 >I<90383F80F89038FFE3FC0003EBFFFED807E0133E390F803E1C1508391F001F00A248 1480A56C1400A2380F803EA23807E0FCEBFFF8000C13E0EB3F80001CC8FCA2121E121F38 0FFFF814FF6C14C015E04814F0391F0007F8003C1301007CEB00FC48147CA5007C14F86C EB01F06CEB03E0390FE01FC06CB512800001EBFE0038003FF01F2D7E9D23>I<EA03E012 FFA3120F1207ABEC3F80ECFFE001E37F9038E781F89038EE00FC13FC49137E5BA25BB2B5 380FFFF0A3242E7FAD27>I<EA0780EA0FC0EA1FE0A4EA0FC0EA0780C7FCA8EA03E012FF A3120F1207B3A4EAFFFEA30F2E7FAD13>I<EA03E012FFA3120F1207B3B3A2B5FCA3102E 7FAD13>108 D<2607E07FEB0FE03BFFE1FFC03FF801E36D487E903AE783F0F07E3B0FEE 01F9C03F3B07F800FB001F03FF1480495BA2495BB2B53A1FFFE3FFFCA3361E7E9D3B>I< 3903E03F8000FFEBFFE001E37F9038E781F8390FEE00FCEA07FC49137E5BA25BB2B5380F FFF0A3241E7F9D27>I<EB1FE0EB7FF83801FFFE3803F03F3907C00F80390F8007C0391F 0003E04814F0A2007EEB01F8A300FE14FCA9007E14F8A26CEB03F0A26C14E0EB8007390F C00FC03907F03F803901FFFE0038007FF8EB1FE01E207E9E23>I<3803E07F39FFE3FFE0 01E77F9038FF03F8390FFC00FC6C48137E497F491480151F16C0A2ED0FE0A816C0151FA2 1680153F6DEB7F006D137E6D5B9038EF03F89038E7FFF001E313C0D9E07FC7FC91C8FCA9 B5FCA3232B7F9D27>I<90381FC01C90387FF83C3801FFFC3803F81E3907E0077C390FC0 03FC001F1301EA3F8090C7FC5AA2127E12FEA7127E127FA27EEB8001001F1303EA0FC038 07E00F3803F83E3801FFFC6C13F0EB1FC090C7FCA991381FFFE0A3232B7E9D25>I<3807 C1F038FFC7FCEBCFFEEBDE7FEA0FD8EA07F8EBF03E141C14005BB17FB51280A3181E7E9D 1C>I<3801FE183807FFB8001F13F8EA3F01EA7C000078137800F81338A37E6C13006C7E EA7FFC383FFF806C13E06C13F06C13F8C613FC1307EB00FE0060137E00E0133EA27EA26C 133C147C6C137838FF01F038F7FFE000E313C000C0130017207E9E1C>I<13E0A41201A3 1203A21207120F121FB512F0A33807E000AE1438A70003137013F03801F8E0EA00FFEB7F C0EB1F00152A7FA81B>I<D803E0133E00FFEB0FFEA3000F13000007147EB115FEA30003 13019038F0037F2601F80E13F03800FFFCEB7FF890391FE07C00241F7F9D27>I<3AFFFE 03FFC0A33A07F000FE00157800031470A26C6C5BA2EBFC0100005CA290387E0380A2EB7F 07013F90C7FCA2148FEB1F8EA2EB0FDCA214FC6D5AA26D5AA36D5AA2221E7F9C25>I<3B FFFE7FFE0FFFA33B0FF007F003F83B07E003E001E015F0D803F015C0A29039F807F80300 011680A29039FC0FFC070000D90E7C1300A290397E1E7E0EEC1C3E153FD93F3C5BEC381F 169C90391FF00FB8A216F86D486C5AA36D486C5AA36D486C5A301E7F9C33>I<3AFFFE0F FF80A33A07F807F8000003EB03E0D801FC138000001307D97E0FC7FCEB7F1EEB3FBCEB1F B8EB0FF85C1307130380497E497EEB1E7EEB1C3F90383C1F8001787FEBF00F48486C7E00 036D7E390FF007F83AFFF80FFFC0A3221D7F9C25>I<3AFFFE03FFC0A33A07F000FE0015 7800031470A26C6C5BA2EBFC0100005CEBFE03017E5BA26D48C7FCA2148FEB1F8EA2EB0F DCA214FC6D5AA26D5AA36D5AA25CA213035C1238D87C07C8FC12FE130E131E485AEA7878 EA7FF06C5AEA0F80222B7F9C25>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr6 6 3 /Fh 3 51 df<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F8131F AD0078131EA2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227DA0 1E>48 D<137013F0120712FFA212F91201B3A6B512E0A313217AA01E>I<EA01FC380FFF 804813C0383C0FE0387003F0387C01F8EAFE0014FCA2147C127C003813FCC7FC14F81301 14F0EB03E0EB07C01480EB0F00131E13385B3801E01CEA038038070038120E5A383FFFF8 5AB512F0A316217CA01E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi12 14.4 2 /Fi 2 77 df<0207B812E04A17FE747E1BE0DA000790C87F6F48ED3FF8F20FFC747E4B48 82861C80864B5A1CC0A34B5AA44B481780A2621C004B485D631A0F634B484B5A505A505A 505A4A90C8485B4F90C7FCF10FFCF13FF84A48EDFFE0060F138092B700FCC8FCA24AEEFF 8003FCC8EA7FE0F11FF8F107FC4A486F7E737E87737F4A5A747EA2874A5AA44A5AA44A48 5F1AFFA2634990C95A4F5BA24F90C7FC49484C5AA24F5A4F5A4948EEFFF04E5B4E138006 0F90C8FC4948ED3FFED93FFC913803FFF8007FB912E0BA128006FCC9FC6C17C052527AD1 59>66 D<0207B612FE5CA260DA000701E0C8FC6F138094C9FCA24B5AA44B5AA44B5AA44B 5AA44B5AA44B5AA44A5BA44A90CAFCA44A5AA44A5AA44A5AA31A184A481638A31A704A5A 1AF01AE0A24A48ED01C0A219031A8049491507A2F10F00614990C9FC193E197E19FE4948 4B5A1803180FF03FF84948EC01FFD93FFE021F5B007FB9FCA2BA5A7E45527BD150>76 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy7 7 1 /Fj 1 1 df<B712FEA327037A8F34>0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmbx10 10 45 /Fk 45 123 df<B61280A919097F9620>45 D<EA0780EA1FE0EA3FF0EA7FF8A2EAFFFCA4 EA7FF8A2EA3FF0EA1FE0EA07800E0E7A8D1B>I<143C14FC1301130F137FB5FCA4139FEA 001FB3B3A4007FB61280A5213779B630>49 D<EB0FFC90387FFFC048B512F0000714FC48 80391FC03FFF263F800F1380267FC00313C06D6C13E0EAFFF06D6C13F0A2157F16F8A26C 5A6C5A6C5A6C5AC8FC16F015FF16E0A24A13C016804A13005D4A5A4A5A4A5A4A5A5D4AC7 FC14FE495AD903F013F8495A495A495A90383E00014914F05B4848130348B6FC5A5A5A5A 5AB712E0A525377BB630>I<EB03FF011F13F0017F13FC90B57E3901FE07FF2603F00113 80D807E014C0260FF80013E07F486C14F0A6000F15E06C5A6C485AD801E014C0C714805C 16004A5AEC3FF890380FFFF015C015F015FE90380007FF020113C06E13E0ED7FF016F8A2 ED3FFCA2D80FC014FEEA3FF0487EA2487EA316FCA25B007FEC7FF85B6C48EBFFF06C4848 13E0260FF80713C06CB612806C1500C614FC013F13F0010790C7FC27387CB630>I<ED0F C0151F153FA2157F15FF5C5CA25C5C5CA2143E147E5C495A5C495A1307495A495A5C133E 137E5B485A5B1203485A485A5B48C7FC5A127E5AB81280A5C70001EBC000A90103B61280 A529377DB630>I<000C15C0D81F80130701F8137F90B6FC168016005D5D5D15E05D92C7 FC14F8010EC8FC90C9FCA7EB03FE90381FFFC0017F13F890B57E9038FC07FE9038E003FF D980011380010014C0000E6D13E0C8FC16F0A316F8A2EA1F80487E487E487EA316F0A25B 6C4814E0495A010014C06C4913806C6C481300390FF01FFE6CB55A6C5C6C14E06C6C1380 D91FF8C7FC25387BB630>I<EC0FF8ECFFFE0103EBFF80010F14C090391FF80FE090397F E003F09038FF800748EB000F4848EB1FF84848133FA2120F5B001FEC1FF0ED0FE0003FEC 07C04990C7FC127FA2EC3FC0ECFFF8D8FFF113FE01F37FD9F78113809039FF007FC049EB 3FE04914F0ED1FF85B16FCA34914FEA4127FA5123F16FCA2121F6D14F8120FED3FF06C6C 14E06C6C137F3A01FF81FFC06C90B512806DEBFE006D5B010F13F00101138027387CB630 >I<123E123F13E090B71280A54816005E5E5E5E5E007EC7EA07C0007C140F4B5A00FC4A C7FC48147E5D5D4A5AC712034A5A4A5AA24A5A4AC8FCA25C147E14FE13015C1303A31307 A2495AA3131FA4133FA96D5A6D5A6D5A293A7BB830>I<49B47E010F13F0013F13FC497F 9038FE01FF3A01F0003F804848EB1FC00007EC0FE05B000FEC07F0A2121F7FA213F87F01 FFEB0FE014C09138F01FC06C13F89138FE3F806C90B512005D6C14F86C806C806DEBFF80 6D14C090B612E04815F0EA07FD260FF07F13F848486C13FC383FC00F14034848C613FE15 7F48C7121F15071503A21501A216FCA26C7EED03F86C7E6DEB07F0D81FF8EB1FE0D80FFE EBFFC06CB612806C1500C614FC013F13F00103138027387CB630>I<EA0780EA1FE0EA3F F0EA7FF8A2EAFFFCA4EA7FF8A2EA3FF0EA1FE0EA0780C7FCA9EA0780EA1FE0EA3FF0EA7F F8A2EAFFFCA4EA7FF8A2EA3FF0EA1FE0EA07800E257AA41B>58 D<ED03E04B7EA24B7EA3 4B7EA24B7EA34B7EA292B57EA34A8015F702038015E715E302078015C1020F801580A202 1F80ED007F4A80023E133FA2027E80027C7F02FC814A7FA20101824A7F49B77EA34982A2 90270FC000017F4A7FA2011F8291C8127F4982013E153FA2017E82017C81B500F8010FB6 1280A5413A7DB948>65 D<DB3FFC14C00203B5EAC003021FECF00791B6EAFC0F0103EDFE 1F499039FC00FFBF011F01C0EB1FFF4948C71207D97FF8804948804849804849157F4849 153F91C9FC48171F485A180F123F5B1807127FA25B95C7FC12FFAB127FA26DEE07C0A212 3FA27F001FEF0F80A26C7E6CEF1F00806C6D153E6C6D157E6C6D5D6D6C4A5AD93FFEEC07 F090261FFFC0EB1FE0010701FCEBFFC06D90B6C7FC01005D021F14F8020314C09126003F FEC8FC3A3B7BB945>67 D<B87E17F817FF18C018F028007FFC00077F9338007FFEEF1FFF 050713807113C0837113E019F0187F19F8183F19FC181FA219FEA419FFAC19FEA419FC18 3FA219F8187F19F0F0FFE05F4D13C04D1380051F1300EF7FFE933803FFFCB912F018C095 C7FC17FC178040397DB849>I<DB3FFC14C00203B5EAC003021FECF00791B6EAFC0F0103 EDFE1F499039FC00FFBF011F01C0EB1FFF4948C71207D97FF8804948804849804849157F 4849153F91C9FC48171F485A180F123F5B1807127FA25B95C8FC12FFAA047FB61280127F A27FA2003FDB0007EBC000A27F121FA26C7E7E806C7F6C7F6C7F6D7ED93FFE5C90261FFF C05B010701FC90B5FC6D90B5EAFE7F0100EDFC1F021FECF0070203ECC0009126003FFEC9 FC413B7BB94B>71 D<B612FEA539007FFC00B3B3ABB612FEA51F397EB824>73 D<B77EA526007FFCC9FCB3AAEF0F80A5171F1800A35FA35F5FA25E16075EEE7FFEB8FCA5 31397DB839>76 D<B500FC0407B512F06E5E6E5EA3C66C6D4BEBE000A2017D6D157DA201 7C6D15F9A36E6CEC01F1A26E6CEC03E1A26E6CEC07C1A36E6CEC0F81A26E6CEC1F01A26E 6D133EA36E6D137CA26E6D13F8A292397FF001F0A392393FF803E0A292391FFC07C0A292 390FFE0F80A3923907FF1F00A26F13BEA26F13FCA36F5BA2705AA2705AB500FE0303B612 F0A2EE1FC0A2EE0F8054397DB85B>I<B8FC17F017FE8318C028007FFC000713E0040113 F09338007FF8EF3FFCA218FE171FA218FFA718FEA2173F18FCA2EF7FF8933801FFF0040F 13E091B71280180017FC17E002FCC9FCB3A2B612FEA538397DB841>80 D<B712FCEEFFE017FC17FF18C028007FFC000F7F04017F706C7E717EA2717EA284A76017 3F604D5A4D5A4C5B040F5B91B7C8FC17FC5F17FE913AFC003FFF80040F7F707F82707FA2 707FAB1AE0F101F0A2EF7FFC1AE0B600FE90383FFE0394381FFF0771EBFFC00503148005 001400CBEA0FFC443A7DB848>82 D<D903FF1306013FEBE00E90B5EAF83E48ECFE7E48EC FFFE3807FE01390FF0003F4848130F484813031501485A1500167E12FFA2163E7F7F7F6D 91C7FCEBFF806C13F8ECFFC015FC6C14FF6C8116E06C816C816C816C816C7E011F801307 D9003F14801403EC001F030713C0818100788012F8167FA36C1680A27EEEFF007E6D5C01 E0130101F8495A9039FF801FF891B55A00FC5DD8F83F1480D8E00749C7FC39C0007FF02A 3B7BB935>I<B600F80103B512E0A5C601FCC83807E000017F5F6E150FA26D6C5E181F6D 94C7FC6F5C6D163E6F147E6D167C8118FC6D5E6F13016D5E6F13036D5E811707027F5D6F 130F023F5D6F131F6E92C8FC16805F6E143EEEC07E6E147CEEE0FC6E5C16F016F16E5C16 FB6E5C16FF6F5BA36F5BA26F90C9FCA26F5AA36F5AA26F5AA26F5A433A7EB848>86 D<EB3FFE0003B512E04814F84880391FF007FE393FF801FF6E7F82157F6C4880A26C5AEA 0380C8FCA291B5FC130F90B6FC0003EBF87F481380381FFC00485A5B485A485AA515FF6C 6C5A6C6C487F903AFC0FBFFFC06CB5123F0007497E6CEBF80F39007FC0032A257DA42E> 97 D<13FFB5FCA512077EAEEDFF80020F13F8023F7F91B6FCDAFE031380DAF00013C002 C0EB3FE04A14F091C7121F17F8A2EE0FFCA317FEA917FCA3EE1FF8A217F06EEB3FE06E13 7F02F0EBFFC09026FDFC07138001F8B5EAFE00023F5BD9F00F13F0D9E00190C7FC2F3A7E B935>I<903801FFC0010F13FC017F7F90B6FC48018013802607FE0113C0EA0FFC13F8EA 1FF0003F6D1380A24848EB7F00151C92C7FC12FFA9127F7FA2123F6DEB03E0121F6C6CEB 07C07F6C6CEB0F8000019038E03F006CEBFFFE6D5B010F13F00101138023257DA42A>I< EE7F80ED7FFFA5150381AEEB03FF011F13F1017F13FD48B7FC48EBC07F3907FE000F4848 7F484813015B123FA2485AA312FFA9127FA36C7EA2121F6D5B000F140FD807FE4913C06C 6C6CB512FE6CEBFFFD6C6C13F9011F13C1903803FE012F3A7DB935>I<49B47E011F13F0 017F13FC90B57E0003903881FF803907FE007F4848EB3FC04848EB1FE05B003FEC0FF0A2 485A16F8150712FF90B6FCA401E0C8FCA4127FA27F123F16F86C7E6C6C13016DEB03F06C 6CEB07E00001EBE01F6C90B51280013F1400010F13FC010013C025257DA42C>I<EC0FF8 ECFFFC010713FF491480EB3FF8D97FE113C0EBFFC114815A1401486D1380ED7F00153E92 C7FCA8B6FCA5000390C8FCB3A9B512FEA5223A7DB91D>I<163FD907FEEBFF8090267FFF E113C048B512FB489138FFDFE02607FC03131F380FF000001F159F484890387FCFC0EEC7 80007FEDE000A7003F5DA26C6C495A000F92C7FC3907FC03FE90B55A485C6D13E0261E07 FEC8FC90CAFCA2123FA213C06CB512F8EDFF8016E06C81826C816C81000F815A273FC000 0F13804848130148C8127FA56C6CECFF006C6C495AD81FF0EB07FCD80FFEEB3FF86CB65A 000115C06C6C91C7FC010713F02B377DA530>I<13FFB5FCA512077EAEED7FC0913803FF F84A7F021F7F91383F03FFDA7801138014F04A6C13C05C5CA391C7FCB3A2B5D8FC3F13FF A5303A7DB935>I<13F0EA03FC487E487EA2481380A46C1300A26C5A6C5AEA00F090C7FC A813FF127FA512077EB3A7B512F8A5153B7DBA1B>I<13FFB5FCA512077EAE92383FFFE0 A592380FF0004B5A4B5A4BC7FC15FEEC03FC4A5A4A5AEC1FC0143F4A7E4A7E81A281ECCF FEEC87FF1407496C7F6E7F6E7FA26F7E6F7E6F7EA26F7EB539F83FFFF8A52D3A7EB932> 107 D<13FFB5FCA512077EB3B3AAB512FCA5163A7DB91B>I<01FED97FE0EB0FFC00FF90 2601FFFC90383FFF8002076D90B57E021FD9FF0380DA3F03903987E07FF0DA78009039CF 001FF800074914DE6C6C48D97FFC6D7E4A5C4A5CA391C75BB3A2B5D8FC1FB50083B512F0 A54C257DA451>I<01FEEB7FC000FF903803FFF84A7F021F7F91383F03FFDA7801138000 0713F06C6C486C13C05C5CA391C7FCB3A2B5D8FC3F13FFA530257DA435>I<903801FFC0 010F13F8017F13FF90B67E0003018013E03A07FE003FF0D80FF8EB0FF8001F8149130700 3F81491303007F81A300FF1680A9007F1600A36C6C495AA2001F5D6D130F6C6C495A6C6C 495A6C6C6CB45A6C90B55A6C6C91C7FC011F13FC010113C029257DA430>I<01FFEBFF80 B5000F13F8023F7F91B6FCDAFE071380DAF00113C000039039C0007FE04A14F091C7123F 17F8161F17FCA2160F17FEA917FC161FA217F8163F17F06EEB7FE06E13FFDAF00113C0DA FC07138091B5EAFE00023F5B020F13F0020190C7FC91C9FCABB512FCA52F357EA435>I< 9038FE07F000FFEB1FFC4A7E4A7E02F8138002E113C0EA07FF6C13C1A202801380A2ED7F 00151C91C8FCB3B512FEA522257EA427>114 D<90383FF0383903FFFEF84813FF121F38 3FC00FEB0003007E1301140012FEA27E6D130013F8EBFFE06C13FC14FF6C14C06C14E06C 14F0000314F8C614FC131F9038007FFE14070078130112F814007EA26C14FC6C13010180 13F89038F00FF090B512E000FD14C000F0140038E01FF81F257DA426>I<131FA55BA45B A25BA25A5A5A001FEBFFE0B6FCA4000390C7FCB115F8A86CEB01F014816CEBC3E090387F FFC06D13806D1300EB03FC1D357EB425>I<01FFEC3FC0B5EB3FFFA5000714016C80B3A2 5DA25D6C5C4B13E06CD9C03E13FF90387FFFFC6D5B6D13E00103130030257DA435>I<B5 39F003FFF8A5000390C7EA3E006E137E6C157C6E13FC6C5DECE001017F5CA2ECF003013F 5CECF807011F5CECFC0F010F5C151FD907FE90C7FCA26E5A6D133E15FE6D5BA26D5BA36E 5AA26E5AA26E5AA26E5A2D257EA432>I<B500F1B538807FFFA50007903B000FF80007E0 6C6E6C14C0180F6C0180168082DAC00F141F6C6F14004B5CD97FE0153E17806E48147E01 3F90393E7FC07C037E14FC90271FF87C3F5B17E0DAFCFC13E1010F9039F81FF1F014FD91 39FFF00FF36DEDFBE017FF6D496C5BA24B7E6D5EA26D496C90C7FCA36EC75AA2023E147C 40257EA445>I<B539F01FFFF0A500019039C007F0006C5D6D6C485A6D6C485AECF83F6D 6C48C7FC010F13FE6DB45A6D5B5D6D5B7F147F6E7E814A7E14FF497F903803FBFFD907F3 7F02E17FEB0FC049486C7E013F6D7ED97F007F01FE6D7E49130FB539803FFFF8A52D257E A432>I<003FB612C0A4D9F0031380D9C007130049485AEB001F007E5C4A5A147F007C49 5A5D495B5B495B000091C7FC495A131F5C90393FF807C0137FEBFFF014E04813C048140F 48138014004848EB1F80121F49133F4848137F397FF003FFB7FCA422257DA42A>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmmi7 7 6 /Fl 6 111 df<017FB512F016FEEEFF80903A03F0003FC0161FEE0FE04948130717F0A3 494814E0160F17C0161F4948EB3F80EE7F0016FEED03F849B512E0A291380003F8ED00FC 017E147E821780A25BA44848EC7F00A216FE4B5A4848495AED0FF0ED3FE0B712804BC7FC 15F02C287CA732>66 D<90387FFFE0815DD903F0C7FCA3495AA4495AA4495AA449C8FCA4 137EA216301638491470A216E0A24848130116C01503ED07804848130F153F913801FF00 B7FC5DA225287CA72E>76 D<017FB512C016F882903903F000FFEE3F80161F4948EB0FC0 A44948131FA3EE3F8049481400167E5E4B5A90393F000FE091B5C7FC5DEC001F017E6D7E 6F7E82A25BA44848130FA3173048481570A21760B5D8800713E0923803E1C0923801FF80 C9EA7E002C297CA732>82 D<000FB712C017E04816C09039800FC00FD81E0014035A0038 EB1F80A2481680A2EC3F005AA2481600C7007E90C7FCA45CA4495AA4495AA4495AA4495A A3000FB57E48806C5C2B287DA727>84 D<1407EC0F80141FA21500140E91C7FCA7EB03E0 EB07F8EB1E3C1318EB303E137013E0A248485AA2C7FCA25CA4495AA4495AA4495AA4495A A21238D87C1FC7FC12FC133E485AEA70F8EA7FE0EA1F80193380A61B>106 D<3907801FC0390FE07FF03918F0E0F83938F1807C3830FF00D870FE133C5BA2D8E1F813 7C5B1201A248485BA34A5AEA07C016E0EC03E0A23A0F8007C1C0A2EDC38002031300D81F 0013C7EC01FE000EEB00F8231B7D9929>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr7 7 9 /Fm 9 117 df<EB3F803801FFF03803E0F83807803C48487E001E7F003E1480A2003C13 07007C14C0A400FC14E0AE007C14C0A36CEB0F80A36CEB1F006C131E6C6C5A3803E0F86C B45A38003F801B277EA521>48 D<13381378EA01F8121F12FF12FE12E01200B3AAB512F8 A315267BA521>I<13FF000313E0000F7F381E07F8383801FC486C7E0078137F00FC7F6C 1480A2141FA2127CC7123F1500A2147EA25C5C495A495AEB078049C7FC131E5B13709038 E00380EA01C0EA03803907000700120E1218003FB5FC5AB55AA319267DA521>I<133F38 01FFC03803C1F038078078380F003C121E003E133E141E5A141F12FCA2B6FCA200FCC7FC A4127C127E003E1307A26C130E7E3807801C3803E0783800FFE0EB3F80181C7E9A1E> 101 D<EB3F80EBFFE03803E0F83807803C48487E001E7F003E1480A248EB07C0A300FC14 E0A7007C14C0A2007E130F003E1480001E1400001F5B380F803E3803E0F86CB45A38003F 801B1C7E9A21>111 D<380F83FC38FF9FFF9038BC0FC09038F007E0390FC003F0EB8001 15F8EC00FCA2157C157EA7157C15FCA2EC01F801C013F0EC03E09038F007C09038BC1F80 90389FFF00EB83F80180C7FCA7EAFFF8A31F257E9925>I<380F07C038FF1FF0EB38F813 71EA1F61EA0FC1EBC0F014005BAFEAFFFCA3151A7E991A>114 D<3803F8C0EA0FFFEA3C 0FEA78031270EAF001A312F86CC7FCEA7FC013FC6CB4FC6C1380000713C0C613E0130738 E003F0130113007EA214E0EAF80100FC13C038FE078038E7FF00EAC1FC141C7E9A1A>I< EA01C0A51203A31207A2120F383FFFE0B5FCA23807C000AB1470A7000313E013E03801F1 C03800FF80EB3E0014257FA31A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmex10 10 24 /Fn 24 68 df<1438147814F0EB01E0EB03C0A2EB0780EB0F005B131E133E5BA25BA248 5AA212035B1207A2485AA3485AA4123F90C7FCA35AA5127EA212FEB3A2127EA2127FA57E A37F121FA46C7EA36C7EA212037F1201A26C7EA2137CA27F131E131F7FEB0780EB03C0A2 EB01E0EB00F0147814381562738226>0 D<126012F012787E7EA27E6C7E7F12037F6C7E A26C7EA2137CA2137E133E133FA2EB1F80A3EB0FC0A414E01307A314F0A51303A214F8B3 A214F0A21307A514E0A3130F14C0A4EB1F80A3EB3F00A2133E137E137CA25BA2485AA248 5A5B12075B48C7FC121EA25A5A5A126015627D8226>I<151E15FE1407EC1FF0EC7FC0EC FF00EB01FC495A495AA2495AB3AC495A5C133F49C7FCEA01FEEA03F8EA0FF0EA7FC000FE C8FCA2EA7FC0EA0FF0EA03F8EA01FEEA007F6D7E131F806D7EB3AC6D7EA26D7E6D7E6DB4 FCEC7FC0EC1FF0EC07FE1400151E1F62788230>8 D<127012FEEA7FC0EA1FF0EA07FCEA 01FEEA007F6D7E6D7EA26D7EB3AC6D7E1303806D7E6DB4FCEC3F80EC1FE0EC07FCEC00FE A2EC07FCEC1FE0EC3F80ECFF00EB01FC495A5C1307495AB3AC495AA2495A49C7FCEA01FE EA07FCEA1FF0EA7FC000FEC8FC12701F62788230>I<151E153E157C15F8EC01F0EC03E0 15C0EC0780140FEC1F00143E143C147C5C13015C495AA2495A130F5C131F91C7FC5BA213 7EA25BA3485AA3485AA212075BA3120F5BA3121FA25BA2123FA490C8FCA25AA7127EA212 FEB3A6127EA2127FA77EA27FA4121FA27FA2120FA37F1207A37F1203A26C7EA36C7EA313 7EA27FA27F80130F8013076D7EA26D7E801300147C143C143E80EC0F801407EC03C015E0 EC01F0EC00F8157C153E151E1F94718232>16 D<127012F8127C7E7E6C7E12076C7E7F6C 7E6C7E1378137C7F133F7F6D7EA26D7E80130380130180A26D7EA2147EA380A3EC1F80A2 15C0140FA315E01407A315F0A21403A215F8A41401A215FCA71400A215FEB3A615FCA214 01A715F8A21403A415F0A21407A215E0A3140F15C0A3141F1580A2EC3F00A3147EA35CA2 495AA25C13035C13075C495AA249C7FC5B133E5B137813F8485A485A5B485A120F48C8FC 123E5A5A12701F947D8232>I<160F161F163E167C16F8ED01F0ED03E016C0ED0780150F ED1F00153E5DA25D4A5A14035D14074A5A5D141F4AC7FCA2147EA25CA2495AA2495AA249 5AA2130F5C131F5C133FA291C8FC5BA2137E13FEA3485AA3485AA4485AA4120F5BA3121F A35BA2123FA55BA2127FA990C9FC5AB3AC7E7FA9123FA27FA5121FA27FA3120FA37F1207 A46C7EA46C7EA36C7EA3137E137FA27F80A2131F80130F801307A26D7EA26D7EA26D7EA2 147EA280A26E7E140F816E7E14038114016E7E157CA28181ED0F801507ED03C016E0ED01 F0ED00F8167C163E161F160F28C66E823D>I<127012F8127C7E7E6C7E6C7E12036C7E7F 6C7E137C7FA27F6D7E801307806D7E1301806D7EA2147EA280A26E7EA26E7EA26E7EA281 140381140181A2140081A2157E157FA3ED3F80A3ED1FC0A4ED0FE0A416F01507A316F8A3 1503A216FCA51501A216FEA9150016FFB3AC16FE1501A916FCA21503A516F8A21507A316 F0A3150F16E0A4ED1FC0A4ED3F80A3ED7F00A3157E15FEA25D1401A25D14035D14075DA2 4A5AA24A5AA24AC7FCA2147EA25CA2495A5C1303495A5C130F5C49C8FC133EA25B5B485A 5B485A1207485A48C9FC123E5A5A127028C67E823D>I<161E167EED01FE1503ED0FF8ED 1FE0ED3FC0ED7F80EDFE004A5A4A5A14074A5A4A5AA24A5AA24A5AA44AC7FCB3B3B3A35C 1301A3495AA25C1307495A5C131F495A5C49C8FC13FEEA03FCEA07F0485AEA3FC048C9FC 12FCA2127FEA3FC0EA0FE06C7EEA03FCC67E137F6D7E806D7E130F806D7E130380A26D7E A3130080B3B3B3A36E7EA46E7EA26E7EA26E7E6E7E14036E7E6E7EED7F80ED3FC0ED1FE0 ED0FF8ED03FE1501ED007E161E27C675823E>26 D<127012FCB4FCEA7FC06C7EEA0FF86C 7EEA01FE6C7E6D7E6D7E6D7E6D7EA26D7E6D7EA2130180A36D7EB3B3B3A38081A36E7EA2 141F81140F816E7E14036E7E6E7E157FED3F80ED1FC0ED0FF0ED03F8ED01FEED007EA2ED 01FEED03F8ED0FF0ED1FC0ED3F80ED7F0015FE4A5A4A5A14074A5A5D141F5D143FA24A5A A392C7FC5CB3B3B3A3495AA35C1303A2495A495AA2495A495A495A49C8FC485AEA07FC48 5AEA3FE0485A48C9FC12FC127027C675823E>I<EE01E0A2EE03C0EE0780EE0F00161E16 3E5E5EA24B5A4B5A15075E4B5A151F93C7FC153E157E157C15FC4A5AA24A5AA24A5AA24A 5A141F5D143F92C8FC5CA214FEA25C1301A2495AA213075CA2495AA3131F5CA2133F5CA2 137FA291C9FC5BA35B1201A3485AA5485AA5120F5BA4121FA35BA4123FA55BA3127FAA5B A312FFB3B2127FA37FAA123FA37FA5121FA47FA3120FA47F1207A56C7EA56C7EA312007F A37F80A2133FA280131FA280130FA36D7EA2801303A26D7EA2130080A2147FA28081141F 81140F6E7EA26E7EA26E7EA26E7E157C157E153E8182150F6F7E8215036F7E6F7EA2167C 82161E82EE0780EE03C0EE01E0A22BF86C8242>32 D<127012F012787E7E7E7F6C7E6C7E A26C7E6C7E7F137C7F133F7F6D7E801307806D7EA26D7EA26D7EA2147E147F8081141F81 A26E7EA2140781A26E7EA2811401A26E7EA38181A282153FA282A2151F82A3150F82A36F 7EA56F7EA5821501A482A381A41780A5167FA317C0AA163FA317E0B3B217C0A3167FAA17 80A316FFA51700A45DA35EA415035EA54B5AA54B5AA35E151FA35E153FA25EA2157F93C7 FCA25D5DA34A5AA214035DA24A5AA25D140FA24A5AA25D143F92C8FC5C147E5CA2495AA2 495AA2495A5C130F5C49C9FC5B133E5B13FC5B485A485AA2485A485A90CAFC121E5A5A5A 12702BF87E8242>I<177CA217F8EE01F01603EE07E017C0EE0F80161FEE3F00A2167E5E A24B5A15035E4B5A150F5E151F5E153F93C7FC5D15FEA24A5AA24A5AA214075D140F5D14 1FA25D143F5D147FA292C8FC5CA2495AA3495AA3495AA3495AA3131F5CA2133FA25CA213 7FA25C13FFA44890C9FCA5485AA512075BA4120FA25BA5121FA35BA5123FA65BA3127FAD 5BA212FFB3A62E95688149>48 D<127812F8127C7E123F6C7E120F6C7E7F6C7EA26C7E6C 7EA2137E137F7F6D7E80130F801307801303806D7EA26D7EA2147FA281143F81141F81A2 140F81140781A2140381A26E7EA36E7EA36F7EA36F7EA382151FA282A2150FA282A21507 82A46F7EA56F7EA5178081A417C0A2167FA517E0A3163FA517F0A6161FA317F8AD160FA2 17FCB3A62E957E8149>I<EC01F01407140F143F147FECFFC0491380903803FE00495A49 5A495A495A495A13FF485B91C7FC5A5B1207485AA2485AA25B123FA25B127FA4485AB3B3 A81C4B607E4A>56 D<127812FE7E13C07FEA3FF0EA1FFCEA07FE6C7E6C7F6C7F6D7E6D7E 131F6D7E806D7E1303807F15807F15C0A2EC7FE0A3EC3FF0A4EC1FF8B3B3A81D4B737E4A >I<EAFFC0B3B3A86C7EA4123F7FA2121F7FA26C7EA26C7E12037F7E806C7F137F6D7E6D 7E6D7E6D7E6D7E903801FF806D13C0EC7FF0143F140F140714011C4B60804A>I<EC1FF8 B3B3A8EC3FF0A4EC7FE0A3ECFFC0A215805B15005B5C1307495A5C495A133F495A495A48 5B4890C7FC485AEA1FFCEA3FF0B45A5B90C8FC5A12781D4B73804A>I<EC1FF8B3B3A715 F0143FA415E0147FA215C014FF1580A25B15005B5C495AA2495A5C495A133F495A5C49C7 FC485A485AEA0FF8EA1FE0485AB45A48C8FC5A7E6C7EEA3FC06C7EEA0FF8EA03FC6C7E6C 7E6D7E806D7E131F6D7E806D7EA26D7E807F15807FA215C0147F15E0A2143F15F0A4141F 15F8B3B3A71D9773804A>I<EAFFC0B3B3A7127F7FA4123F7FA2121FA27F120F7F1207A2 6C7EA26C7E7E806D7E133F6D7E6D7E6D7E6D7E6D7E6D7EEC7F80EC3FC0EC1FF014071403 1407141FEC3FC0EC7F80ECFF00495A495A495A495A495A495A137F495A91C7FC5A485AA2 485AA2120F5B121F5BA2123FA25B127FA45B12FFB3B3A71C9760804A>I<EAFFC0B3A612 7FA27FAD123FA37FA6121FA57FA3120FA57FA21207A47F1203A56C7EA56C7FA4137F80A2 133FA280A2131FA280130FA36D7EA36D7EA36D7EA36D7EA28081A2143F81141F81A2140F 811407811403A26E7EA26E7EA2157F8182151F82150F8215076F7E8215016F7EA2167E82 A2EE1F80160FEE07C017E0EE03F01601EE00F8177CA22E95688349>64 D<EE0FFCB3A617F8A2161FAD17F0A3163FA617E0A5167FA317C0A516FFA21780A45D1700 A54B5AA54B5AA45E150FA25EA2151FA25EA2153F5EA34B5AA34BC7FCA34A5AA34A5AA25D 1407A25D140F5D141FA25D143F5D147F92C8FCA214FEA2495AA2495A5C13075C130F5C13 1F5C49C9FC5B137E5BA2485A485AA2485A5B485A121F48CAFC123E5A5A12782E957E8349 >I<EAFFC0B3B3B00A34688049>I<EAFFC0B3B3B00A345A8049>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmti10 10 20 /Fo 20 120 df<387FFFF8A3B512F0A2150579941E>45 D<14F8EB07FE90380FFF1C9038 3F07BE90387C03FEEBF801EA01F000035CEBE0001207485A4A5AEA1F80123FA249485A5A A300FE495AA448495AEDC1C0A391381F8380A2143F127C91387F8700007E13FF393E03CF 8E381F0F8F390FFF07FC3907FC03F83901F000F0222677A42A>97 D<133FEA0FFF5A7EEA007EA45BA4485AA4485AA4485A14F8EBE3FEEBEFFF390FDF0F8090 38FC07C001F013E0EBE003EA1FC015F01380A2EA3F00A4007E1307A448EB0FE0A315C048 131FA21580EC3F00A2147E147C14FC007C5B495A383C03E0381E0FC06CB4C7FC6C5AEA01 F01C3B77B926>I<ED01F8157F15FF157FED03F0A4ED07E0A4ED0FC0A4ED1F80A4ED3F00 14F8EB07FEEB0FFF90383F07FEEB7C03EBF801EA01F000035CEBE0001207485A4A5AEA1F 80123FA249485A5AA300FE495AA448495AEDC1C0A391381F8380A2143F127C91387F8700 007E13FF393E03CF8E381F0F8F390FFF07FC3907FC03F83901F000F0253B77B92A>100 D<147F903803FFC0010F13E090381F81F0EB7E00491378485A485A485AA2485A001F14F0 90388001E0003FEB07C0EC3F8048B5120014FC14E090C8FC12FEA65AA2007C1410153800 7E147015F0003EEB01E06CEB07C0EC1F00380F80FE3807FFF86C13E0C690C7FC1D2677A4 26>I<ED07C0ED0FF0ED1FF8ED3C3CED78FC15F8EC01F915F1EDF0F80203137016005D14 07A44A5AA54A5AA2010FB5128016C016809039001F80004AC7FCA5147EA55CA5495AA549 5AA5495AA45C130FA35C131FA391C8FCA2EA1C3E123E127EEAFE3C5B12FCEA78F0EA7FE0 6C5AEA0F80264C82BA19>I<EC07C0EC3FF091387FF8E0903901F83DF0903803E01F9038 07C00FEB0F80011F14E090383F0007A2137E01FEEB0FC05B1201A249EB1F801203A34848 EB3F00A449137EA45DA20003130114036C6C485A140F3800F83DEB7FF96D485AEB0FC3EB 0003A24A5AA44A5A121C003E495A127E00FE49C7FC147E485B387803F8383FFFE06C1380 D803FEC8FC24367CA426>I<EB03F013FF5A7EEB07E0A4495AA4495AA449C8FCA4137EEC 07F0EC1FFCEC7FFE9038FCF83F9039FDE01F80EBFF80EC000F485A16C05B49EB1F801203 5BA34848EB3F00A3157E485AA35D485A913801F81CA33A3F0003F038A3EDE070127E16E0 A2EDE3C048903801FF80007C6D13000038143C263B7BB92A>I<EB01C0EB03E0EB07F0A2 14E014C0EB038090C7FCAB13F0EA03FC487EEA0F1F121CA21238A2485AA3EAE07EA25B12 00A2485AA3485AA3485AA214E0EA0FC0A2381F81C0A3EB0380A2EB0700A2131EEA0FFC6C 5AEA01E0143879B619>I<EB0FC0EA03FF5A7E38001F80A4EB3F00A4137EA45BA4485AA4 485AA4485AA4485AA4485AA448C7FCA4127E130EA3485AA45BA2EA7C70A2EA3FE06C5A6C 5A123B79B915>108 D<D801E001FEEB07F03C07F803FF801FFC260FFC0F9038C07FFE3C 1E3E1F07E0F83F001C903B3C03F1E01F8026383F70EBF380913AE001F7000F02C013FE00 7018C002805B4A4848EB1F80485A017E5CA21200494948EB3F00A3187E4848495AA36048 48495A943801F81CA34848903A3F0003F038A3F0E0704848137E19E0A2F0E1C048484990 3801FF80000F027C6D13006CC70038143C3E2679A444>I<D801E013FE3A07F803FF8026 0FFC0F7F3A1E3E1F07E0001C90383C03F038383F70ECE00114C000708114804A485A485A 137EA2120049495AA34B5A485AA34B5A485A92383F0380A3484890387E0700A3ED7C0E48 5A5EA25E4848EB3FF0000F6E5A6CC7EA0780292679A42F>I<147F903803FFC04913E090 381FC1F090383E00F849137C49137E4848133E12034848133F485AA2485AA2003F147F90 C7FC5AA300FE14FEA315FC14014814F8A2EC03F0A2EC07E0007CEB0FC01580007EEB1F00 003E133E6C5B380F83F83807FFE06C5BC648C7FC202677A42A>I<9039078007C090390F E01FF090391FF07FF8903938F8F87C9138F9E03E903970FF803F4B7E495A17805CA23801 C1F8A3EA00014948133FA44948EB7F00A3167E494813FEA25E4B5AEB1F804B5A5EECC007 013F5C4B5A6E48C7FCECF07E90387E7FF86E5AEC0F8091C9FC5BA4485AA4485AA3387FFF C0B5FC6C5B293580A42A>I<3903C003F03907F00FFC390FF83FFE391C7C7C1FECF00F3A 387FC01F80153F00709038807F001400153E017E131C484890C7FCA31200485AA4485AA4 485AA4485AA4485AA448C9FC7E120E212679A423>114 D<14FE903807FF804913C09038 1F03E090383C00F049137001F813F0EBF00100011303A215E00003EB01C06DC7FC7F3801 FFC014F8806C7F6D7E6D1380130F010013C0141F140FA2123C127E00FE1480A248140048 5B00E0131E00705B0078137C383E03F06CB45A6C5BD801FEC7FC1C267AA422>I<1307EB 0F80131FA4EB3F00A4137EA45BA2B512FC14FE14FC3801F800A3485AA4485AA4485AA448 5AA448C7FC1438A21470127E14E0A2EB01C0A2EB0380383E0700131EEA1FFC6C5AEA03E0 173578B31C>I<13F8D801FEEB01C0486CEB03E039070F8007120E121C1238ED0FC0EA30 1F1270A23AE03F001F80A3EA007EED3F005BA34848137EA448485B160EA3913801F81CA3 1638140300011307020E13702600F83C13F090397FF87FE090393FF03FC090390FC00F00 272679A42D>I<01F0130ED803FC131F486C5BD80F1FEB7F80121C153F0038141FA24848 130FA21600D8E07E7FA3C6485B150E485AA348485BA35D485AA25DA2156015E05D14014A 5A000349C7FC6D5A3801F81E3800FFFCEB3FF0EB0FC0212679A426>I<01F01507D803FC 903903800F80486C903807C01FD80E1F010FEB3FC0121C171F0038160FED1F8048481507 A21880D8E07E90383F0003A3C6481507037E1400485AA3484849130EA4484848485BA35F A25FA202035C1203D9F007495A000190390EFC03803A00FC1E7E0F90277FFC3FFFC7FC90 393FF00FFC90390FE003F0322679A437>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmsy10 10 10 /Fp 10 95 df<007FB81280B912C0A26C17803204799641>0 D<121E123FEA7F80EAFFC0 A4EA7F80EA3F00121E0A0A7A9917>I<0060150600F0150F6C151F007C153E6C157C6C15 F86C6CEB01F06C6CEB03E06C6CEB07C06C6CEB0F806C6CEB1F00017C133E6D5B6D5B9038 0F81F0903807C3E0903803E7C06DB45A6D90C7FC147EA214FF497F903803E7C0903807C3 E090380F81F049C67E013E137C497F497F4848EB0F804848EB07C04848EB03E04848EB01 F048C812F8003E157C48153E48151F48150F00601506282874A841>I<EB0FE0EB7FFC48 B5FC4814804814C04814E04814F0A24814F8A24814FCA2B612FEA86C14FCA26C14F8A26C 14F0A26C14E06C14C06C14806C140038007FFCEB0FE01F207BA42A>15 D<147014F0A3495AA3495AA2495AA249CCFC5B131E5B5B5B1203EA07C0EA1F80007FBA12 FCBB12FEA26C19FCD81F80CCFCEA07C0EA03F0120013787F7F131F7F6D7EA26D7EA26D7E A36D7EA31470472C7AAA53>32 D<181C181EA384A3727EA2727EA2727E85180019788585 F11F80F107C0F103F0007FBA12FCBCFCA26C19FCCCEA03F0F107C0F11F80F11E00616161 1801614E5AA24E5AA24EC7FCA3181EA3181C482C7BAA53>I<1738173CA283A283717EA2 717E717E8417001878007FB812FCB97E727E6C84CBEA03E0F001F8F0007EF13F80F10FE0 F107F8F101FFA2F107F8F10FE0F13F80F17E00F001F8F003E0007FB95ABA5A4EC7FC6C5F CB1278601701604D5A4D5AA24DC8FC171EA25FA2173848307BAC53>41 D<EE0180EE03C0A2EE0780A2EE0F00A2161EA25EA25EA25EA24B5AA24B5AA24B5A150F93 C7FC151EA25DA25DA25DA24A5AA24A5AA24A5AA24AC8FCA2141EA25CA25CA25CA2495AA2 495AA2495AA249C9FCA2131EA25B137C13785BA2485AA2485AA2485AA248CAFCA2121EA2 5AA25AA25AA212602A4E75BB00>54 D<007FB712F8B812FCA27ECA123CB217182E177C9D 37>58 D<14034A7EA24A7EA24A7EA3EC3CF0A2EC7878A24A7EA349487EA249487EA24948 6C7EA249486C7EA3011E6D7EA2496D7EA2491478A34980A2484880A2484880A24848EC07 80A348C8EA03C0A2001EED01E0A248ED00F0A3481678A248163CA2006016182E347CB137 >94 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmmi10 10 14 /Fq 14 121 df<133F14C080EB07F06D7E6D7EA2130080A2147E147F80A281141F81A214 0F81A2140781A2140381A2140181A2140081140114034A7E4A7EEC1E3F023C13801478EC F01F010114C0EB03E0903807C00FD90F8013E0EB1F00013E13074914F001FC1303484814 F8485A48481301484814FC485A003F140048C812FE12FE48157E48153F0070151F283B7C B930>21 D<121E123FEA7F80EAFFC0A4EA7F80EA3F00121E0A0A7A8917>58 D<121E123FEA7F80EAFFC0A213E0A2127F123F121E1200A4EA01C0A3EA0380A2EA0700A2 120E5A123C123812100B1A7A8917>I<150C151EA2153CA31578A315F0A3EC01E0A3EC03 C0A3EC0780A3EC0F00A3141EA35CA35CA35CA3495AA3495AA2495AA349C7FCA3131EA35B A35BA35BA3485AA3485AA3485AA348C8FCA3121EA35AA35AA35AA212601F537BBD2A>61 D<0103B77E4916F06D16FC903B0007FC0003FE4BEB00FFF07F80F03FC04A5A19E0A2181F 4A5A183FA34A4815C0187F198018FF4A4815004D5A4D5A4D5A4AC7EA1FF0EF7FC04C485A 92B500FCC7FC5B17FF4AC7EA7FC0EF1FE049486E7E84170784495A1703A349481407A449 484A5AA260171F49484A5A4D5A604C48C7FC4948495A4C5A01FFEC3FF0007F90B65AB8C8 FC16F83B397DB83F>66 D<DCFF801340030F01F013C0037FEBFC01913A01FF803E03913A 07FC000F07DA0FF09038038F80DA3FC0EB01DF4AC713FF02FE80D903FCED7F00EB07F049 5A011F824948153E5C49C9FC5B4848163CA2485A1207491638120F5B001F94C7FCA2485A A3485AA45B12FF93B512FE5D8104001380177FA24DC7FC127FA34C5A123FA27F001F4B5A A26C6C14076C7E00034B5A6C6C141DD800FE1478903A7F8001F078903A3FF00FE030010F B5EA8010010349C9FC9038003FE03A3D7DBA41>71 D<0103B6FC5B7F90260007FEC8FC15 F8A34A5AA44A5AA44A5AA44A5AA44AC9FCA4495AA4495AA4494815C01701A2EF0380495A A2EF0700A249485C170E171EA249485C177C17FC4C5A49481307EE1FF001FF14FF007F90 B6FCB85AA232397DB839>76 D<0103B7FC4916E06D16F8903B0007FC0007FC4BEB01FEEF 007F19804A48EC3FC0A319E04A5AA44A48EC7FC0A319804A4814FF19004D5AA24AC7485A 4D5AEF0FE0EF3FC0494849B4C7FC91B612FC17F094C8FCD903FCCAFCA4495AA4495AA449 5AA4495AA4495AA213FF007F13FFB67E92CAFC3B397DB835>80 D<91B500FC90B512E049 5D83D9000301C090381FFC006E90C7EA0FE0198096C7FC6E6D131E60606F6C5B4D5A6F6C 485A604DC8FC92381FF00E5F030F133C705A5F923807FDE0EEFFC05F6F90C9FC5E6F7EA3 6F7F5D5D4B7F167F92380E3FE0151C15389238781FF015F0913801E00FDA03C07FEC0780 4A486C7E140E5C4A6D7E5C02F06D7E495A495A49486D7F49C8FC49157F017F4B7E2603FF 8001037F007F01F0011FEBFF80B56C5B6C497F43397EB845>88 D<133FEA0FFF5AA2EA00 7EA45BA4485AA4485AA4485A14FCEBE3FF01E713C0390FDF07E09038FC01F013F801F013 F8381FE0004913FC5BA248C7FCA4007E1301A448EB03F8A315F014074814E0A2EC0FC0A2 EC1F801500007C133E147E003C5B383E01F0381F07E06CB45A6C90C7FCEA01FC1E3B7CB9 24>98 D<EB0FC0EA03FF5A7E38001F80A4EB3F00A4137EA45BA4485AA4485AA4485AA448 5AA4485AA448C7FCA4127E1307A3EAFC0EA35BA3EA7C381378EA3FF06C5AEA0780123B7D B919>108 D<D803E0017F14FE3D07F803FFC007FF80486C48D9F00F13E03D1C3E1F81F8 3F03F0DA3C00EB780126383F78D9FCF07F4AEBFDE000704990387DC0004AEB7F804A91C7 FC91C700FE1301485A017E5CA21200494948495AA34E5A4848495AA24E5AA24848495A95 381F80E0A34848494890383F01C0A395383E03804848495AF10700190EF01E1E484849C7 EA1FFC000F6E6E5A6CC7000EEC03E043267EA449>I<D803E0137F3A07F803FFC0486C48 13F03A1C3E1F81F8EC3C0026383F787F5C007049137C5C5C91C712FC485A137EA2120049 495AA34B5A485AA24B5AA2485A92380FC070A3484890381F80E0A3EE01C0485AEE0380EE 07006F5A484814FE000FEC07F86CC7EA01F02C267EA432>I<90390FC007E090393FF00F F890397FF83FFC9039F07C783C3A01C03EF03E3A03801FE07ED80700EBC0FE000E14C115 815AED80FC4890383F00701600A2C7FC147EA45CA4495A1638A2121C267E03F0137012FF 16E0130700FEEC01C0D8FC0FEB03803AF81EF8070039783C7C1E397FF83FFC393FE01FF8 390FC007E027267DA42F>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr10 10 64 /Fr 64 123 df<EC07F8EC3FFE49B57E903903FC07C090390FE001E090381FC003494848 7E90387F000F137E13FE496D5A00016E5A6F5A92C8FCA9ED07F0B7FCA33901FC001F1507 B3AA486C497E267FFFE0B512C0A32A3B7FBA2E>12 D<001E130F003FEB1F80397F803FC0 39FFC07FE0A201E013F0A2007F133F003F131F001EEB0F7000001300A4484813E0A33903 8001C0A23907000380A2000EEB070048130E003C131E0038131C001013081C1A7DB92A> 34 D<14E01301EB03C0EB0780EB0F00131EA25B5B13F85B12015B12035B1207A2485AA2 121F90C7FCA25AA2123EA2127EA5127C12FCB2127C127EA5123EA2123FA27EA27F120FA2 6C7EA212037F12017F12007F13787F7FA27FEB0780EB03C0EB01E01300135278BD20>40 D<126012F012787E7E7EA26C7E6C7E7F12017F12007F1378137CA27FA2133F7FA21480A2 130FA214C0A5130714E0B214C0130FA51480A2131FA21400A25B133EA25BA2137813F85B 12015B12035B485A48C7FCA2121E5A5A5A126013527CBD20>I<121E123FEA7F80EAFFC0 A213E0A2127F123F121E1200A4EA01C0A3EA0380A2EA0700A2120E5A123C123812100B1A 7A8917>44 D<B512FCA616067F941C>I<121E123FEA7F80EAFFC0A4EA7F80EA3F00121E 0A0A7A8917>I<150C151EA2153CA31578A315F0A3EC01E0A3EC03C0A3EC0780A3EC0F00 A3141EA35CA35CA35CA3495AA3495AA2495AA349C7FCA3131EA35BA35BA35BA3485AA348 5AA3485AA348C8FCA3121EA35AA35AA35AA212601F537BBD2A>I<EB03F8EB1FFF017F13 C09038FE0FE03901F803F048486C7E48486C7E4848137EA248487FA2003F1580A290C712 1F4815C0A64815E0B36C15C0A56C6CEB3F80A3001F1500A26C6C137EA26C6C5B6C6C485A 6C6C485A3900FE0FE090387FFFC0011F90C7FCEB03F8233A7DB72A>I<EB01C013031307 131F13FFB5FCA2131F1200B3B3A8497E007FB512F0A31C3879B72A>I<EB0FF0EB7FFE48 B57E3903E03FC03907800FE0390E0007F0486D7E48806E7E5A6E7E127CB4FC16807F157F A26CC7FCA2001C14FFC8FC1600A25C5D5D14035D4A5A4A5A5D4A5A4AC7FC143E5C5C495A 495A495A5C49C8FC011EEB03805B5B491307484814005B485A48C75A48B6FC5A5A485CB6 FCA321387CB72A>I<EB07F8EB3FFF4913C03901F80FE03903E003F0D807807F496C7E48 8013C0486C6C7EA4120F495AD803805BC7FCA25D14035D4A5A5D4A5AEC7F80D91FFEC7FC 5CECFF809038000FE0EC03F06E7E6E7E81157F1680A2ED3FC0A216E0A2123E127F487EA3 16C090C7FC48147F007C158012706CECFF006C5C000F495A3907C003F83903F80FF06CB5 12C06C6C90C7FCEB07F8233A7DB72A>I<157815F8A214011403A21407A2140F141FA214 3F147B147314F3EB01E314C31303EB07831403130F131E131C133C1378137013F0EA01E0 13C012031380EA07005A120E121E5A123812785AB712F8A3C73803F800AB4A7E0103B512 F8A325397EB82A>I<0004140C000F143C9038F003FC90B55A5D5D158092C7FC14FC000E 13F090C9FCABEB07F8EB1FFE90387FFF809038F80FC0390FE007E090388003F0496C7E00 0E6D7E1206C87EA2157FA31680A31218123E127F5AA316005A00785C00705CA26C495A5D 6C495A6C495A6C6C485A3903E03F806CB5C7FC38007FFCEB1FE0213A7CB72A>I<EC3FC0 903801FFF0010713FC90380FE03E90381F000F013E5B49EB3F8049137F485A485A120749 EB3F00000F141E001F91C7FC5BA2123FA3387F0020EB03FF4913C0497F39FF1C03F09038 3000F849137C157E497FA2491480151F16C0A290C7FC16E0A47EA57E6D14C0A2121FED3F 80120F6D14000007147E6C6C137C6D5B6C6C485A3900FE07E090383FFFC06D90C7FCEB03 FC233A7DB72A>I<1238123C123F90B612E0A316C0481580A216000078C7120E00705CA2 5D00F05C5A5D4A5AC7FC4A5A4AC7FC140EA25C143C14381478147014F0A2495AA21303A2 495AA2130FA3131F5CA3133FA5137FA96DC8FC131E233B7BB82A>I<EB03F8EB1FFF4913 C09038FC07E03901F001F03903C000F8157C48487F120F90C7FC815AA37FA26D133EA2D8 0FF0133C6D137CD807FE5B6D5B6CEBC3E06CEBE7C06CEBFF806D48C7FC6D7E010F7F013F 13E0497FD801F813FC3903E03FFE3807C01F390F8007FF48486C1380001E1300003EEC7F C048141FED0FE0A2481407A21503A416C0127CED07807EED0F006C141E6C6C5BD807E013 F83903FC03F0C6B55A013F1380D907FCC7FC233A7DB72A>I<EB03F8EB1FFF017F7F9038 FC07E03901F001F048486C7E48487F000F147C4848137E153E003F143F48C7FC1680A248 141F16C0A516E0A47E153FA27E7F001F147FA26C6C13DF12073903E0019F3901F8071F26 00FFFE13C0EB7FFCEB1FF8EB00801400ED3F80A31600A2000F147E487E486C5B5D14015D 49485A6C48485A001EEB1F80260FC07FC7FC3807FFFC000113F038003FC0233A7DB72A> I<121E123FEA7F80EAFFC0A4EA7F80EA3F00121EC7FCB0121E123FEA7F80EAFFC0A4EA7F 80EA3F00121E0A247AA317>I<121E123FEA7F80EAFFC0A4EA7F80EA3F00121EC7FCB012 1E123FEA7F8012FF13C0A3127F123F121F1201A4EA0380A312071300A2120E121E121C5A A212100A347AA317>I<007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17 F836167B9F41>61 D<1538157CA315FEA34A7EA34A7FA34A7F153FA2020F7FEC0E1FA202 1E7FEC1C0FA2023C7FEC3807A202787FEC7003A202F07FECE001A20101804A7EA2010381 4A137FA201078191C7123F91B6FC4981A2010EC7121F011E81011C140FA2013C81013814 07A201788101701403A201F08116011201486C81D80FFE02071380B500C090B512FEA337 3C7DBB3E>65 D<B712E016FC16FF0001903980007FC06C90C7EA1FE0707E707E707EA270 7EA283A65FA216035F4C5A160FEE1FE0EE7FC04B485A91B548C7FCA2707E91C7EA3FE0EE 0FF0707E707E707E707EA21880177F18C0A7188017FFA24C13005F16034C5AEE1FF8486D EB7FF0B812C094C7FC16F832397DB83B>I<B812F8A30001903880001F6C90C71203EE01 FC1600177C173CA2171CA4171E170EA2ED0380A21700A41507A2150F153F91B5FCA3EC00 3F150F1507A21503A692C8FCAD4813C0B612C0A32F397DB836>70 D<B648B512FEA30001902680000313006C90C76C5AB3A491B6FCA391C71201B3A6486D49 7EB648B512FEA337397DB83E>72 D<B612C0A3C6EBC0006D5AB3B3AD497EB612C0A31A39 7EB81E>I<B612E0A3000101C0C8FC6C90C9FCB3AD1738A517781770A417F0A21601A216 031607160FEE3FE04890388001FFB8FCA32D397DB834>76 D<B56C92380FFFF8A300016D 92381FFC006C60D9EFE0153BA3D9E7F01573A3D9E3F815E3A2D9E1FCEC01C3A3D9E0FEEC 0383A3027FEC0703A26E6C130EA36E6C131CA36E6C1338A26E6C1370A36E6C13E0A29139 01FC01C0A3913900FE0380A392387F0700A2ED3F8EA3ED1FDCA3ED0FF8A26F5A487E487E D80FFE6D48497EB500E00203B512F8A2ED01C045397DB84C>I<B591380FFFFE80A2C66D 010013E06EEC3F80EF1F00D9EFF0140E8013E7EBE3FC8013E1EBE0FF81147F81143F6E7E 81140F6E7E8114036E7E8180ED7F8016C0153FED1FE016F0150FED07F816FC1503ED01FE 16FF81EE7F8E17CE163FEE1FEE17FE160FA216071603A216011600A2486C157E486C153E EA0FFEB500E0141E170EA237397DB83E>I<B612FEEDFFE016F8000190388007FE6C90C7 6C7EEE3FC0707E707E707EA2707EA283A65FA24C5AA24C5A4C5A4C5AEEFF80DB07FEC8FC 91B512F816E0A291380007F8ED01FC6F7E167F707E83161F83A683A560F00380EE0FF8A3 486D0107EB0700B6EB03FC933801FE0E933800FFFCCAEA3FF8EF07F0393B7DB83D>82 D<D90FF813C090383FFE0190B512813901F80FE33907E001F7390F80007F90C7123F4814 1F003E140FA2481407A200FC1403A415017EA27E6C91C7FC7F13E0EA3FF8EBFF806C13F8 6CEBFF806C14E06C14F86C806C80013F7F01071480D9007F13C0020713E0EC007FED1FF0 150F150716F81503126012E01501A47E16F0A26C14036C15E0A26CEC07C06CEC0F80D8FB C0EB1F00D8F9F0133ED8F0FF13FC39E03FFFF8010F13E0D8C00190C7FC253D7CBA2E>I< 003FB812E0A3D9E003EB003F90260001FE1307007EEE03F0007C160100781600A3007017 70A400F01778481738A4C71600B3B0913807FF80011FB612E0A335397DB83C>I<B69038 0FFFFEA300010180010013E06C90C8EA3F80EF1F00170EB3B27F5FA280013F5DA26D6C5C 130F6E5C01074A5A6D6C13036D6C495AD900FE011FC7FC91383F807C6EB45A020713E002 001380373B7DB83E>I<B500FC91B51280A30003018091381FF8006C90C8EA07E0606C70 5AA26D6C4AC7FCA280013F150EA26E141E011F151CA26E143C010F1538A26D6C5CA28001 035DA26E130101015DA26E13036D5DA26E6C48C8FCA215C0023F130EA2EDE01E021F131C A2EDF03C020F1338A26E6C5AA215FC02035BA215FF6E5BA36E5BA26FC9FCA3153EA3151C 393B7EB83E>I<B5D8FE01B5D8FC01B512C0A300039026C0000790C7381FFC006C90C76C 48EC07F06C735A04015E1A016D6C6E4A5AA36D6C4DC7FC4C7FA26E5F011F9126073FC013 0EA26E171E010F020F6D131CEE0E1FA26D6C011E6D5BEE1C0FA26D6C5F4C6C7EA26E17F0 01014A6C6C5BA36D6C01F06D485AEEE001A2DA7F804B5A923981C000FFA203C115079126 3FC380D97F87C8FCA203E3158FDA1FE715CE93C7123FA26EB415FC4B141FA202075E4B14 0FA36E486E5AA302015E4B1403A202005E4B1401523B7FB855>I<EAFFF8A4EAF000B3B3 B3B3A3EAFFF8A40D5378BD17>91 D<48148039038001C00007130301001380000EEB0700 48130EA2485BA2485BA3485BA400EFEB778039FF807FC001C013E001E013F0A2007F133F A2393FC01FE0391F800FC0390F0007801C1A76B92A>I<EAFFF8A4EA0078B3B3B3B3A3EA FFF8A40D537FBD17>I<13101338137C13FE487E3803EF803807C7C0380F83E0381F01F0 383E00F848137C00F0131E0060130C170D77B92A>I<EB3FE0EBFFFC000313FF3907C03F 80390F800FC0486C6C7E01E07F6E7EA2380FC001D807807FEA0300C7FCA414FF130FEB7F F13801FF01EA07F8EA0FF0EA1FE0EA3FC0EA7F80A2D8FF00141CA41403A2387F80071406 3A3FC01C7E383A1FF0787FF83A0FFFF03FF0000301E013E03A007F000F8026277DA52A> 97 D<EA03F812FFA3120F1203B0EC0FE0EC7FFC9038F9FFFE9039FBE03F809039FF800F C09039FE0007E049EB03F0A249EB01F816FCA216FE1500A216FFA916FEA3150116FCA2ED 03F86D14F0ED07E06DEB0FC09039E7801F809039E3E07F009038C1FFFE9038807FF8C7EA 1FC0283B7EB92E>I<EB03FC90381FFF80017F13E09038FE01F03901F800F83903F001FC 3807E003EA0FC0121F90388001F8003FEB00F01560007F140090C8FCA25AA97EA27FA200 3F140E7F001F141C6C7E000714386C7ED801FC13F03900FF03E090387FFFC0011F1300EB 07F81F277DA525>I<ED1FC0EC07FFA3EC007F151FB0EB07F8EB1FFE90387FFF9F9038FE 07DF3901F800FF4848137F4848133F4848131F121F485AA2127F90C7FCA35AA97EA27F12 3FA2121F6C6C133F157F6C6C13FF3A03F001DFF03A01FC079FFF39007FFF1FEB3FFED907 F013C0283B7DB92E>I<EB07F8EB1FFF017F13809038FC0FC03901F003E03903E001F000 0714F8390FC000FC121F4913FE003F147EA248C7127FA35A90B6FCA390C8FCA57EA27F12 3F15076C7E150E6C7E0007141C6C6C133CD801FC13783900FF01F090383FFFC06D138090 3803FC0020277EA525>I<147E903803FF80010F13C090381FC7E090383F07F0EB7E0F13 FE13FC0001EB07E09038F803C0000390C7FCADB512FCA3D803F8C7FCB3AB487EB512F8A3 1C3B7FBA19>I<ED03E090390FF00FF090393FFC1FF890B5EA3C7C3A01F81FF0FC3A03E0 07C07C3A07C003E038000FECF000001F80EB8001003F80A7001F5CEBC003000F5C00075C 6C6C485A9038F81F800006B5C7FCEB3FFC380E0FF090C9FC121EA3121F6C7E90B512C015 F86C14FE6CECFF8016C04815E0391F80007F48C7EA0FF0007E140316F8481401A5007EEC 03F0A26CEC07E06C6CEB0FC0D80FE0EB3F803A07FC01FF000001B512FC6C6C13F0010790 C7FC26387EA52A>I<EA03F812FFA3120F1203B0EC07F0EC1FFCEC7FFF9138F07F809038 F9C01FD9FB807F9038FF000F49805BA35BB3A4486C497EB500E1B51280A3293A7EB92E> I<EA03C0487E487E487EA46C5A6C5A6C5AC8FCA9EA01F812FFA312071203B3AA487EB512 E0A313387EB718>I<EA03F812FFA3120F1203B1913803FFFCA36E13C0913800FE005D15 F04A5A4A5A4A5A4AC7FC141E5C5C14FCEBF9FE13FBEBFF7F496C7EEBFC1F01F87F6E7E6E 7EA26E7E6E7EA26E7E157FA2ED3F8016C0486CEB7FF0B500E1B5FCA3283A7EB92C>107 D<EA03F812FFA3120F1203B3B3AD487EB512E0A3133A7EB918>I<2703F807F8EB0FF000 FFD91FFEEB3FFCDA7FFFEBFFFE913AF03F81E07F3D0FF9C00FC3801F802603FB80D9E700 7F020013E601FED907FC6D7EA2495CA2495CB3A4486C496C497EB500E1B500C3B51280A3 41257EA446>I<3903F807F000FFEB1FFCEC7FFF9138F07F80390FF9C01F2603FB807F90 38FF000F49805BA35BB3A4486C497EB500E1B51280A329257EA42E>I<EB03FE90380FFF 80013F13E09038FE03F83901F800FC4848137E48487F4848EB1F80001F15C049130F003F 15E0A248C7EA07F0A44815F8A96C15F0A26D130F003F15E0A26C6CEB1FC0000F15806D13 3F6C6CEB7F006C6C13FE3900FE03F890387FFFF0011F13C0D903FEC7FC25277EA52A>I< 3903F80FE000FFEB7FFC9038F9FFFE9039FBE07F803A07FF801FC03A03FE000FE049EB07 F0A249EB03F816FC150116FEA3ED00FFA916FE1501A316FC150316F86DEB07F0ED0FE06D 14C09039FF803F809039FBE07F009038F9FFFE9038F87FF8EC1FC091C8FCAB487EB512E0 A328357EA42E>I<903903F801C090381FFE03EB7FFF9038FE07873901FC01C73903F000 EF0007147F4848133F485A003F141F5B127FA390C7FC5AA96C7EA3123F7F001F143F6C7E 157F6C6C13FF3903F801DF3901FE0F9F39007FFF1FEB3FFCEB07F090C7FCABED3FE00207 B5FCA328357DA42C>I<3803F03F00FFEB7F809038F1FFE014C7390FF38FF03803F70F13 F69038FE07E09038FC03C0EC018091C7FCA25BB3A3487EB512F8A31C257EA421>I<EBFF 83000313E7000F13FFEA1F80383E003F487F00787FA200F87FA37E6C90C7FC6C7EEA7FF8 EBFF806C13E06C13F86C7F6C7F00017F6C7E01031380EB003F0060EB1FC000E0130FA26C 1307A37E15806C130F1500B4131EEBC07C38F3FFF800E15B38C07F801A277DA521>I<13 1CA5133CA3137CA213FC120112031207121FB6FCA3D801FCC7FCB2EC01C0A93900FE0380 A2017E13006D5AEB1FFE6D5AEB03F01A347FB220>I<D803F8EB0FE000FFEB03FFA3000F EB003F0003140FB3A5151FA2153F1201156F6C6CEBEFF8903A7E03CFFF8090383FFF8F6D 130FD903FCEBE00029267EA42E>I<B538C07FFEA33A0FFC001FF0D803F8EB07C016807F 00011500A26D5B0000140EA2017F5BA2EC803C013F1338A26D6C5AA214E0010F5BA214F1 01075BA2903803FB80A214FF6D90C7FCA36D5AA2147CA3143827257EA32C>I<B53A8FFF F07FFEA3260FF8009038800FF8000791397F0007E0EF03C000031780816DEC8007000102 7F1400A26D6E5A000002FF130E15EF6DECE01ED97F01141C15C70281EBF03CD93F831438 1583D91FC36D5AECC701A2D90FE76D5AECEE00A2D907FEEBFFC04A137FA201035D4A133F A2010192C7FC4A7FA20100141E4A130E37257EA33C>I<B500C0B5FCA300039038007FF0 6C48EB3F806C6C013EC7FC6D133CEB3F806E5A011F5B6D6C5A14F1903807FBC06DB45A6D 90C8FCA26D5A147F818114FF497FECCFF0EB03C790380783F890380F03FCEB0E0190381E 00FE497F496D7E01F8133F000181D80FFC497EB548B51280A329247FA32C>I<B538C07F FEA33A07FC001FF06C48EB07C016807F00011500A26C6C130EA26D131E6D131CA26D6C5A A2ECC078011F1370A2ECE0F0010F5B14F0903807F1C0A214FB01035BA26DB4C7FCA36D5A A2147CA31438A214781470A25CA21301007C5BEAFE035C130749C8FCEAFC1EEA783CEA3F F86C5AEA0FC027357EA32C>I<003FB512FCA39038C007F8D83E0013F0003C130FEC1FE0 003814C00078EB3F80147F0070EBFF005C1301495A5CEA0007495A5C495A133F90387F80 0E14005B485A5B0003141E485A5B4848131C001F143C4848137C4913FC007F1303B6FCA3 1F247EA325>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmbx12 14.4 24 /Fs 24 119 df<153C157EEC01FE1407141F14FF13070003B5FCB6FCA3147F13F8EAFC00 C7FCB3B3B3A4007FB712FEA62F4E76CD43>49 D<EC3FFC49B512E0010F14F8013F14FE90 B77E4816E00007D9007F7FD80FF8010F7F01E06D7F484801017FD83FF06D7F01FC168048 6C806D6E13C0B5FC6E6D13E0A318F082A26C90C7FCA26C5A6C5A6C5AEA01C0C94813E0A3 18C05E18805E18004C5AA24B5B5F4B5B4B5B5F4B90C7FC4B5A4B5A4B5AEDFFE05E4A5B4A 90C8FCEC07FC4A48EB03F04A5A4A5A4A5A4AC7EA07E0495A495A495A5C4948140FEB1F80 49C8121F49B7FC90B812C05A5A5A5A5A5A5AA2B91280A5344E79CD43>I<91380FFF8091 B512F8010314FF010F15C0013F15F090267FF8077F9026FF80017F4848C76C7ED803F881 D807FE6E13807F486D15C06E7F486D15E0A76C5D4A15C06C5B6C90C71480EA00FC90C848 1300A24C5A5F4B5B4B5B4B5B4B5B92B5C7FC91B512FC16F016C016FCEEFF80DA000713E0 030113F89238007FFE707E70138018C07013E018F0A27013F818FCA3D801E016FEEA07F8 EA1FFE487E487FA2B57EA218FCA44A4913F87E91C714F06C5D01FC16E06C484A13C0494A 1380D80FFE91B512002707FFE0075B000190B65A6C16F0013F15C0010F92C7FC010114F8 D9001F90C8FC374F7ACD43>I<171F4D7E4D7EA24D7EA34C7FA24C7FA34C7FA34C7FA24C 7FA34C80A24C80167E8304FE804C7E03018116F8830303814C7E03078116E083030F814C 7E031F81168083033F8293C7FC4B82157E8403FE824B800201835D840203835D02076F7F 5D844AB87EA24A83A34A8492C812014A6F80A2027E8202FE845C0101717FA24A82010385 5C0107717FA24A82010F855C496C707FB600F8021FB712E0A65B547BD366>65 D<BA7E19FCF1FF801AF01AFC1AFFD8000701F8C7000F14C0060080071F13F807077F737F 07007F747F747F86747F747F8886888688A2747FA3751380A31DC0A51DE0A287AB63A21D C0A51D80A298B5FC1D00A3505BA26462505B6462505B505B505B97B5C7FC4F5B07075B07 1F5B96B55A060F14C0BCC8FC621AF01AC007FCC9FC19805B527CD167>68 D<93380FFFC00303B6FC031F15E092B712FC020316FF020FD9FC0014C0023F01C0010F13 F04A90C700037F902601FFFC020013FE49496F7E49496F7F49496F7F49496F7F49496F7F 4990C96C7F4948707FA24849707F48864A83481B804A83481BC0A2481BE04A83A2481BF0 A348497113F8A5B51AFCAF6C1BF86E5FA46C1BF0A26E5F6C1BE0A36C6D4D13C0A26C6D94 B51280A26C1B006C6D4C5B6E5E6C626D6D4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D 01FE4AB5C7FC6D6D4A5B6D02E0011F5B023F01FC90B512F0020F90B712C0020394C8FC02 0016FC031F15E0030392C9FCDB001F13E0565479D265>79 D<BAFC19F819FF1AC01AF01A FCD8000701F8C7001F7F0603EBFF80060014C0073F13E07313F0851BF8851BFCA27313FE A31BFFA91BFEA34F13FCA21BF8611BF04F13E04F13C096B5128006031400061F5B92B812 F8621A804FC7FC19E003FCCBFCB3ACB812E0A650527CD15C>I<EC3FFF0103B512F0011F 14FE017F6E7E90B712E048D9C0077F48D9800113FC486D6C6C7E163F486D6D7E848284A2 826C49816C5BA2C648C7FC90C8FCA44BB5FC4AB6FC143F49B7FC130F013FEBF80790B512 800003EBFC004813F0485B485B485B91C7FC485AA2485AA55EA26C6C5C5E6C6C147D6E01 FD13F86C9026C001F9EBFFE06C9026F00FF014F06C90B5487E0001ED803F6C6C49487E01 1F01F8010313E0010101C090C8FC3C387CB641>97 D<913801FFF0021F13FF91B612E001 0315F8010F81499038800FFE4948486C7ED9FFF8491380485B4A4913C0485B5A485BA25A 91C76C1380486F1300A2EE01FC484891C8FCA412FFAB127FA27FA27EA26C6DEC07E0A27E 6EEC0FC06C7FEF1F806C6D143F6C6DEC7F006C01FE14FE6D6C495A011F9038E00FF86D90 B55A01035D01001580021F01FCC7FC020113E033387CB63C>99 D<943801FFC00407B5FC A6EE001F1707B3A3913801FFC0021F13F891B6FC010315C7010F15F749D9C01FB5FC9039 7FFE0003494813004801F08048498048498083485BA24890C8FCA25AA2485AA412FFAB12 7FA4123F7FA27EA26C7F5F6C6D5C6C6D5C6C6D91B5FC6C6D010314F06D6C49ECFFC09039 3FFF803F6D90B512CF0107150F010114FCD9003F13F00203018049C7FC42547CD24B>I< 913803FF80023F13F891B512FE01036E7E010F15E0013F01017F903A7FFC003FF8D9FFF0 6D7E48496D7E48496D7E5A4A6D13805A91C76C13C05AA24817E082485AA218F0A212FFA2 90B8FCA418E049CAFCA5127FA36C7EA36CEE01E0EF03F06C7FEF07E06C7F6C6DEC0FC06C 161F6C6DEC3F80D97FFCEC7F006DB4EB01FE6D9038E00FFC010790B55A010115E06D6C14 80021F49C7FC020013E034387CB63D>I<ED0FFE92B51280020714E0021F14F0027F14F8 9138FFF83F499038E07FFC01071380923800FFFE5B495A5C133FEE7FFC5C017FEC3FF8EE 1FF0EE07C093C7FCADB712E0A626007FFCC8FCB3B3A5007FB6FCA62F547CD329>I<DA1F FE14FE49B539E007FF800107DAF81F13C0011FDAFE7F13E0017F91B612F09026FFF80714 3F48D9E001EBF07F48902680007F133F4804F813E04890C7383FFC1F19C0484891391FFE 070095C7FCA2003F82A8001F5EA36C6C4A5AA26C6D495A6C5E6C9039E001FFE06CD9F807 5B4890B65A484BC8FC01E714F8D807C114E09026C01FFEC9FC91CBFC120FA27FA27F7F7F 90B7FC17F06C16FE717E18E06C836C83846C83488312075AD81FFCC7000114804848EC00 1F484815077113C0485A83A56C6C4B1380A26C6C4B13006D5D6C6C4B5A6C6C4B5A6C01E0 49B45A6C01FE011F5BC690B712C0013F93C7FC010F15FC010115E0D9000F01FCC8FC3C4F 7CB543>I<EB3FF8B5FCA61203C6FCB3A3EE0FFC93B57E030314E0030F14F84B8092393F C07FFE92387E003F4B6D7EECF9F0DAFBE0814B7F14FF4B8192C7FCA25CA35CB3ACB6D8FC 0FB612C0A642537BD24B>I<133FEBFFC0487F487F487FA2487FA66C5BA26C5B6C5B6C5B 013FC7FC90C8FCACEB1FF8B5FCA612017EB3B3A4B612F0A61C547BD326>I<EB1FF8B5FC A612017EB3B3B3AFB612F8A61D537BD226>108 D<D93FF8D90FFEED7FF0B591267FFFC0 903803FFFE4BB500F0010F6D7E030702FC013F14E0031F6E90B67E922A3FC07FFF01FE03 7F92267F001F903803F800000302FC6DD987E06D7EC6D9F9F8ED8FC0DAFBE003DFC77F4B 6D01DE143F02FF16FE4B4B8192C75CA24A5EA34A5EB3ACB6D8FC07B6D8E03FB6FCA66836 7BB571>I<D93FF8EB0FFCB591B57E030314E0030F14F84B8092393FC07FFE92387E003F 00034A6D7EC6EBF9F0DAFBE0814B7F14FF4B8192C7FCA25CA35CB3ACB6D8FC0FB612C0A6 42367BB54B>I<913801FFE0021F13FE91B612C0010315F0010F15FC499038807FFE903B 7FFC000FFF8049486D7F4801E001017F48496D7FA248496E7E488391C8123F4883A24883 4981A2007F1880A400FF18C0AC007F1880A3003F18006D5DA26C5FA26C6D4A5AA26C6D4A 5A6C5F6C6D495B6C01F801075B6D6C495B90273FFF807F90C7FC010F90B512FC6D5D0100 15C0023F91C8FC020113E03A387CB643>I<D93FF8EB7FF0B50103B5FC031F14E0037F14 F802F9B612FE02FB01007FDAFFF8011F7F000102E06D7F6C4A6D7F92C76C7F4A6E7F4A82 83727EA28584A21A80A284A21AC0AB1A8060A31A006061A24E5AA26E4A5B6E5C6103C049 5B6F011F5B6F495B9227FE01FFFEC7FC02FDB65ADAFC7F14F0031F14C0030749C8FC0300 13E093CAFCB0B612FCA6424D7CB54B>I<90393FF003FCB5EB1FFF4B13C092B512F002F1 14F89238FC7FFCECF3F000039039F7E0FFFEC65CECFF805DA25CEE7FFCEE3FF85CEE0FE0 93C7FCA35CB3AAB612FEA62F367CB537>114 D<903901FFE007011FEBFC1F017FEBFF7F 48B7FC1207390FFE003FD81FF0130749130148487F5B007F81A200FF81A37F7F01F891C7 FC13FEEBFFF06CEBFF8015FC6C14FF16C06C15F06C816C15FE6C817E6C6C1580011F15C0 1303D9003F14E01400030F13F01501007C8000FC157F163F6C151FA37E17E07F7FEE3FC0 01F0147F6DECFF806D4913009039FF800FFE91B55A013F5CD8FC1F14E0D8F803148027E0 007FF8C7FC2C387CB635>I<147EA614FEA41301A31303A21307A2130F131F133F137F13 FF1203000F90B6FCB8FCA5C66C48C8FCB3A8EE0FC0AB013FEC1F808017006D5C6DEB807E EDE0FE6DEBFFFC01015C6D5C023F13C0DA03FEC7FC2A4D7ECB34>I<B600F8010FB5FCA6 C649C8EA7F801900017F167E6E15FE6D5E6F1301A26D6D5C17036D5E6F13076D5E6F130F 6D5E6F131FA26D6D5C173F6D93C7FC6F5B027F147E6F13FE6E5C1681A26EEBC1F816C36E 5C16E76E5C16FFA26E5CA26E5CA26E91C8FCA26F5AA36F5AA26F5AA26F5AA26F5A40367D B447>118 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr12 12 9 /Ft 9 117 df<121E123FEA7F80EAFFC0A313E0127F123F121E1200A5EA01C0A4EA0380 A2EA0700A2120E121E121C5A5A12200B1D78891B>44 D<121E123FEA7F80EAFFC0A4EA7F 80EA3F00121E0A0A78891B>46 D<14FF010713E0011F13F890383F81FC9038FE007FD801 F8EB1F8049130F000315C04848EB07E0491303000F15F0A24848EB01F8A2003F15FCA390 C8FC4815FEA54815FFB3A46C15FEA46D1301A2003F15FCA3001F15F8A26C6CEB03F0A26D 1307000715E06C6CEB0FC06D131F6C6CEB3F806C6CEB7F0090383F81FC6DB45A010713E0 010090C7FC28447CC131>48 D<1438147814F81303130F137FB5FCA213F71387EA0007B3 B3AFEB1FFE007FB61280A4214278C131>I<EB03FE90381FFFC0017F13F090B57E3903F0 0FFE3907C003FF48C71380000EEC7FC048EC3FE0003C141F003815F048140F16F8127CB4 14076D14FC7FA46C5AA26CC7FCC8FC16F8150FA216F0151F16E0ED3FC0A2ED7F80160015 FEA24A5A4A5A4A5A5D4A5A4AC7FC143E5C5C495A5C495A4948131C49C7FC131E4914385B 5B485A491478485A48C812F048B6FC5A5A5AB7FC16E0A326427BC131>I<49B4FC010F13 E0013F13F84913FE3901FE01FF3A03F0007F8001C0EB3FC048C7EA1FE0487E01E014F048 6C130F6D14F8A46C5AA26C5AC813F0151FA216E0A2ED3FC01680ED7F0015FE5DEC03F8EC 1FF090380FFFC092C7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0ED07F816FC A216FE1503A216FFA2121F487E487E487EA316FE15075B6C4814FC007EC7FC0038EC0FF8 003C15F06CEC1FE06C6CEB3FC0D807E0EB7F803A03FC01FF006CB55A6C6C13F8011F13E0 010190C7FC28447CC131>I<ED1FFC4AB512C0020714F091391FF007FC91397F8000FFD9 01FEC7EA3FC049486E7ED907F0EC07F049486E7E49486E7E49486E7E49C9127F49834848 707E4848707EA24848707EA2000F84491607001F84A24848707EA3007F84A24982A300FF 1980AD6C6C4C1300A4003F60A26D1607001F60A26C6C4C5AA20007606D161F0003606D16 3F6C6C4C5A6C606D6C4BC7FC6E5C6D6C4A5A6D6C4A5AD907F8EC0FF06D6C4A5A6DB4EC7F C09028007FC001FFC8FC91391FF007FC0207B512F0020114C09126001FFCC9FC41487BC5 4C>79 D<EC7FC0903803FFFC010F7F90383F803F90397E000F8049EB1FC0484814E04848 133F485A120F49EB1FC0001FEC0F804848EB070092C7FCA2127FA290C9FC5AAA7EA27FA2 123F6D1470121FA26C6C14E06C7E0003EC01C06C6CEB03806C6C1307017FEB0F0090383F C07E90381FFFF8010713E001001380242E7DAC2B>99 D<130EA6131EA4133EA2137EA213 FE120112031207001FB512F0B6FCA3C648C7FCB3A3151CAA017E1338137FA26D1370EB1F 8090380FE1E0903807FFC06D13809038007E001E3E7EBC26>116 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmsy10 12 1 /Fu 1 1 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmmi12 17.28 2 /Fv 2 77 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr17 17.28 6 /Fw 6 85 df<EA0F80EA1FC0EA3FE0EA7FF0EAFFF8A5EA7FF0EA3FE0EA1FC0EA0F80C7FC B3B2EA0F80EA1FC0EA3FE0EA7FF0EAFFF8A5EA7FF0EA3FE0EA1FC0EA0F800D3E75BD24> 58 D<007FBD1280BE12C0A36C1C80D1FCB3A3007FBD1280BE12C0A36C1C805A1F7AB267> 61 D<B500FC071FB51280A36E61C6F58000013F51C7FCD91DFFF177FCA2011C6D18E7A3 6E6CEF01C7A36E6CEF0387A36E6CEF0707A26E6C170EA36E6C171CA36E6C1738A36E6C17 70A26E6D16E0A36F6CED01C0A36F6CED0380A36F6CED0700A26F6C150EA36F6C5DA36F6C 5DA36F6C5DA26F6D5CA3706C495AA3706C495AA3706C49C7FCA2706C130EA3706C5BA370 6C5BA3706C5BA3706D5AA294387FC1C0A394383FE380A3DD1FF7C8FCA3EF0FFEA2013E6F 5AA2137F715A2601FFC04F7E000701F896383FFF80B66C6D48023FB61280A3715A696278 E17A>77 D<B812FCEFFFE018FCF0FF80C601FCC7000F13E0D93FF8020013F8011FEE1FFC F007FF06017F727F737E737E737E86737EA2737EA287A28587A96361A298C8FCA24F5A62 4F5AA24F5A4F5AF1FF80060390C9FC4E5AF01FF8F0FFE0050F138091B700FCCAFC18E0A2 02F8C7EA1FFCEF03FE943800FF80F03FC0727E727E727E727E85727EA2727FA286197FA2 86A586A786A67414E0193FA4736CEB01C0190FA2496C706CEB038090B582B76F90388007 0073EBC00E97387FF03C97381FFFF8CD00035B9738007FC05B6479E162>82 D<DA07FF1403023F01F05B49B512FC010702FF5B90260FFC0013C0D91FE090380FE01F01 7FC7EA03F001FE913800F83FD801F8153C48486F5A496F5A000782484881484881A248CA FCA2845A007E83A200FE83A384A37EA284A26C7EA27F6D93C7FC123F7FEA1FFC7F6C6C7E 14E06C13FC6CEBFFC06C14F86CECFF806D14F86DECFF806D15E0010715F86D15FE010081 021F15C0020181DA003F80030380DB003F7F16039338007FFEEF1FFF1707711380837113 C0187F19E0183F181FA2006018F000E0170FA31807A37EA46C18E0A36C170F19C07E181F 6C18807FF03F006D167E7F6D5ED8FE7C4B5A486C4B5A6D6C4A5A26F80FE0EC1FC0D903FC EC7F8029F001FFE003FFC7FC6D6CB512FC48011F14F0020314C0489026001FFEC8FC3C66 7AE349>I<003FBC12F8A49126C000039038C0000301FCC76C49EB007F01F0190F01C019 074848F103FC90C81701007E1A00007C1B7CA300781B3CA400701B1CA600F01B1E481B0E A7C91800B3B3B3A54C7FA2041F13F84AB87EA457627CE160>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: Letter letter %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 1380 289 a Fw(TRSM:)43 b Fv(B)j Fw(:=)40 b Fv(L)2276 237 y Fu(\000)p Ft(1)2406 289 y Fv(B)1660 724 y Ft(Oct.)k(21,)32 b(2003)0 1040 y Fs(1)135 b(Description)43 b(of)j(Op)t(eration)0 1222 y Fr(Let)28 b(us)f(consider)f(the)i(computation)d Fq(B)j Fr(:=)22 b Fq(LB)32 b Fr(where)27 b Fq(L)g Fr(is)g(an)g Fq(m)19 b Fp(\002)f Fq(m)27 b Fr(lo)n(w)n(er)e(triangular)f(matrix)h(and)j Fq(B)k Fr(is)26 b(a)h(general)0 1322 y Fq(m)c Fp(\002)f Fq(n)35 b Fr(matrix.)54 b(This)34 b(op)r(eration)e(is)h(t)n(ypically)e (referred)i(to)i(as)e(a)h Fo(triangular)j(solve)g(with)f(multiple)h (right-hand-sides)0 1421 y Fr(\(TRSM\).)29 b(The)e(reason)f(for)h(this) g(is)g(that)125 1559 y Fp(\017)41 b Fr(It)24 b(is)f(generally)e (implemen)n(ted)g(as)j(the)g(computation)e(of)j(the)f(solution)e(of)i Fq(LX)29 b Fr(=)23 b Fq(B)t Fr(,)i(where)f Fq(X)31 b Fr(o)n(v)n(erwrites)21 b Fq(B)t Fr(;)k(and)125 1711 y Fp(\017)41 b Fr(By)27 b(partitioning)d Fq(X)34 b Fr(and)27 b Fq(B)32 b Fr(in)n(to)26 b(columns:)966 1859 y Fq(X)j Fr(=)1152 1791 y Fn(\000)1231 1858 y Fq(x)1278 1870 y Fm(1)p 1356 1888 4 100 v 1399 1858 a Fq(x)1446 1870 y Fm(2)p 1524 1888 V 1567 1858 a Fp(\001)14 b(\001)g(\001)p 1703 1888 V 83 w Fq(x)1794 1870 y Fl(n)1881 1791 y Fn(\001)2016 1859 y Fr(and)82 b Fq(B)28 b Fr(=)2410 1791 y Fn(\000)2490 1858 y Fq(b)2526 1870 y Fm(1)p 2603 1888 V 2646 1858 a Fq(b)2682 1870 y Fm(2)p 2758 1888 V 2802 1858 a Fp(\001)14 b(\001)g(\001)p 2938 1888 V 82 w Fq(b)3017 1870 y Fl(n)3104 1791 y Fn(\001)208 2006 y Fr(w)n(e)27 b(recognize)e(that)690 2153 y Fq(L)761 2086 y Fn(\000)840 2152 y Fq(x)887 2164 y Fm(1)p 965 2183 V 1008 2152 a Fq(x)1055 2164 y Fm(2)p 1132 2183 V 1176 2152 a Fp(\001)14 b(\001)g(\001)p 1312 2183 V 82 w Fq(x)1402 2164 y Fl(n)1490 2086 y Fn(\001)1551 2153 y Fr(=)1638 2086 y Fn(\000)1718 2152 y Fq(Lx)1822 2164 y Fm(1)p 1899 2183 V 1942 2152 a Fq(Lx)2046 2164 y Fm(2)p 2123 2183 V 2166 2152 a Fp(\001)g(\001)g(\001)p 2303 2183 V 83 w Fq(Lx)2450 2164 y Fl(n)2537 2086 y Fn(\001)2598 2153 y Fr(=)2686 2086 y Fn(\000)2765 2152 y Fq(b)2801 2164 y Fm(1)p 2878 2183 V 2921 2152 a Fq(b)2957 2164 y Fm(2)p 3034 2183 V 3077 2152 a Fp(\001)g(\001)g(\001)p 3214 2183 V 83 w Fq(b)3293 2164 y Fl(n)3379 2086 y Fn(\001)208 2301 y Fr(so)26 b(that)i(eac)n(h)f(column)f(of)h Fq(X)7 b Fr(,)27 b Fq(x)1234 2313 y Fl(j)1270 2301 y Fr(,)g(equals)f(the)i (solution)d(of)j(the)g(triangular)c(system)i Fq(Lx)3034 2313 y Fl(j)3092 2301 y Fr(=)d Fq(b)3216 2313 y Fl(j)3250 2301 y Fr(.)0 2511 y Fk(Step)32 b(1:)41 b(Precondition)32 b(and)g(P)m(ostcondition)0 2664 y Fr(In)c(our)f(discussion,)e(w)n(e)i (will)e(let)1082 2643 y(^)1063 2664 y Fq(B)32 b Fr(denote)27 b(the)h(original)23 b(con)n(ten)n(ts)k(of)h Fq(B)k Fr(so)27 b(that)g(\(the)i(most)d(imp)r(ortan)n(t)f(parts)i(of)6 b(\))29 b(the)0 2764 y(precondition)c(and)i(p)r(ostcondition)f(are)g (giv)n(en)g(b)n(y)1686 2911 y Fq(P)1739 2923 y Fm(pre)1858 2911 y Fr(:)e(\()p Fq(B)j Fr(=)2134 2890 y(^)2115 2911 y Fq(B)t Fr(\))0 3058 y(and)1574 3158 y Fq(P)1627 3170 y Fm(p)r(ost)1778 3158 y Fr(:)d(\()p Fq(B)j Fr(:=)c Fq(L)2115 3124 y Fj(\000)p Fm(1)2223 3137 y Fr(^)2203 3158 y Fq(B)5 b Fr(\))p Fq(;)0 3286 y Fr(resp)r(ectiv)n(ely)-7 b(.)0 3555 y Fs(2)135 b(Algorithms)42 b(Deriv)l(ed)j(from)g(a)g(P)l (artitioning)f(of)h Fi(L)0 3737 y Fr(Let)28 b(us)f(start)g(b)n(y)h (partitioning)23 b(the)28 b(triangular)c(matrix)h Fq(L)p Fr(:)1543 3939 y Fq(L)e Fp(!)1729 3822 y Fn(\022)1834 3879 y Fq(L)1891 3891 y Fl(T)9 b(L)p 2030 3909 V 2047 3909 V 2151 3879 a Fr(0)p 1790 3912 506 4 v 1790 3929 V 1831 3998 a Fq(L)1888 4010 y Fl(B)s(L)p 2030 4028 4 100 v 2047 4028 V 2090 3998 a Fq(L)2147 4010 y Fl(B)s(R)2296 3822 y Fn(\023)0 4142 y Fr(where)27 b(the)h(blo)r(c)n(ks)e(on)h(the)h (diagonal)c(are)j(tak)n(en)g(to)g(b)r(e)h(square.)125 4242 y(Substituting)h(this)h(partitioned)e(matrix)g(in)n(to)i(the)g(p)r (ostcondition,)g(w)n(e)g(\014nd)h(that)g Fq(B)k Fr(and)3096 4221 y(^)3076 4242 y Fq(B)g Fr(m)n(ust)30 b(b)r(e)g(partitioned)0 4341 y(b)n(y)d(ro)n(ws)281 4311 y Fm(1)317 4341 y Fr(:)1267 4511 y Fq(B)g Fp(!)1463 4394 y Fn(\022)1568 4450 y Fq(B)1631 4462 y Fl(T)p 1524 4484 204 4 v 1524 4500 V 1566 4570 a Fq(B)1629 4582 y Fl(B)1727 4394 y Fn(\023)1885 4511 y Fr(and)2122 4490 y(^)2102 4511 y Fq(B)g Fp(!)2298 4369 y Fn( )2428 4429 y Fr(^)2408 4450 y Fq(B)2471 4462 y Fl(T)p 2364 4484 V 2364 4500 V 2425 4558 a Fr(^)2406 4579 y Fq(B)2469 4591 y Fl(B)2567 4369 y Fn(!)0 4709 y Fr(so)g(that)1206 4763 y Fn(\022)1311 4819 y Fq(L)1368 4831 y Fl(T)9 b(L)p 1508 4849 4 100 v 1524 4849 V 1629 4819 a Fr(0)p 1267 4852 506 4 v 1267 4869 V 1309 4939 a Fq(L)1366 4951 y Fl(B)s(L)p 1508 4969 4 100 v 1524 4969 V 1568 4939 a Fq(L)1625 4951 y Fl(B)s(R)1773 4763 y Fn(\023)1857 4880 y Fr(=)1945 4738 y Fn( )2074 4798 y Fr(^)2054 4819 y Fq(B)2117 4831 y Fl(T)p 2011 4852 204 4 v 2011 4869 V 2072 4927 a Fr(^)2052 4948 y Fq(B)2115 4960 y Fl(B)2214 4738 y Fn(!)2279 4755 y Fj(\000)p Fm(1)2369 4763 y Fn(\022)2474 4819 y Fq(B)2537 4831 y Fl(T)p 2430 4852 V 2430 4869 V 2471 4939 a Fq(B)2534 4951 y Fl(B)2633 4763 y Fn(\023)3794 4880 y Fr(\(1\))p 0 5023 1560 4 v 92 5084 a Fh(1)127 5107 y Fg(Notice)25 b(that)g Ff(B)i Fg(and)735 5090 y(^)719 5107 y Ff(B)g Fg(could)e(b)r(e)f(partitioned)j (in)n(to)e(quadran)n(ts.)32 b(This)24 b(will)h(b)r(e)f(discussed)h(in)g (a)e(future)h(rep)r(ort.)1929 5356 y Fr(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 660 -115 a Fn(\022)765 -58 y Fq(B)828 -46 y Fl(T)p 721 -25 204 4 v 721 -8 V 763 61 a Fq(B)826 73 y Fl(B)924 -115 y Fn(\023)1008 2 y Fr(=)1096 -139 y Fn( )1203 -58 y Fq(L)1260 -46 y Fl(T)9 b(L)1358 -94 y Fj(\000)p Fm(1)1466 -79 y Fr(^)1447 -58 y Fq(B)1510 -46 y Fl(T)p 1162 -25 442 4 v 1162 -8 V 1342 49 a Fr(^)1323 70 y Fq(B)1386 82 y Fl(B)1603 -139 y Fn(!)p 1785 106 4 250 v 1802 106 V 1853 -115 a(\022)1958 -58 y Fq(B)2021 -46 y Fl(T)p 1914 -25 204 4 v 1914 -8 V 1956 61 a Fq(B)2019 73 y Fl(B)2118 -115 y Fn(\023)2202 2 y Fr(=)2289 -139 y Fn( )2619 -58 y Fq(L)2676 -46 y Fl(T)g(L)2774 -94 y Fj(\000)p Fm(1)2882 -79 y Fr(^)2863 -58 y Fq(B)2926 -46 y Fl(T)p 2355 -25 888 4 v 2355 -8 V 2416 49 a Fr(^)2397 70 y Fq(B)2460 82 y Fl(B)2535 70 y Fp(\000)18 b Fq(L)2675 82 y Fl(B)s(L)2777 70 y Fr(\()p Fq(L)2866 82 y Fl(T)9 b(L)2964 34 y Fj(\000)p Fm(1)3073 49 y Fr(^)3053 70 y Fq(B)3116 82 y Fl(T)3168 70 y Fr(\))3242 -139 y Fn(!)p 1785 206 4 100 v 1802 206 V 1121 276 a Fr([1])p 1785 306 V 1802 306 V 1328 w([2])p 1785 405 V 1802 405 V 542 408 2816 4 v 542 425 V 1785 525 4 100 v 1802 525 V 592 532 a Fn(\000)672 599 y Fq(B)735 611 y Fl(L)p 824 629 V 841 629 V 884 599 a Fq(B)947 611 y Fl(R)1043 532 y Fn(\001)1104 600 y Fr(=)1192 532 y Fn(\000)1271 604 y Fq(L)1347 582 y Fr(^)1328 603 y Fq(B)1391 615 y Fl(L)p 1480 633 4 109 v 1497 633 V 1560 582 a Fr(^)1540 603 y Fq(B)1603 615 y Fl(R)1699 532 y Fn(\001)p 1785 633 V 1802 633 V 271 w(\000)2088 599 y Fq(B)2151 611 y Fl(L)p 2240 629 4 100 v 2257 629 V 2300 599 a Fq(B)2363 611 y Fl(R)2459 532 y Fn(\001)2520 600 y Fr(=)2608 532 y Fn(\000)2707 582 y Fr(^)2687 603 y Fq(B)2750 615 y Fl(L)p 2840 633 4 109 v 2856 633 V 2900 604 a Fq(L)2976 582 y Fr(^)2957 603 y Fq(B)3020 615 y Fl(R)3115 532 y Fn(\001)p 1785 733 4 100 v 1802 733 V 1121 803 a Fr([3])p 1785 832 V 1802 832 V 1328 w([4])p 1785 932 V 1802 932 V 1190 1185 a(Figure)26 b(1:)37 b(Lo)r(op)27 b(in)n(v)-5 b(arian)n(ts)24 b(for)j Fq(B)g Fr(:=)c Fq(L)2531 1155 y Fj(\000)p Fm(1)2639 1164 y Fr(^)2620 1185 y Fq(B)t Fr(.)0 1449 y Fk(Step)32 b(2:)41 b(Deriv)-5 b(ation)32 b(of)g(the)g(Lo)s(op-In)m(v)-5 b(arian)m(ts)0 1602 y Fr(Since)1026 1641 y Fn(\022)1131 1697 y Fq(L)1188 1709 y Fl(T)9 b(L)p 1328 1727 V 1344 1727 V 1449 1697 a Fr(0)p 1087 1730 506 4 v 1087 1747 V 1129 1817 a Fq(L)1186 1829 y Fl(B)s(L)p 1328 1846 4 100 v 1344 1846 V 1387 1817 a Fq(L)1444 1829 y Fl(B)s(R)1593 1641 y Fn(\023)1654 1648 y Fj(\000)p Fm(1)1766 1758 y Fr(=)1854 1616 y Fn( )2155 1697 y Fq(L)2212 1661 y Fj(\000)p Fm(1)2212 1721 y Fl(T)g(L)p 2543 1727 4 103 v 2560 1727 V 2664 1697 a Fr(0)p 1920 1730 889 4 v 1920 1747 V 1961 1820 a Fp(\000)p Fq(L)2083 1784 y Fj(\000)p Fm(1)2083 1844 y Fl(B)s(R)2190 1820 y Fq(L)2247 1832 y Fl(B)s(L)2349 1820 y Fq(L)2406 1784 y Fj(\000)p Fm(1)2406 1844 y Fl(T)g(L)p 2543 1850 4 103 v 2560 1850 V 2603 1820 a Fq(L)2660 1784 y Fj(\000)p Fm(1)2660 1844 y Fl(B)s(R)2808 1616 y Fn(!)0 1981 y Fr(the)28 b(equalit)n(y)d(in)i(\(1\))h(yields)d(the)j(p)r (ostcondition,)d(in)i(terms)f(of)i(the)g(blo)r(c)n(k)n(ed)e(op)r (erands,)g(giv)n(en)g(b)n(y)1064 2119 y Fn(\022)1169 2175 y Fq(B)1232 2187 y Fl(T)p 1125 2209 204 4 v 1125 2225 V 1166 2295 a Fq(B)1229 2307 y Fl(B)1328 2119 y Fn(\023)1412 2236 y Fr(=)1500 2094 y Fn( )1989 2175 y Fq(L)2046 2187 y Fl(T)9 b(L)2143 2139 y Fj(\000)p Fm(1)2252 2154 y Fr(^)2232 2175 y Fq(B)2295 2187 y Fl(T)p 1566 2209 1205 4 v 1566 2225 V 1607 2304 a Fq(L)1664 2316 y Fl(B)s(R)1771 2268 y Fj(\000)p Fm(1)1860 2304 y Fr(\()1912 2283 y(^)1892 2304 y Fq(B)1955 2316 y Fl(B)2031 2304 y Fp(\000)18 b Fq(L)2171 2316 y Fl(B)s(L)2273 2304 y Fr(\()p Fq(L)2362 2316 y Fl(T)9 b(L)2460 2268 y Fj(\000)p Fm(1)2569 2283 y Fr(^)2549 2304 y Fq(B)2612 2316 y Fl(T)2664 2304 y Fr(\)\))2770 2094 y Fn(!)3794 2236 y Fr(\(2\))125 2487 y(The)26 b(idea)f(no)n(w)h(is)g(that,)h(in)f(terms)f(of)i(the)g (blo)r(c)n(k)n(ed)e(matrices,)f Fo(al)t(l)k Fr(computation)c(to)i(b)r (e)h(p)r(erformed)e(to)i(compute)f(the)0 2586 y(\014nal)d(answ)n(er)g (is)g(giv)n(en)g(b)n(y)g(the)i(righ)n(t-hand)d(side)h(in)h(\(2\).)36 b(A)n(t)24 b(an)g(in)n(termediate)d(step)k(\(at)f(the)h(top)f(of)g(the) h(lo)r(op\),)f(the)g(state)0 2686 y(of)h(matrix)e Fq(B)29 b Fr(can)c(b)r(e)h(describ)r(ed)e(b)n(y)h(a)g Fo(subset)f Fr(of)h(the)h(op)r(erations)d(in)h(\(2\))i(ha)n(ving)d(b)r(een)j(p)r (erformed.)34 b(Suc)n(h)25 b(in)n(termediate)0 2786 y(states)k(are)f (called)g Fo(lo)l(op-invariants)p Fr(.)44 b(They)29 b(represen)n(t)f (the)i(state)f(that)h Fq(B)k Fr(should)28 b(b)r(e)i(in)f(at)g(the)h (follo)n(wing)25 b(four)k(p)r(oin)n(ts)0 2885 y(in)e(the)h(algorithm:) 101 3050 y(1.)42 b(Before)26 b(the)i(lo)r(op)e(commenses;)101 3215 y(2.)42 b(A)n(t)27 b(the)h(top)g(of)g(the)g(b)r(o)r(dy)f(of)h(the) g(lo)r(op;)101 3380 y(3.)42 b(A)n(t)27 b(the)h(b)r(ottom)f(of)h(the)g (b)r(o)r(dy)g(of)f(the)h(lo)r(op;)f(and)101 3546 y(4.)42 b(Immediately)23 b(after)k(the)h(lo)r(op)f(is)f(exited.)125 3710 y(Since)31 b(the)i(paren)n(theses)d(in)i(\(2\))g(indicate)e(an)i (order)f(in)h(whic)n(h)f(computations)f(should)h(b)r(e)h(p)r(erformed,) g(reasonable)0 3810 y(in)n(termediate)24 b(states)j(for)g Fq(B)32 b Fr(are)27 b(giv)n(en)f(in)g(Fig.)h(1,)g(In)n(v)-5 b(arian)n(ts)25 b([1])j(and)f([2].)0 4042 y Fe(2.1)112 b(V)-9 b(arian)m(t)38 b(1)0 4195 y Fr(Let)21 b(us)g(start)f(b)n(y)h (deriving)d(an)i(algorithm,)e(V)-7 b(arian)n(t)19 b(1,)j(b)n(y)f(pic)n (king)d(the)j(lo)r(op-in)n(v)-5 b(arian)n(t)16 b(to)21 b(equal)e(that)i(giv)n(en)e(as)h(In)n(v)-5 b(arian)n(t)0 4295 y([1])27 b(in)g(Fig.)g(1:)1427 4352 y Fn(\022)1532 4409 y Fq(B)1595 4421 y Fl(T)p 1488 4442 204 4 v 1488 4458 V 1530 4528 a Fq(B)1593 4540 y Fl(B)1691 4352 y Fn(\023)1775 4469 y Fr(=)1863 4327 y Fn( )1970 4409 y Fq(L)2027 4421 y Fl(T)9 b(L)2125 4372 y Fj(\000)p Fm(1)2233 4388 y Fr(^)2214 4409 y Fq(B)2277 4421 y Fl(T)p 1929 4442 442 4 v 1929 4458 V 2109 4516 a Fr(^)2090 4537 y Fq(B)2153 4549 y Fl(B)2370 4327 y Fn(!)2450 4469 y Fq(:)0 4755 y Fk(Step)32 b(3:)41 b(Determination)30 b(of)i(the)g(Lo)s (op-Guard)0 4908 y Fr(The)i(next)g(step)h(in)e(the)i(pro)r(cess)d(is)h (to)h(determine)f(the)h(condition)e(under)i(whic)n(h)f(the)h(\014nal)f (result)g(has)h(not)g(y)n(et)g(b)r(een)0 5008 y(computed,)27 b(whic)n(h)g(means)g(that)h(the)g(lo)r(op)f(cannot)g(y)n(et)h(b)r(e)g (terminated.)36 b(W)-7 b(e)28 b(will)d(call)h(this)h(condition)f(the)i (lo)r(op-guard)0 5107 y(and)f(denote)h(it)f(b)n(y)g(the)h(predicate)e Fq(G)p Fr(.)1929 5356 y(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 125 -60 a Fr(Notice)26 b(that)i(the)g(lo)r(op)e (is)h(exited)f(when)i(this)f(condition)e(b)r(ecomes)i Fo(false)p Fr(.)38 b(Th)n(us,)27 b Fq(G)h Fr(needs)g(to)f(b)r(e)h(c)n (hoses)e(so)h(that)1581 123 y(\()p Fq(P)1666 135 y Fm(pre)1782 123 y Fp(^)18 b(:)p Fq(G)p Fr(\))24 b Fp(\))f Fq(P)2190 135 y Fm(p)r(ost)0 306 y Fr(or,)k(for)g(our)g(sp)r(eci\014c)f(case,)868 416 y Fn( )999 441 y(\022)1105 498 y Fq(B)1168 510 y Fl(T)p 1061 531 204 4 v 1061 547 V 1102 617 a Fq(B)1165 629 y Fl(B)1264 441 y Fn(\023)1348 558 y Fr(=)1436 416 y Fn( )1543 498 y Fq(L)1600 510 y Fl(T)9 b(L)1697 461 y Fj(\000)p Fm(1)1806 477 y Fr(^)1786 498 y Fq(B)1849 510 y Fl(T)p 1501 531 442 4 v 1501 547 V 1682 605 a Fr(^)1662 626 y Fq(B)1725 638 y Fl(B)1943 416 y Fn(!!)2093 558 y Fp(^)19 b(:)p Fq(G)2287 416 y Fn(!)2376 558 y Fp(\))2482 466 y Fn(\020)2532 558 y Fq(B)27 b Fr(:=)c Fq(L)2790 524 y Fj(\000)p Fm(1)2898 537 y Fr(^)2878 558 y Fq(B)2945 466 y Fn(\021)3009 558 y Fq(:)0 832 y Fr(Recalling)d(that)j(the)h (partitionings)19 b(of)k Fq(L)p Fr(,)h Fq(B)t Fr(,)g(and)1641 811 y(^)1621 832 y Fq(B)k Fr(need)23 b(to)g(b)r(e)h(suc)n(h)f(that)g (the)h(v)-5 b(arious)21 b(matrix)g(m)n(ultiplications)c(mak)n(e)0 932 y(sense,)26 b(w)n(e)h(\014nd)g(that)f(if)h Fq(L)833 944 y Fl(T)9 b(L)957 932 y Fr(equals)24 b(all)h(of)h Fq(L)p Fr(,)h(then)g Fq(B)1773 944 y Fl(T)1852 932 y Fr(m)n(ust)e(equal)h(all)e(of)i Fq(B)31 b Fr(and)2756 911 y(^)2737 932 y Fq(B)2800 944 y Fl(T)2879 932 y Fr(m)n(ust)25 b(equal)g(all)g(of)3529 911 y(^)3509 932 y Fq(B)t Fr(.)37 b(Letting)0 1031 y Fq(m)p Fr(\()p Fq(X)7 b Fr(\))27 b(denote)h(the)g (ro)n(w)e(dimension)f(of)i(a)h(matrix,)d(and)i(noticing)e(that)j Fq(L)2376 1043 y Fl(T)9 b(L)2501 1031 y Fr(is)27 b(square,)f(w)n(e)h (can)h(argue)e(that)3522 1001 y Fm(2)845 1239 y Fr(\()p Fq(m)p Fr(\()p Fq(L)1039 1251 y Fl(T)9 b(L)1137 1239 y Fr(\))23 b(=)g Fq(m)p Fr(\()p Fq(L)p Fr(\)\))g Fp(\))1635 1147 y Fn(\020)1685 1239 y Fr(\()p Fq(L)1774 1251 y Fl(T)9 b(L)1895 1239 y Fr(=)22 b Fq(L)p Fr(\))c Fp(^)h Fr(\()p Fq(B)2258 1251 y Fl(T)2334 1239 y Fr(=)j Fq(B)t Fr(\))d Fp(^)2613 1147 y Fn(\020)2682 1218 y Fr(^)2663 1239 y Fq(B)2726 1251 y Fl(T)2801 1239 y Fr(=)2908 1218 y(^)2889 1239 y Fq(B)2956 1147 y Fn(\021)o(\021)0 1442 y Fr(so)27 b(that)598 1453 y Fn( )729 1478 y(\022)834 1535 y Fq(B)897 1547 y Fl(T)p 790 1568 204 4 v 790 1585 V 832 1654 a Fq(B)895 1666 y Fl(B)994 1478 y Fn(\023)1078 1595 y Fr(=)1165 1453 y Fn( )1273 1535 y Fq(L)1330 1547 y Fl(T)9 b(L)1427 1499 y Fj(\000)p Fm(1)1536 1514 y Fr(^)1516 1535 y Fq(B)1579 1547 y Fl(T)p 1231 1568 442 4 v 1231 1585 V 1412 1642 a Fr(^)1392 1663 y Fq(B)1455 1675 y Fl(B)1673 1453 y Fn(!!)1823 1595 y Fp(^)19 b Fr(\()p Fq(m)p Fr(\()p Fq(L)2091 1607 y Fl(T)9 b(L)2188 1595 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))2558 1453 y Fn(!)2646 1595 y Fp(\))2752 1503 y Fn(\020)2802 1595 y Fq(B)27 b Fr(:=)c Fq(L)3060 1561 y Fj(\000)p Fm(1)3168 1574 y Fr(^)3149 1595 y Fq(B)3216 1503 y Fn(\021)3279 1595 y Fq(:)0 1836 y Fr(The)30 b(fact)f(that)h Fq(B)583 1848 y Fl(B)670 1836 y Fr(and)853 1815 y(^)833 1836 y Fq(B)896 1848 y Fl(B)983 1836 y Fr(con)n(tain)e(no)h(ro)n(ws)g (can)g(b)r(e)h(depicted)f(b)n(y)g(the)h(double)f(lines)e(reac)n(hing)g (the)j(b)r(ottom)f(of)h(the)0 1936 y(resp)r(ectiv)n(e)c(partitioned)f (matrices)g(in)i(the)h(lo)r(op-in)n(v)-5 b(arian)n(t:)853 2046 y Fn( )984 2071 y(\022)1089 2128 y Fq(B)1152 2140 y Fl(T)p 1045 2161 204 4 v 1045 2177 V 1087 2247 a Fq(B)1150 2259 y Fl(B)1248 2071 y Fn(\023)1333 2188 y Fr(=)1420 2046 y Fn( )1528 2128 y Fq(L)1585 2140 y Fl(T)9 b(L)1682 2091 y Fj(\000)p Fm(1)1791 2107 y Fr(^)1771 2128 y Fq(B)1834 2140 y Fl(T)p 1486 2161 442 4 v 1486 2177 V 1667 2235 a Fr(^)1647 2256 y Fq(B)1710 2268 y Fl(B)1928 2046 y Fn(!!)2078 2188 y Fp(^)19 b Fr(\()p Fq(m)p Fr(\()p Fq(L)2346 2200 y Fl(T)9 b(L)2443 2188 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))2813 2046 y Fn(!)936 2471 y Fp(\))1102 2329 y Fn( )1233 2353 y(\022)1336 2410 y Fq(B)1399 2422 y Fl(T)p 1294 2443 199 4 v 1294 2460 V 1493 2353 a Fn(\023)1577 2471 y Fr(=)1665 2329 y Fn( )1772 2414 y Fq(L)1829 2379 y Fj(\000)p Fm(1)1829 2439 y Fl(T)9 b(L)1946 2393 y Fr(^)1926 2414 y Fq(B)1989 2426 y Fl(T)p 1730 2448 353 4 v 1730 2464 V 2083 2329 a Fn(!!)2233 2471 y Fp(^)19 b Fr(\()p Fq(m)p Fr(\()p Fq(L)2501 2483 y Fl(T)9 b(L)2598 2471 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))2968 2329 y Fn(!)936 2753 y Fp(\))1102 2611 y Fn( )1168 2636 y(\022)1270 2692 y Fq(B)p 1229 2725 151 4 v 1229 2742 V 1379 2636 a Fn(\023)1463 2753 y Fr(=)1551 2611 y Fn( )1658 2697 y Fq(L)1715 2667 y Fj(\000)p Fm(1)1823 2676 y Fr(^)1804 2697 y Fq(B)p 1616 2730 296 4 v 1616 2746 V 1912 2611 a Fn(!!)936 2985 y Fp(\))1102 2893 y Fn(\020)1151 2985 y Fq(B)28 b Fr(=)22 b Fq(L)1386 2951 y Fj(\000)p Fm(1)1495 2964 y Fr(^)1475 2985 y Fq(B)1542 2893 y Fn(\021)1605 2985 y Fq(:)0 3193 y Fr(Th)n(us,)27 b(the)h(lo)r(op-guard)d(can)i(b)r (e)h(c)n(hosen)f(as)g Fq(G)c Fr(:)g Fp(:)14 b Fr(\()q Fq(m)p Fr(\()p Fq(L)1825 3205 y Fl(T)9 b(L)1922 3193 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))28 b(or,)f (simpli\014ed,)d(as)1573 3375 y Fq(G)f Fr(:)g Fq(m)p Fr(\()p Fq(L)1869 3387 y Fl(T)9 b(L)1967 3375 y Fr(\))23 b Fp(6)p Fr(=)g Fq(m)p Fr(\()p Fq(L)p Fr(\))p Fq(:)0 3591 y Fk(Step)32 b(4:)41 b(Initialization)0 3745 y Fr(Next,)28 b(w)n(e)e(m)n(ust)h(initialize)22 b(the)27 b(partitionings)c(of)k(the)h (matrices)c(so)j(that)g Fq(B)k Fr(con)n(tains)25 b(the)j(correct)d(con) n(ten)n(ts)i(b)r(efore)g(the)0 3844 y(lo)r(op)f(commenses.)34 b(Notice)27 b(that)h(the)g(partitionings)775 4097 y Fq(L)22 b Fp(!)960 3980 y Fn(\022)1065 4036 y Fq(L)1122 4048 y Fl(T)9 b(L)p 1262 4066 4 100 v 1279 4066 V 1383 4036 a Fr(0)p 1022 4070 506 4 v 1022 4086 V 1063 4156 a Fq(L)1120 4168 y Fl(B)s(L)p 1262 4186 4 100 v 1279 4186 V 1322 4156 a Fq(L)1379 4168 y Fl(B)s(R)1527 3980 y Fn(\023)1602 4097 y Fq(;)97 b(B)27 b Fp(!)1918 3980 y Fn(\022)2024 4036 y Fq(B)2087 4048 y Fl(T)p 1980 4070 204 4 v 1980 4086 V 2021 4156 a Fq(B)2084 4168 y Fl(B)2183 3980 y Fn(\023)2258 4097 y Fq(;)97 b Fr(and)2614 4076 y(^)2594 4097 y Fq(B)28 b Fp(!)2791 3955 y Fn( )2920 4015 y Fr(^)2900 4036 y Fq(B)2963 4048 y Fl(T)p 2856 4070 V 2856 4086 V 2918 4144 a Fr(^)2898 4165 y Fq(B)2961 4177 y Fl(B)3060 3955 y Fn(!)0 4371 y Fr(where)k Fq(L)302 4383 y Fl(T)9 b(L)431 4371 y Fr(is)32 b(0)21 b Fp(\002)g Fr(0,)33 b(and)f Fq(B)996 4383 y Fl(T)1081 4371 y Fr(and)1266 4350 y(^)1247 4371 y Fq(B)1310 4383 y Fl(T)1394 4371 y Fr(con)n(tain)f(0)h(ro)n(ws,)g (ha)n(v)n(e)f(the)h(prop)r(ert)n(y)f(that)i(they)f(lea)n(v)n(e)e(the)j (v)-5 b(ariables)29 b(in)i(a)0 4470 y(state)c(where)g(the)h(lo)r(op-in) n(v)-5 b(arian)n(t)1445 4523 y Fn(\022)1550 4579 y Fq(B)1613 4591 y Fl(T)p 1506 4613 V 1506 4629 V 1548 4699 a Fq(B)1611 4711 y Fl(B)1710 4523 y Fn(\023)1794 4640 y Fr(=)1881 4498 y Fn( )1989 4579 y Fq(L)2046 4591 y Fl(T)9 b(L)2143 4543 y Fj(\000)p Fm(1)2252 4558 y Fr(^)2232 4579 y Fq(B)2295 4591 y Fl(T)p 1947 4613 442 4 v 1947 4629 V 2128 4687 a Fr(^)2108 4708 y Fq(B)2171 4720 y Fl(B)2389 4498 y Fn(!)p 0 4797 1560 4 v 92 4851 a Fh(2)127 4874 y Fg(Note:)39 b Ff(m)p Fg(\()p Ff(L)469 4885 y Fd(T)8 b(L)560 4874 y Fg(\))26 b(=)f Ff(n)p Fg(\()p Ff(L)811 4885 y Fd(T)8 b(L)902 4874 y Fg(\),)28 b(where)g Ff(n)p Fg(\()p Ff(X)5 b Fg(\))27 b(equals)i(the)f(n)n(um)n(b)r(er)f(of)g(columns)h(in)g (matrix)g Ff(X)5 b Fg(,)27 b(since)i Ff(L)2991 4885 y Fd(T)8 b(L)3108 4874 y Fg(is)28 b(square.)41 b(F)-6 b(or)27 b Ff(L)3620 4885 y Fd(T)8 b(L)3738 4874 y Fg(to)27 b(b)r(e)0 4962 y(able)g(to)g(m)n(ultiply)542 4945 y(^)525 4962 y Ff(B)578 4973 y Fd(T)627 4962 y Fg(,)f(the)h(n)n(um)n(b)r(er)g(of)e (ro)n(ws)h(in)1406 4945 y(^)1390 4962 y Ff(B)1443 4973 y Fd(T)1518 4962 y Fg(m)n(ust)g(equal)i Ff(n)p Fg(\()p Ff(L)2002 4973 y Fd(T)8 b(L)2093 4962 y Fg(\))27 b(so)f(that)h Ff(m)p Fg(\()p Ff(L)2528 4973 y Fd(T)8 b(L)2620 4962 y Fg(\))24 b(=)g Ff(m)p Fg(\()2856 4945 y(^)2839 4962 y Ff(B)2892 4973 y Fd(T)2941 4962 y Fg(\).)39 b(Also,)27 b(the)g(n)n(um)n(b)r(er)g(of)f(ro)n(ws)f(in)0 5041 y Ff(B)53 5052 y Fd(T)125 5041 y Fg(m)n(ust)f(equal)h Ff(m)p Fg(\()p Ff(L)623 5052 y Fd(T)8 b(L)739 5041 y Fg(if)23 b(w)n(e)h(are)g(to)g(set)g(the)h(left-hand)g(side)f(equal)i(to)e(the)g (righ)n(t-hand)h(side,)f(so)g(that)h Ff(m)p Fg(\()p Ff(B)3068 5052 y Fd(T)3118 5041 y Fg(\))20 b(=)f Ff(m)p Fg(\()p Ff(L)3376 5052 y Fd(T)8 b(L)3468 5041 y Fg(\).)1929 5356 y Fr(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 0 -60 a Fr(is)26 b Fo(true)p Fr(.)37 b(This)26 b(is)h(b)r(ecause)g Fq(B)932 -48 y Fl(T)1012 -60 y Fr(and)g Fq(L)1230 -95 y Fj(\000)p Fm(1)1230 -35 y Fl(T)9 b(L)1347 -81 y Fr(^)1328 -60 y Fq(B)1391 -48 y Fl(T)1471 -60 y Fr(con)n(tain)26 b(no)n(w)h(ro)n(ws)f(so)h(that)g (the)h(lo)r(op-in)n(v)-5 b(arian)n(t)23 b(reduces)k(to)1546 54 y Fn(\022)p 1607 144 204 4 v 1607 160 V 1649 230 a Fq(B)1712 242 y Fl(B)1810 54 y Fn(\023)1895 171 y Fr(=)1982 29 y Fn( )p 2048 139 V 2048 156 V 2109 214 a Fr(^)2090 235 y Fq(B)2153 247 y Fl(B)2251 29 y Fn(!)2331 171 y Fq(:)0 415 y Fr(whic)n(h)22 b(is)g(equiv)-5 b(alen)n(t)21 b(to)i Fq(B)k Fr(=)997 394 y(^)977 415 y Fq(B)t Fr(,)d(the)g (precondition,)d(giv)n(en)h(ho)n(w)g(the)i(matrices)c(w)n(ere)j (partitioned)d(in)j(the)g(initialization.)0 625 y Fk(Step)32 b(5:)41 b(Mo)m(ving)32 b(through)g(the)f(op)s(erands)0 779 y Fr(The)d(next)f(question)g(is)f(ho)n(w)h(to)h(mo)n(v)n(e)d (through)i(the)h(op)r(erands)e(so)h(that)h(w)n(e)g(progress)d(from)h (the)i(original)23 b(state)k(where)1565 884 y Fn(\022)p 1626 974 V 1626 991 V 1667 1060 a Fq(B)1730 1072 y Fl(B)1829 884 y Fn(\023)1913 1001 y Fr(=)2001 859 y Fn( )p 2066 970 V 2066 986 V 2128 1044 a Fr(^)2108 1065 y Fq(B)2171 1077 y Fl(B)2270 859 y Fn(!)0 1224 y Fr(to)g(the)h(\014nal)f(state)g (where)1474 1261 y Fn(\022)1577 1317 y Fq(B)1640 1329 y Fl(T)p 1535 1350 199 4 v 1535 1367 V 1733 1261 a Fn(\023)1817 1378 y Fr(=)1905 1236 y Fn( )2012 1322 y Fq(L)2069 1286 y Fj(\000)p Fm(1)2069 1346 y Fl(T)9 b(L)2186 1301 y Fr(^)2167 1322 y Fq(B)2230 1334 y Fl(T)p 1971 1355 353 4 v 1971 1371 V 2323 1236 a Fn(!)2403 1378 y Fq(:)0 1579 y Fr(In)28 b(considering)c(the)k(partitioned)d(matrices)1057 1685 y Fn(\022)1162 1741 y Fq(L)1219 1753 y Fl(T)9 b(L)p 1359 1771 4 100 v 1375 1771 V 1480 1741 a Fr(0)p 1118 1775 506 4 v 1118 1791 V 1160 1861 a Fq(L)1217 1873 y Fl(B)s(L)p 1359 1891 4 100 v 1375 1891 V 1418 1861 a Fq(L)1475 1873 y Fl(B)s(R)1624 1685 y Fn(\023)1699 1802 y Fq(;)1819 1685 y Fn(\022)1924 1741 y Fq(B)1987 1753 y Fl(T)p 1880 1775 204 4 v 1880 1791 V 1921 1861 a Fq(B)1984 1873 y Fl(B)2083 1685 y Fn(\023)2158 1802 y Fq(;)97 b Fr(and)2508 1660 y Fn( )2638 1720 y Fr(^)2618 1741 y Fq(B)2681 1753 y Fl(T)p 2574 1775 V 2574 1791 V 2635 1849 a Fr(^)2616 1870 y Fq(B)2679 1882 y Fl(B)2777 1660 y Fn(!)0 2025 y Fr(w)n(e)25 b(notice)g(that)g(w)n(e)h(need)f(to)h(gro)n(w)e(the)i (submatrices)c Fq(L)1804 2037 y Fl(T)9 b(L)1902 2025 y Fr(,)26 b Fq(B)2014 2037 y Fl(T)2066 2025 y Fr(,)g(and)f Fq(B)2337 2037 y Fl(B)2395 2025 y Fr(,)h(while)d(shrinking)g(the)j (other)f(submatrices)e(in)0 2124 y(these)28 b(partitionings.)125 2224 y(T)-7 b(o)28 b(do)h(so,)f(w)n(e)h(will)d(repartition)f(the)30 b(matrices,)c(exp)r(osing)h(submatrices)f(that)j(can)f(then)i(b)r(e)f (mo)n(v)n(ed)e(to)i(other)f(parts)0 2323 y(of)g(the)g(partitionings.)0 2534 y Fk(2.1.1)94 b(V)-8 b(arian)m(t)33 b(1:)42 b(un)m(blo)s(c)m(k)m (ed)32 b(algorithm)0 2687 y Fr(First,)c(w)n(e)g(will)d(repartition)h (the)j(matrices)c(so)j(that)h(the)g(submatrices)c(that)k(are)e(mo)n(v)n (ed)g(b)r(et)n(w)n(een)h(the)h(v)-5 b(arious)26 b(parts)i(of)0 2787 y(the)f(matrix)e(are)h(individual)d(elemen)n(ts,)i(ro)n(ws,)g (and/or)h(columns.)34 b(This)26 b(will)e(lead)i(to)h(so-called)c(un)n (blo)r(c)n(k)n(ed)i(algorithms.)125 2887 y(Repartition)143 3043 y Fn(\022)248 3099 y Fq(L)305 3111 y Fl(T)9 b(L)p 445 3129 4 100 v 461 3129 V 566 3099 a Fr(0)p 204 3132 506 4 v 204 3149 V 246 3219 a Fq(L)303 3231 y Fl(B)s(L)p 445 3248 4 100 v 461 3248 V 504 3219 a Fq(L)561 3231 y Fl(B)s(R)710 3043 y Fn(\023)794 3160 y Fp(!)900 2993 y Fn(0)900 3142 y(@)1014 3048 y Fq(L)1071 3060 y Fm(00)p 1181 3077 V 1198 3077 V 1279 3048 a Fr(0)p 1399 3077 V 164 w(0)p 973 3081 639 4 v 973 3097 V 1030 3167 a Fq(l)1057 3137 y Fl(T)1055 3188 y Fm(10)p 1181 3197 4 100 v 1198 3197 V 1241 3167 a Fq(\025)1289 3179 y Fm(11)p 1399 3197 V 1485 3167 a Fr(0)p 973 3200 639 4 v 1014 3270 a Fq(L)1071 3282 y Fm(20)p 1181 3300 4 100 v 1198 3300 V 1253 3270 a Fq(l)1278 3282 y Fm(21)p 1399 3300 V 1443 3270 a Fq(L)1500 3282 y Fm(22)1611 2993 y Fn(1)1611 3142 y(A)1697 3160 y Fq(;)1817 3043 y Fn(\022)1922 3099 y Fq(B)1985 3111 y Fl(T)p 1878 3132 204 4 v 1878 3149 V 1920 3219 a Fq(B)1983 3231 y Fl(B)2082 3043 y Fn(\023)2166 3160 y Fp(!)2272 2993 y Fn(0)2272 3142 y(@)2386 3048 y Fq(B)2449 3060 y Fm(0)p 2344 3081 184 4 v 2344 3097 V 2392 3167 a Fq(b)2428 3137 y Fl(T)2428 3188 y Fm(1)p 2344 3200 V 2386 3270 a Fq(B)2449 3282 y Fm(2)2528 2993 y Fn(1)2528 3142 y(A)2614 3160 y Fq(;)97 b Fr(and)2965 3018 y Fn( )3094 3078 y Fr(^)3074 3099 y Fq(B)3137 3111 y Fl(T)p 3030 3132 204 4 v 3030 3149 V 3092 3207 a Fr(^)3072 3227 y Fq(B)3135 3239 y Fl(B)3234 3018 y Fn(!)3322 3160 y Fp(!)3428 2968 y Fn(0)3428 3114 y(B)3428 3167 y(@)3562 3022 y Fr(^)3543 3043 y Fq(B)3606 3055 y Fm(0)p 3501 3076 184 4 v 3501 3092 V 3546 3150 a Fr(^)3549 3172 y Fq(b)3585 3142 y Fl(T)3585 3193 y Fm(1)p 3501 3205 V 3562 3263 a Fr(^)3543 3284 y Fq(B)3606 3296 y Fm(2)3684 2968 y Fn(1)3684 3114 y(C)3684 3167 y(A)0 3433 y Fr(to)n(w)n(ards)26 b(the)i(top)f(of)h(the)g (lo)r(op)e(and)h(\\mo)n(v)n(e)e(the)j(double)f(lines")143 3589 y Fn(\022)248 3645 y Fq(L)305 3657 y Fl(T)9 b(L)p 445 3675 4 100 v 461 3675 V 566 3645 a Fr(0)p 204 3678 506 4 v 204 3695 V 246 3765 a Fq(L)303 3777 y Fl(B)s(L)p 445 3795 4 100 v 461 3795 V 504 3765 a Fq(L)561 3777 y Fl(B)s(R)710 3589 y Fn(\023)794 3706 y Fp( )900 3539 y Fn(0)900 3688 y(@)1014 3594 y Fq(L)1071 3606 y Fm(00)p 1181 3623 V 1263 3594 a Fr(0)p 1383 3623 V 1400 3623 V 180 w(0)p 973 3627 639 4 v 1030 3697 a Fq(l)1057 3667 y Fl(T)1055 3717 y Fm(10)p 1181 3727 4 100 v 1224 3697 a Fq(\025)1272 3709 y Fm(11)p 1383 3727 V 1399 3727 V 1485 3697 a Fr(0)p 973 3730 639 4 v 973 3746 V 1014 3816 a Fq(L)1071 3828 y Fm(20)p 1181 3846 4 100 v 1236 3816 a Fq(l)1261 3828 y Fm(21)p 1383 3846 V 1399 3846 V 1443 3816 a Fq(L)1500 3828 y Fm(22)1611 3539 y Fn(1)1611 3688 y(A)1697 3706 y Fq(;)1817 3589 y Fn(\022)1922 3645 y Fq(B)1985 3657 y Fl(T)p 1878 3678 204 4 v 1878 3695 V 1920 3765 a Fq(B)1983 3777 y Fl(B)2082 3589 y Fn(\023)2166 3706 y Fp( )2272 3539 y Fn(0)2272 3688 y(@)2386 3594 y Fq(B)2449 3606 y Fm(0)p 2344 3627 184 4 v 2392 3697 a Fq(b)2428 3667 y Fl(T)2428 3717 y Fm(1)p 2344 3730 V 2344 3746 V 2386 3816 a Fq(B)2449 3828 y Fm(2)2528 3539 y Fn(1)2528 3688 y(A)2614 3706 y Fq(;)97 b Fr(and)2965 3564 y Fn( )3094 3624 y Fr(^)3074 3645 y Fq(B)3137 3657 y Fl(T)p 3030 3678 204 4 v 3030 3695 V 3092 3753 a Fr(^)3072 3774 y Fq(B)3135 3786 y Fl(B)3234 3564 y Fn(!)3322 3706 y Fp( )3428 3514 y Fn(0)3428 3660 y(B)3428 3713 y(@)3562 3568 y Fr(^)3543 3589 y Fq(B)3606 3601 y Fm(0)p 3501 3622 184 4 v 3546 3680 a Fr(^)3549 3702 y Fq(b)3585 3671 y Fl(T)3585 3722 y Fm(1)p 3501 3735 V 3501 3751 V 3562 3809 a Fr(^)3543 3830 y Fq(B)3606 3842 y Fm(2)3684 3514 y Fn(1)3684 3660 y(C)3684 3713 y(A)0 3979 y Fr(to)n(w)n(ards)31 b(the)i(b)r(ottom)f(of)h(the)h(lo)r(op.)51 b(The)33 b(idea)e(here)i(is) f(that)h(the)g(double)f(lines)f(ha)n(v)n(e)g(seman)n(tic)g(meaning)f (and)j(sho)n(w)0 4078 y(that)28 b(up)r(on)g(repartitioning)399 4254 y Fq(L)456 4266 y Fl(T)9 b(L)577 4254 y Fp(!)23 b Fq(L)740 4266 y Fm(00)p 955 4283 4 101 v 971 4283 V 1121 4254 a Fq(L)1178 4266 y Fl(T)9 b(R)1303 4254 y Fp(!)1409 4186 y Fn(\000)1489 4253 y Fr(0)p 1570 4283 4 100 v 82 w(0)1696 4186 y Fn(\001)p 253 4287 1631 4 v 253 4303 V 294 4426 a Fq(L)351 4438 y Fl(B)s(L)477 4426 y Fp(!)583 4308 y Fn(\022)701 4373 y Fq(l)728 4343 y Fl(T)726 4394 y Fm(10)p 644 4406 210 4 v 685 4476 a Fq(L)742 4488 y Fm(20)854 4308 y Fn(\023)p 955 4506 4 203 v 972 4506 V 1014 4426 a Fq(L)1071 4438 y Fl(B)s(R)1201 4426 y Fp(!)1307 4308 y Fn(\022)1410 4373 y Fq(\025)1458 4385 y Fm(11)p 1569 4403 4 100 v 1655 4373 a Fr(0)p 1369 4406 412 4 v 1422 4476 a Fq(l)1447 4488 y Fm(21)p 1569 4506 4 100 v 1612 4476 a Fq(L)1669 4488 y Fm(22)1780 4308 y Fn(\023)1883 4365 y Fq(;)2149 4253 y(B)2212 4265 y Fl(T)2288 4253 y Fp(!)23 b Fq(B)2457 4265 y Fm(0)p 2003 4286 638 4 v 2003 4303 V 2044 4425 a Fq(B)2107 4437 y Fl(B)2187 4425 y Fp(!)2294 4308 y Fn(\022)2402 4373 y Fq(b)2438 4343 y Fl(T)2438 4393 y Fm(1)p 2355 4406 184 4 v 2396 4476 a Fq(B)2459 4488 y Fm(2)2538 4308 y Fn(\023)2640 4365 y Fq(;)97 b Fr(and)3148 4213 y(^)3128 4234 y Fq(B)3191 4246 y Fl(T)3267 4234 y Fp(!)3393 4213 y Fr(^)3373 4234 y Fq(B)3436 4246 y Fm(0)p 2977 4268 647 4 v 2977 4284 V 3038 4409 a Fr(^)3019 4430 y Fq(B)3082 4442 y Fl(B)3162 4430 y Fp(!)3268 4288 y Fn( )3378 4356 y Fr(^)3381 4378 y Fq(b)3417 4348 y Fl(T)3417 4398 y Fm(1)p 3334 4411 184 4 v 3395 4469 a Fr(^)3375 4490 y Fq(B)3438 4502 y Fm(2)3517 4288 y Fn(!)3624 4365 y Fq(:)147 b Fr(\(3\))0 4652 y(T)-7 b(o)n(w)n(ards)25 b(the)j(end)g(of)g(the)g(lo)r(op)e(the)i (quadran)n(ts)e(of)i(the)g(partitioned)d(matrix)g(are)h(rede\014ned)i (lik)n(e)347 4879 y Fq(L)404 4891 y Fl(T)9 b(L)524 4879 y Fp( )630 4762 y Fn(\022)733 4827 y Fq(L)790 4839 y Fm(00)p 899 4857 4 100 v 981 4827 a Fr(0)p 691 4860 412 4 v 749 4930 a Fq(l)776 4900 y Fl(T)774 4950 y Fm(01)p 899 4960 4 100 v 943 4930 a Fq(\025)991 4942 y Fm(11)1103 4762 y Fn(\023)p 1204 4960 4 203 v 1221 4960 V 1264 4879 a Fq(L)1321 4891 y Fl(T)g(R)1446 4879 y Fp( )1552 4762 y Fn(\022)1655 4827 y Fr(0)p 1613 4860 125 4 v 1655 4930 a(0)1738 4762 y Fn(\023)p 305 4963 1536 4 v 305 4979 V 379 5050 a Fq(L)436 5062 y Fl(B)s(L)561 5050 y Fp( )667 4983 y Fn(\000)747 5049 y Fq(L)804 5061 y Fm(20)p 914 5079 4 100 v 957 5049 a Fq(l)982 5061 y Fm(21)1093 4983 y Fn(\001)p 1204 5080 4 101 v 1221 5080 V 1321 5050 a Fq(L)1378 5062 y Fl(B)s(R)1508 5050 y Fp( )23 b Fq(L)1671 5062 y Fm(22)1840 4939 y Fq(;)2002 4879 y(B)2065 4891 y Fl(T)2140 4879 y Fp( )2246 4762 y Fn(\022)2349 4827 y Fq(B)2412 4839 y Fm(0)p 2307 4860 184 4 v 2355 4930 a Fq(b)2391 4900 y Fl(T)2391 4951 y Fm(1)2490 4762 y Fn(\023)p 1960 4963 633 4 v 1960 4980 V 2102 5050 a Fq(B)2165 5062 y Fl(B)2245 5050 y Fp( )g Fq(B)2414 5062 y Fm(2)2593 4939 y Fq(;)97 b Fr(and)2991 4854 y(^)2971 4875 y Fq(B)3034 4887 y Fl(T)3110 4875 y Fp( )3216 4733 y Fn( )3343 4801 y Fr(^)3323 4822 y Fq(B)3386 4834 y Fm(0)p 3281 4855 184 4 v 3326 4913 a Fr(^)3329 4935 y Fq(b)3365 4905 y Fl(T)3365 4956 y Fm(1)3465 4733 y Fn(!)p 2930 4982 643 4 v 2930 4999 V 3096 5056 a Fr(^)3076 5077 y Fq(B)3139 5089 y Fl(B)3219 5077 y Fp( )3345 5056 y Fr(^)3325 5077 y Fq(B)3388 5089 y Fm(0)3572 4939 y Fq(:)199 b Fr(\(4\))1929 5356 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 0 -60 a Fk(Step)32 b(6:)41 b(State)33 b(after)f(repartitioning)0 93 y Fr(The)20 b(next)f(question)f(b)r (ecomes)g(what)i(the)g(state)f(is)f(of)i(the)g(op)r(erands,)g(in)e (terms)h(of)g(the)h(submatrices)d(that)i(w)n(ere)g(exp)r(osed)g(as)0 193 y(part)k(of)h(the)g(repartitioning.)32 b(Notice)22 b(that)j(b)n(y)e(substituting)g(the)h(exp)r(osed)f(submatrices)e(in)i (\(3\))h(in)n(to)f(the)h(lo)r(op-in)n(v)-5 b(arian)n(t)0 293 y(w)n(e)27 b(\014nd)h(that)649 369 y Fn( )714 393 y(\022)819 450 y Fq(B)882 462 y Fl(T)p 776 483 204 4 v 776 500 V 817 569 a Fq(B)880 581 y Fl(B)979 393 y Fn(\023)1063 511 y Fr(=)1150 369 y Fn( )1258 450 y Fq(L)1315 462 y Fl(T)9 b(L)1412 414 y Fj(\000)p Fm(1)1521 429 y Fr(^)1501 450 y Fq(B)1564 462 y Fl(T)p 1216 483 442 4 v 1216 500 V 1397 557 a Fr(^)1377 578 y Fq(B)1440 590 y Fl(B)1658 369 y Fn(!!)1812 511 y Fp(\))1919 319 y Fn(0)1919 465 y(B)1919 518 y(@)1991 344 y(0)1991 493 y(@)2208 398 y Fq(B)2271 410 y Fm(0)p 2064 432 389 4 v 2064 448 V 2105 453 a Fn(\022)2214 518 y Fq(b)2250 488 y Fl(T)2250 539 y Fm(1)p 2166 551 184 4 v 2208 621 a Fq(B)2271 633 y Fm(2)2350 453 y Fn(\023)2452 344 y(1)2452 493 y(A)2548 511 y Fr(=)2636 319 y Fn(0)2636 465 y(B)2636 518 y(@)2784 380 y Fq(L)2841 344 y Fj(\000)p Fm(1)2841 402 y(00)2950 359 y Fr(^)2930 380 y Fq(B)2993 392 y Fm(0)p 2708 413 398 4 v 2708 429 V 2750 433 a Fn( )2860 501 y Fr(^)2863 523 y Fq(b)2899 493 y Fl(T)2899 544 y Fm(1)p 2816 556 184 4 v 2877 614 a Fr(^)2857 635 y Fq(B)2920 647 y Fm(2)2999 433 y Fn(!)3106 319 y(1)3106 465 y(C)3106 518 y(A)3179 319 y(1)3179 465 y(C)3179 518 y(A)0 786 y Fr(whic)n(h,)27 b(up)r(on)g(simpli\014cation,)c(giv)n(es)i(the)j(state)g(of)f(the)h (exp)r(osed)f(submatrices)e(after)i(the)h(repartitioning:)1475 907 y Fn(0)1475 1057 y(@)1589 962 y Fq(B)1652 974 y Fm(0)p 1548 995 V 1548 1012 V 1595 1082 a Fq(b)1631 1051 y Fl(T)1631 1102 y Fm(1)p 1548 1115 V 1589 1184 a Fq(B)1652 1196 y Fm(2)1731 907 y Fn(1)1731 1057 y(A)1826 1074 y Fr(=)1914 882 y Fn(0)1914 1028 y(B)1914 1081 y(@)2028 957 y Fq(L)2085 922 y Fj(\000)p Fm(1)2085 979 y(00)2194 936 y Fr(^)2174 957 y Fq(B)2237 969 y Fm(0)p 1987 990 329 4 v 1987 1007 V 2104 1065 a Fr(^)2107 1086 y Fq(b)2143 1056 y Fl(T)2143 1107 y Fm(1)p 1987 1120 V 2121 1177 a Fr(^)2101 1198 y Fq(B)2164 1210 y Fm(2)2316 882 y Fn(1)2316 1028 y(C)2316 1081 y(A)2402 1074 y Fq(:)1369 b Fr(\(5\))0 1408 y Fk(Step)32 b(7:)41 b(State)33 b(after)f(mo)m(ving)f(the)g(double)g(lines)0 1561 y Fr(After)23 b(the)g(double)e(lines)f(are)i(mo)n(v)n(ed,)f(w)n(e) h(are)f(at)i(the)f(b)r(ottom)g(of)h(the)f(lo)r(op)f(and)h(the)h(lo)r (op-in)n(v)-5 b(arian)n(t)18 b(m)n(ust)j(again)f(b)r(e)j(true.)0 1661 y(Notice)h(that)i(the)f(rede\014nition)e(of)j(the)f(partitionings) c(giv)n(en)j(in)g(\(4\))i(means)d(that)j(in)e(terms)g(of)i(the)f(exp)r (osed)g(submatrices)0 1760 y(the)j(con)n(ten)n(ts)f(of)g(the)h(op)r (erands)f(m)n(ust)g(b)r(e)312 1921 y Fn( )378 1946 y(\022)483 2002 y Fq(B)546 2014 y Fl(T)p 439 2035 204 4 v 439 2052 V 480 2122 a Fq(B)543 2134 y Fl(B)642 1946 y Fn(\023)726 2063 y Fr(=)814 1921 y Fn( )921 2002 y Fq(L)978 2014 y Fl(T)9 b(L)1075 1966 y Fj(\000)p Fm(1)1184 1981 y Fr(^)1164 2002 y Fq(B)1227 2014 y Fl(T)p 879 2035 442 4 v 879 2052 V 1060 2109 a Fr(^)1040 2130 y Fq(B)1103 2142 y Fl(B)1321 1921 y Fn(!!)1476 2063 y Fp(\))1582 1871 y Fn(0)1582 2017 y(B)1582 2070 y(@)1654 1896 y(0)1654 2045 y(@)1768 1886 y(\022)1871 1951 y Fq(B)1934 1963 y Fm(0)p 1830 1984 184 4 v 1877 2054 a Fq(b)1913 2023 y Fl(T)1913 2074 y Fm(1)2013 1886 y Fn(\023)p 1727 2087 389 4 v 1727 2103 V 1871 2173 a Fq(B)1934 2185 y Fm(2)2115 1896 y Fn(1)2115 2045 y(A)2211 2063 y Fr(=)2299 1871 y Fn(0)2299 2017 y(B)2299 2070 y(@)2413 1881 y(\022)2516 1946 y Fq(L)2573 1958 y Fm(00)p 2682 1976 4 100 v 2764 1946 a Fr(0)p 2474 1979 412 4 v 2531 2049 a Fq(l)2558 2019 y Fl(T)2556 2070 y Fm(10)p 2682 2079 4 100 v 2725 2049 a Fq(\025)2773 2061 y Fm(11)2886 1881 y Fn(\023)2947 1897 y Fj(\000)p Fm(1)3050 1856 y Fn( )3177 1925 y Fr(^)3157 1946 y Fq(B)3220 1958 y Fm(0)p 3116 1979 184 4 v 3160 2037 a Fr(^)3163 2058 y Fq(b)3199 2028 y Fl(T)3199 2079 y Fm(1)3299 1856 y Fn(!)p 2371 2106 1035 4 v 2371 2122 V 2858 2180 a Fr(^)2839 2201 y Fq(B)2902 2213 y Fm(2)3406 1871 y Fn(1)3406 2017 y(C)3406 2070 y(A)3479 1871 y(1)3479 2017 y(C)3479 2070 y(A)3565 2063 y Fq(:)0 2364 y Fr(Since)1149 2394 y Fn(\022)1251 2459 y Fq(L)1308 2471 y Fm(00)p 1418 2489 4 100 v 1500 2459 a Fr(0)p 1210 2492 412 4 v 1267 2562 a Fq(l)1294 2532 y Fl(T)1292 2583 y Fm(10)p 1418 2592 4 100 v 1461 2562 a Fq(\025)1509 2574 y Fm(11)1622 2394 y Fn(\023)1683 2410 y Fj(\000)p Fm(1)1795 2511 y Fr(=)1882 2394 y Fn(\022)2134 2459 y Fq(L)2191 2424 y Fj(\000)p Fm(1)2191 2481 y(00)p 2468 2489 4 103 v 2559 2459 a Fr(0)p 1944 2492 747 4 v 1985 2565 a Fp(\000)p Fq(\025)2098 2530 y Fj(\000)p Fm(1)2098 2587 y(11)2187 2565 y Fq(l)2214 2535 y Fl(T)2212 2586 y Fm(10)2282 2565 y Fq(L)2339 2530 y Fj(\000)p Fm(1)2339 2587 y(00)p 2468 2595 4 103 v 2511 2565 a Fq(\025)2559 2530 y Fj(\000)p Fm(1)2559 2587 y(11)2690 2394 y Fn(\023)0 2702 y Fr(the)28 b(state)f(of)h(the)g(exp)r(osed)f(submatrices)e(after) i(mo)n(ving)e(the)j(double)e(lines)g(m)n(ust)g(b)r(e)915 2838 y Fn(0)915 2987 y(@)1029 2828 y(\022)1132 2892 y Fq(B)1195 2904 y Fm(0)p 1090 2926 184 4 v 1138 2995 a Fq(b)1174 2965 y Fl(T)1174 3016 y Fm(1)1273 2828 y Fn(\023)p 988 3029 389 4 v 988 3045 V 1132 3115 a Fq(B)1195 3127 y Fm(2)1376 2838 y Fn(1)1376 2987 y(A)1472 3005 y Fr(=)1559 2813 y Fn(0)1559 2959 y(B)1559 3012 y(@)1674 2823 y(\022)1925 2888 y Fq(L)1982 2853 y Fj(\000)p Fm(1)1982 2910 y(00)p 2259 2918 4 103 v 2350 2888 a Fr(0)p 1735 2921 747 4 v 1776 2994 a Fp(\000)p Fq(\025)1889 2959 y Fj(\000)p Fm(1)1889 3016 y(11)1978 2994 y Fq(l)2005 2964 y Fl(T)2003 3015 y Fm(10)2073 2994 y Fq(L)2130 2959 y Fj(\000)p Fm(1)2130 3016 y(00)p 2259 3024 4 103 v 2302 2994 a Fq(\025)2350 2959 y Fj(\000)p Fm(1)2350 3016 y(11)2481 2823 y Fn(\023)2556 2798 y( )2683 2867 y Fr(^)2663 2888 y Fq(B)2726 2900 y Fm(0)p 2622 2921 184 4 v 2667 2978 a Fr(^)2670 3000 y Fq(b)2706 2970 y Fl(T)2706 3021 y Fm(1)2805 2798 y Fn(!)p 1632 3047 1281 4 v 1632 3064 V 2242 3122 a Fr(^)2222 3143 y Fq(B)2285 3155 y Fm(2)2912 2813 y Fn(1)2912 2959 y(C)2912 3012 y(A)0 3307 y Fr(or,)h(simplifying,)1040 3374 y Fn(0)1040 3523 y(@)1154 3364 y(\022)1257 3429 y Fq(B)1320 3441 y Fm(0)p 1215 3462 184 4 v 1263 3532 a Fq(b)1299 3502 y Fl(T)1299 3552 y Fm(1)1399 3364 y Fn(\023)p 1113 3565 389 4 v 1113 3582 V 1257 3651 a Fq(B)1320 3663 y Fm(2)1501 3374 y Fn(1)1501 3523 y(A)1597 3541 y Fr(=)1685 3349 y Fn(0)1685 3495 y(B)1685 3548 y(@)1799 3335 y( )2149 3424 y Fq(L)2206 3388 y Fj(\000)p Fm(1)2206 3446 y(00)2315 3403 y Fr(^)2295 3424 y Fq(B)2358 3436 y Fm(0)p 1864 3457 816 4 v 1906 3537 a Fr(\()1935 3515 y(^)1938 3537 y Fq(b)1974 3507 y Fl(T)1974 3557 y Fm(1)2045 3537 y Fp(\000)18 b Fq(l)2155 3507 y Fl(T)2153 3557 y Fm(10)2223 3537 y Fq(L)2280 3501 y Fj(\000)p Fm(1)2280 3559 y(00)2388 3516 y Fr(^)2368 3537 y Fq(B)2431 3549 y Fm(0)2469 3537 y Fr(\))p Fq(\025)2549 3501 y Fj(\000)p Fm(1)2549 3559 y(11)2680 3335 y Fn(!)p 1757 3584 1030 4 v 1757 3600 V 2242 3658 a Fr(^)2222 3679 y Fq(B)2285 3691 y Fm(2)2787 3349 y Fn(1)2787 3495 y(C)2787 3548 y(A)0 3816 y Fr(whic)n(h)27 b(means)1232 3853 y Fn(0)1232 4002 y(@)1346 3908 y Fq(B)1409 3920 y Fm(0)p 1304 3941 184 4 v 1352 4011 a Fq(b)1388 3981 y Fl(T)1388 4031 y Fm(1)p 1304 4044 V 1304 4061 V 1346 4130 a Fq(B)1409 4142 y Fm(2)1487 3853 y Fn(1)1487 4002 y(A)1583 4020 y Fr(=)1671 3828 y Fn(0)1671 3974 y(B)1671 4027 y(@)2028 3903 y Fq(L)2085 3867 y Fj(\000)p Fm(1)2085 3925 y(00)2194 3882 y Fr(^)2174 3903 y Fq(B)2237 3915 y Fm(0)p 1743 3936 816 4 v 1785 4016 a Fr(\()1814 3994 y(^)1817 4016 y Fq(b)1853 3985 y Fl(T)1853 4036 y Fm(1)1924 4016 y Fp(\000)18 b Fq(l)2034 3985 y Fl(T)2032 4036 y Fm(10)2102 4016 y Fq(L)2159 3980 y Fj(\000)p Fm(1)2159 4038 y(00)2267 3995 y Fr(^)2247 4016 y Fq(B)2310 4028 y Fm(0)2348 4016 y Fr(\))p Fq(\025)2428 3980 y Fj(\000)p Fm(1)2428 4038 y(11)p 1743 4049 V 1743 4065 V 2121 4123 a Fr(^)2101 4144 y Fq(B)2164 4156 y Fm(2)2559 3828 y Fn(1)2559 3974 y(C)2559 4027 y(A)2645 4020 y Fq(:)1126 b Fr(\(6\))0 4354 y Fk(Step)32 b(8:)41 b(Determining)30 b(the)i(up)s(date)g(to)f (the)h(exp)s(osed)f(submatrices)0 4507 y Fr(Notice)26 b(that)h(after)f(repartitioning,)d(the)k(state)g(of)g(the)g(exp)r(osed) f(submatrices)e(is)i(giv)n(en)f(b)n(y)h(the)i(predicate)d(in)h(\(5\).) 37 b(After)0 4606 y(mo)n(ving)20 b(the)k(double)e(lines,)h(the)g(state) g(of)h(the)f(exp)r(osed)g(submatrices)e(m)n(ust)h(b)r(e)i(as)f(giv)n (en)e(b)n(y)i(the)h(predicate)e(in)g(\(6\).)36 b(Since)0 4706 y(no)27 b(computation)f(happ)r(ens)h(in)g(mo)n(ving)e(the)j (double)e(lines,)g(b)r(efore)h(mo)n(ving)e(the)j(double)e(lines)g(the)i (state)f(m)n(ust)g(b)r(e)1232 4828 y Fn(0)1232 4977 y(@)1346 4882 y Fq(B)1409 4894 y Fm(0)p 1304 4916 184 4 v 1304 4932 V 1352 5002 a Fq(b)1388 4972 y Fl(T)1388 5023 y Fm(1)p 1304 5035 V 1346 5105 a Fq(B)1409 5117 y Fm(2)1487 4828 y Fn(1)1487 4977 y(A)1583 4995 y Fr(=)1671 4803 y Fn(0)1671 4949 y(B)1671 5002 y(@)2028 4878 y Fq(L)2085 4842 y Fj(\000)p Fm(1)2085 4900 y(00)2194 4857 y Fr(^)2174 4878 y Fq(B)2237 4890 y Fm(0)p 1743 4911 816 4 v 1743 4927 V 1785 5007 a Fr(\()1814 4985 y(^)1817 5007 y Fq(b)1853 4977 y Fl(T)1853 5028 y Fm(1)1924 5007 y Fp(\000)18 b Fq(l)2034 4977 y Fl(T)2032 5028 y Fm(10)2102 5007 y Fq(L)2159 4971 y Fj(\000)p Fm(1)2159 5029 y(00)2267 4986 y Fr(^)2247 5007 y Fq(B)2310 5019 y Fm(0)2348 5007 y Fr(\))p Fq(\025)2428 4971 y Fj(\000)p Fm(1)2428 5029 y(11)p 1743 5040 V 2121 5098 a Fr(^)2101 5119 y Fq(B)2164 5131 y Fm(2)2559 4803 y Fn(1)2559 4949 y(C)2559 5002 y(A)2645 4995 y Fq(:)1126 b Fr(\(7\))1929 5356 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 0 -60 a Fr(Th)n(us,)27 b(the)h(up)r(date)g(m)n (ust)f(c)n(hange)g(the)h(state)f(of)g(the)h(submatrices)d(lik)n(e)540 52 y Fn(0)540 198 y(B)540 251 y(@)612 76 y(0)612 226 y(@)726 131 y Fq(B)789 143 y Fm(0)p 685 164 184 4 v 685 181 V 733 251 a Fq(b)769 221 y Fl(T)769 271 y Fm(1)p 685 284 V 726 354 a Fq(B)789 366 y Fm(2)868 76 y Fn(1)868 226 y(A)964 243 y Fr(=)1051 52 y Fn(0)1051 198 y(B)1051 251 y(@)1166 126 y Fq(L)1223 91 y Fj(\000)p Fm(1)1223 148 y(00)1331 105 y Fr(^)1311 126 y Fq(B)1374 138 y Fm(0)p 1124 160 329 4 v 1124 176 V 1242 234 a Fr(^)1245 256 y Fq(b)1281 226 y Fl(T)1281 276 y Fm(1)p 1124 289 V 1258 347 a Fr(^)1238 368 y Fq(B)1301 380 y Fm(2)1453 52 y Fn(1)1453 198 y(C)1453 251 y(A)1526 52 y(1)1526 198 y(C)1526 251 y(A)1621 243 y Fp(\000)-14 b(!)1778 52 y Fn(0)1778 198 y(B)1778 251 y(@)1851 76 y(0)1851 226 y(@)1965 131 y Fq(B)2028 143 y Fm(0)p 1923 164 184 4 v 1923 181 V 1971 251 a Fq(b)2007 221 y Fl(T)2007 271 y Fm(1)p 1923 284 V 1965 354 a Fq(B)2028 366 y Fm(2)2107 76 y Fn(1)2107 226 y(A)2202 243 y Fr(=)2290 52 y Fn(0)2290 198 y(B)2290 251 y(@)2648 126 y Fq(L)2705 91 y Fj(\000)p Fm(1)2705 148 y(00)2813 105 y Fr(^)2793 126 y Fq(B)2856 138 y Fm(0)p 2363 160 816 4 v 2363 176 V 2404 256 a Fr(\()2433 234 y(^)2436 256 y Fq(b)2472 226 y Fl(T)2472 276 y Fm(1)2543 256 y Fp(\000)18 b Fq(l)2653 226 y Fl(T)2651 276 y Fm(10)2721 256 y Fq(L)2778 220 y Fj(\000)p Fm(1)2778 278 y(00)2886 235 y Fr(^)2867 256 y Fq(B)2930 268 y Fm(0)2967 256 y Fr(\))p Fq(\025)3047 220 y Fj(\000)p Fm(1)3047 278 y(11)p 2363 289 V 2740 347 a Fr(^)2720 368 y Fq(B)2783 380 y Fm(2)3178 52 y Fn(1)3178 198 y(C)3178 251 y(A)3251 52 y(1)3251 198 y(C)3251 251 y(A)3337 243 y Fq(:)0 546 y Fr(W)-7 b(e)28 b(conclude)e(that)i(the)g(follo)n(wing)c(up)r(dates)j(m) n(ust)g(b)r(e)h(made)f(to)g(submatrices)e(of)i Fq(B)t Fr(:)1461 729 y Fq(b)1497 695 y Fl(T)1497 750 y Fm(1)1572 729 y Fr(:=)c Fq(\025)1731 694 y Fj(\000)p Fm(1)1731 751 y(11)1821 729 y Fr(\()1850 707 y(^)1853 729 y Fq(b)1889 695 y Fl(T)1889 750 y Fm(1)1959 729 y Fp(\000)18 b Fq(l)2069 695 y Fl(T)2067 750 y Fm(10)2137 729 y Fq(L)2194 694 y Fj(\000)p Fm(1)2194 751 y(00)2303 708 y Fr(^)2283 729 y Fq(B)2346 741 y Fm(0)2383 729 y Fr(\))q Fq(:)0 922 y Fr(Realize)26 b(that)465 900 y(^)468 922 y Fq(b)504 892 y Fl(T)504 943 y Fm(1)584 922 y Fr(and)766 901 y(^)746 922 y Fq(B)809 934 y Fm(0)875 922 y Fr(refer)h(to)h(the)h(original)24 b(con)n(ten)n(ts)j(of)h(matrix)e Fq(B)t Fr(.)39 b(As)28 b(part)g(of)g(the)h(computation,)d Fq(B)32 b Fr(has)c(b)r(een)0 1022 y(partially)f(up)r(dated,)32 b(and)f(th)n(us)g(all)d(or)i(part)g (of)h(the)g(con)n(ten)n(ts)g(of)2140 1001 y(^)2120 1022 y Fq(B)k Fr(are)30 b(no)h(longer)d(a)n(v)-5 b(ailable.)43 b(Notice,)30 b(ho)n(w)n(ev)n(er,)g(that)0 1121 y Fq(B)63 1133 y Fm(0)128 1121 y Fr(curren)n(tly)c(con)n(tains)f Fq(L)866 1086 y Fj(\000)p Fm(1)866 1143 y(00)975 1100 y Fr(^)955 1121 y Fq(B)1018 1133 y Fm(0)1083 1121 y Fr(while)g Fq(b)1335 1091 y Fl(T)1335 1142 y Fm(1)1415 1121 y Fr(con)n(tains)h (the)i(original)23 b(con)n(ten)n(ts)2509 1099 y(^)2512 1121 y Fq(b)2548 1091 y Fl(T)2548 1142 y Fm(1)2600 1121 y Fr(.)37 b(Th)n(us,)27 b(the)h(up)r(date)1546 1304 y Fq(b)1582 1270 y Fl(T)1582 1325 y Fm(1)1657 1304 y Fr(:=)22 b(\()p Fq(b)1835 1270 y Fl(T)1835 1325 y Fm(1)1906 1304 y Fp(\000)c Fq(l)2016 1270 y Fl(T)2014 1325 y Fm(10)2084 1304 y Fq(B)2147 1316 y Fm(0)2184 1304 y Fr(\))p Fq(\025)2264 1268 y Fj(\000)p Fm(1)2264 1326 y(11)0 1487 y Fr(will)25 b(lea)n(v)n(e)g(the)j(submatrices)d(of)i Fq(B)32 b Fr(in)27 b(the)h(desired)e(state.)0 1702 y Fk(Algorithm)0 1856 y Fr(An)31 b(annotated)f(algorithm,)e(in)i(whic)n(h)g(the)h(states)f (of)h(the)g(v)-5 b(ariables)28 b(are)h(presen)n(ted)i(as)f(w)n(ell)e (as)i(the)i(op)r(erations)c(to)j(b)r(e)0 1955 y(executed,)37 b(is)d(presen)n(ted)h(in)f(Fig.)h(2.)59 b(By)35 b(remo)n(ving)d(all)g (annotations)i(and)h(remo)n(ving)c(an)n(y)k(op)r(eration)e(in)n(v)n (olving)e(the)0 2055 y(\014cticious)366 2034 y(^)346 2055 y Fq(B)5 b Fr(,)27 b(the)h(\014nal)f(algorithm)c(is)k(giv)n(en)e (Fig.)i(3.)0 2271 y Fk(2.1.2)94 b(V)-8 b(arian)m(t)33 b(1:)42 b(blo)s(c)m(k)m(ed)32 b(algorithm)0 2424 y Fr(Next,)d(w)n(e)f (will)e(repartition)f(the)k(matrices)d(so)h(that)i(the)g(submatrices)c (that)k(are)f(mo)n(v)n(ed)e(b)r(et)n(w)n(een)i(the)h(v)-5 b(arious)26 b(parts)i(of)0 2524 y(the)g(matrix)d(are)i(submatrices.)34 b(This)26 b(will)f(lead)h(to)i(so-called)c(blo)r(c)n(k)n(ed)i (algorithms.)125 2623 y(Repartition)121 2809 y Fn(\022)226 2865 y Fq(L)283 2877 y Fl(T)9 b(L)p 422 2895 4 100 v 439 2895 V 543 2865 a Fr(0)p 182 2898 506 4 v 182 2915 V 223 2985 a Fq(L)280 2997 y Fl(B)s(L)p 422 3015 4 100 v 439 3015 V 482 2985 a Fq(L)539 2997 y Fl(B)s(R)687 2809 y Fn(\023)772 2926 y Fp(!)878 2759 y Fn(0)878 2908 y(@)992 2814 y Fq(L)1049 2826 y Fm(00)p 1159 2844 V 1175 2844 V 1261 2814 a Fr(0)p 1385 2844 V 168 w(0)p 950 2847 647 4 v 950 2864 V 992 2933 a Fq(L)1049 2945 y Fm(10)p 1159 2963 4 100 v 1175 2963 V 1218 2933 a Fq(L)1275 2945 y Fm(11)p 1385 2963 V 1471 2933 a Fr(0)p 950 2967 647 4 v 992 3036 a Fq(L)1049 3048 y Fm(20)p 1159 3066 4 100 v 1175 3066 V 1234 3036 a Fq(l)1259 3048 y Fm(21)p 1385 3066 V 1428 3036 a Fq(L)1485 3048 y Fm(22)1597 2759 y Fn(1)1597 2908 y(A)1683 2926 y Fq(;)1803 2809 y Fn(\022)1908 2865 y Fq(B)1971 2877 y Fl(T)p 1864 2898 204 4 v 1864 2915 V 1906 2985 a Fq(B)1969 2997 y Fl(B)2067 2809 y Fn(\023)2151 2926 y Fp(!)2257 2759 y Fn(0)2257 2908 y(@)2372 2814 y Fq(B)2435 2826 y Fm(0)p 2330 2847 184 4 v 2330 2864 V 2372 2933 a Fq(B)2435 2945 y Fm(1)p 2330 2967 V 2372 3036 a Fq(B)2435 3048 y Fm(2)2513 2759 y Fn(1)2513 2908 y(A)2600 2926 y Fq(;)97 b Fr(and)2950 2784 y Fn( )3080 2844 y Fr(^)3060 2865 y Fq(B)3123 2877 y Fl(T)p 3016 2898 204 4 v 3016 2915 V 3077 2973 a Fr(^)3058 2994 y Fq(B)3121 3006 y Fl(B)3219 2784 y Fn(!)3308 2926 y Fp(!)3414 2734 y Fn(0)3414 2880 y(B)3414 2933 y(@)3548 2788 y Fr(^)3528 2809 y Fq(B)3591 2821 y Fm(0)p 3487 2843 184 4 v 3487 2859 V 3548 2917 a Fr(^)3528 2938 y Fq(B)3591 2950 y Fm(1)p 3487 2971 V 3548 3029 a Fr(^)3528 3050 y Fq(B)3591 3062 y Fm(2)3670 2734 y Fn(1)3670 2880 y(C)3670 2933 y(A)3756 2926 y Fq(;)0 3249 y Fr(where)27 b Fq(L)297 3261 y Fm(11)395 3249 y Fr(is)f(a)h Fq(b)18 b Fp(\002)g Fq(b)28 b Fr(matrix,)d(and)i Fq(B)1265 3261 y Fm(1)1330 3249 y Fr(and)1511 3228 y(^)1492 3249 y Fq(B)1555 3261 y Fm(1)1619 3249 y Fr(ha)n(v)n(e)g Fq(b)g Fr(ro)n(ws.)36 b(\\Mo)n(ving)25 b(the)j(double)e(lines")f(is)i(represen)n(ted)f(b)n(y) 139 3435 y Fn(\022)244 3492 y Fq(L)301 3504 y Fl(T)9 b(L)p 441 3521 4 100 v 457 3521 V 562 3492 a Fr(0)p 200 3525 506 4 v 200 3541 V 242 3611 a Fq(L)299 3623 y Fl(B)s(L)p 441 3641 4 100 v 457 3641 V 500 3611 a Fq(L)557 3623 y Fl(B)s(R)706 3435 y Fn(\023)790 3552 y Fp( )896 3385 y Fn(0)896 3535 y(@)1010 3440 y Fq(L)1067 3452 y Fm(00)p 1177 3470 V 1263 3440 a Fr(0)p 1387 3470 V 1404 3470 V 184 w(0)p 969 3473 647 4 v 1010 3543 a Fq(L)1067 3555 y Fm(10)p 1177 3573 4 100 v 1220 3543 a Fq(L)1277 3555 y Fm(11)p 1387 3573 V 1403 3573 V 1489 3543 a Fr(0)p 969 3576 647 4 v 969 3593 V 1010 3663 a Fq(L)1067 3675 y Fm(20)p 1177 3692 4 100 v 1220 3663 a Fq(L)1277 3675 y Fm(21)p 1387 3692 V 1403 3692 V 1447 3663 a Fq(L)1504 3675 y Fm(22)1615 3385 y Fn(1)1615 3535 y(A)1701 3552 y Fq(;)1821 3435 y Fn(\022)1926 3492 y Fq(B)1989 3504 y Fl(T)p 1882 3525 204 4 v 1882 3541 V 1924 3611 a Fq(B)1987 3623 y Fl(B)2086 3435 y Fn(\023)2170 3552 y Fp( )2276 3385 y Fn(0)2276 3535 y(@)2390 3440 y Fq(B)2453 3452 y Fm(0)p 2349 3473 184 4 v 2390 3543 a Fq(B)2453 3555 y Fm(1)p 2349 3576 V 2349 3593 V 2390 3663 a Fq(B)2453 3675 y Fm(2)2532 3385 y Fn(1)2532 3535 y(A)2618 3552 y Fq(;)97 b Fr(and)2969 3410 y Fn( )3098 3471 y Fr(^)3078 3492 y Fq(B)3141 3504 y Fl(T)p 3035 3525 204 4 v 3035 3541 V 3096 3599 a Fr(^)3076 3620 y Fq(B)3139 3632 y Fl(B)3238 3410 y Fn(!)3326 3552 y Fp( )3433 3360 y Fn(0)3433 3506 y(B)3433 3560 y(@)3566 3415 y Fr(^)3547 3436 y Fq(B)3610 3448 y Fm(0)p 3505 3469 184 4 v 3566 3526 a Fr(^)3547 3547 y Fq(B)3610 3559 y Fm(1)p 3505 3581 V 3505 3597 V 3566 3655 a Fr(^)3547 3676 y Fq(B)3610 3688 y Fm(2)3688 3360 y Fn(1)3688 3506 y(C)3688 3560 y(A)0 3855 y Fr(to)n(w)n(ards)31 b(the)i(b)r(ottom)f(of)h(the)h(lo)r(op.)51 b(The)33 b(idea)e(here)i(is) f(that)h(the)g(double)f(lines)f(ha)n(v)n(e)g(seman)n(tic)g(meaning)f (and)j(sho)n(w)0 3954 y(that)28 b(up)r(on)g(repartitioning)400 4145 y Fq(L)457 4157 y Fl(T)9 b(L)577 4145 y Fp(!)23 b Fq(L)740 4157 y Fm(00)p 955 4175 4 101 v 972 4175 V 1126 4145 a Fq(L)1183 4157 y Fl(T)9 b(R)1308 4145 y Fp(!)1414 4078 y Fn(\000)1494 4144 y Fr(0)p 1575 4174 4 100 v 82 w(0)1701 4078 y Fn(\001)p 253 4179 1639 4 v 253 4195 V 295 4317 a Fq(L)352 4329 y Fl(B)s(L)477 4317 y Fp(!)583 4200 y Fn(\022)686 4265 y Fq(L)743 4277 y Fm(10)p 644 4298 210 4 v 686 4368 a Fq(L)743 4380 y Fm(20)854 4200 y Fn(\023)p 955 4398 4 203 v 972 4398 V 1015 4317 a Fq(L)1072 4329 y Fl(B)s(R)1202 4317 y Fp(!)1308 4200 y Fn(\022)1411 4265 y Fq(L)1468 4277 y Fm(11)p 1577 4295 4 100 v 1663 4265 a Fr(0)p 1369 4298 420 4 v 1411 4368 a Fq(L)1468 4380 y Fm(21)p 1577 4398 4 100 v 1620 4368 a Fq(L)1677 4380 y Fm(22)1789 4200 y Fn(\023)1891 4257 y Fq(;)2158 4145 y(B)2221 4157 y Fl(T)2296 4145 y Fp(!)23 b Fq(B)2465 4157 y Fm(0)p 2011 4178 638 4 v 2011 4195 V 2053 4317 a Fq(B)2116 4329 y Fl(B)2196 4317 y Fp(!)2302 4200 y Fn(\022)2405 4264 y Fq(B)2468 4276 y Fm(1)p 2363 4298 184 4 v 2405 4367 a Fq(B)2468 4379 y Fm(2)2547 4200 y Fn(\023)2649 4257 y Fq(;)97 b Fr(and)3152 4120 y(^)3132 4140 y Fq(B)3195 4152 y Fl(T)3271 4140 y Fp(!)3397 4120 y Fr(^)3377 4140 y Fq(B)3440 4152 y Fm(0)p 2986 4174 638 4 v 2986 4190 V 3047 4300 a Fr(^)3027 4321 y Fq(B)3090 4333 y Fl(B)3171 4321 y Fp(!)3277 4204 y Fn(\022)3399 4248 y Fr(^)3379 4269 y Fq(B)3442 4281 y Fm(1)p 3338 4302 184 4 v 3399 4360 a Fr(^)3379 4381 y Fq(B)3442 4393 y Fm(2)3521 4204 y Fn(\023)3624 4257 y Fq(:)147 b Fr(\(8\))0 4560 b(o)n(w)n(ards)25 y(T)-7 b(the)j(end)g(of)g(the)g(lo)r(op)e(the)i (quadran)n(ts)e(of)i(the)g(partitioned)d(matrix)g(are)h(rede\014ned)i (lik)n(e)347 4802 y Fq(L)404 4814 y Fl(T)9 b(L)525 4802 y Fp( )631 4685 y Fn(\022)733 4750 y Fq(L)790 4762 y Fm(00)p 900 4780 4 100 v 986 4750 a Fr(0)p 692 4783 420 4 v 733 4853 a Fq(L)790 4865 y Fm(01)p 900 4883 4 100 v 943 4853 a Fq(L)1000 4865 y Fm(11)1112 4685 y Fn(\023)p 1213 4883 4 203 v 1230 4883 V 1272 4802 a Fq(L)1329 4814 y Fl(T)g(R)1454 4802 y Fp( )1561 4685 y Fn(\022)1663 4750 y Fr(0)p 1622 4783 125 4 v 1663 4853 a(0)1746 4685 y Fn(\023)p 306 4886 1544 4 v 306 4903 V 368 4973 a Fq(L)425 4985 y Fl(B)s(L)550 4973 y Fp( )656 4906 y Fn(\000)736 4972 y Fq(L)793 4984 y Fm(20)p 902 5002 4 100 v 946 4972 a Fq(L)1003 4984 y Fm(21)1114 4906 y Fn(\001)p 1213 5003 4 101 v 1230 5003 V 1330 4973 a Fq(L)1387 4985 y Fl(B)s(R)1517 4973 y Fp( )23 b Fq(L)1680 4985 y Fm(22)1849 4862 y Fq(;)2010 4803 y(B)2073 4815 y Fl(T)2149 4803 y Fp( )2255 4685 y Fn(\022)2357 4750 y Fq(B)2420 4762 y Fm(0)p 2316 4783 184 4 v 2357 4853 a Fq(B)2420 4865 y Fm(1)2499 4685 y Fn(\023)p 1969 4886 633 4 v 1969 4903 V 2110 4973 a Fq(B)2173 4985 y Fl(B)2254 4973 y Fp( )g Fq(B)2423 4985 y Fm(2)2602 4862 y Fq(;)97 b Fr(and)3000 4777 y(^)2980 4798 y Fq(B)3043 4810 y Fl(T)3118 4798 y Fp( )3224 4681 y Fn(\022)3347 4725 y Fr(^)3327 4746 y Fq(B)3390 4758 y Fm(0)p 3285 4779 184 4 v 3347 4837 a Fr(^)3327 4858 y Fq(B)3390 4870 y Fm(1)3469 4681 y Fn(\023)p 2938 4891 633 4 v 2938 4907 V 3100 4965 a Fr(^)3080 4986 y Fq(B)3143 4998 y Fl(B)3223 4986 y Fp( )3349 4965 y Fr(^)3329 4986 y Fq(B)3392 4998 y Fm(0)3571 4862 y Fq(:)200 b Fr(\(9\))1929 5356 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 353 383 3195 4 v 351 470 4 87 v 403 446 a Fg(Step)p 588 470 V 605 470 V 117 w(Annotated)26 b(Algorithm:)64 b Ff(B)23 b Fg(:=)c Ff(L)1640 429 y Fg(^)1623 446 y Ff(B)p 3545 470 V 353 473 3195 4 v 353 490 V 351 589 4 100 v 436 556 a Fg(1a)p 588 589 V 605 589 V 656 493 a Fn(\010)705 556 y Ff(B)k Fg(=)872 539 y(^)855 556 y Ff(B)912 493 y Fn(\011)p 3545 589 V 353 593 3195 4 v 351 878 4 286 v 454 709 a Fg(4)p 588 878 V 605 878 V 167 w Fc(P)n(artition)47 b Ff(L)20 b Fb(!)1182 596 y Fn(\022)1286 659 y Ff(L)1334 670 y Fd(T)8 b(L)p 1467 682 4 79 v 1484 682 V 1583 659 a Fg(0)p 1243 686 473 4 v 1243 702 V 1284 757 a Ff(L)1332 768 y Fd(B)r(L)p 1467 781 4 79 v 1484 781 V 1527 757 a Ff(L)1575 768 y Fd(B)r(R)1715 596 y Fn(\023)1776 709 y Fg(,)23 b Ff(B)h Fb(!)1986 596 y Fn(\022)2090 659 y Ff(B)2143 670 y Fd(T)p 2047 686 189 4 v 2047 702 V 2088 757 a Ff(B)2141 768 y Fd(B)2236 596 y Fn(\023)2320 709 y Fg(and)2474 692 y(^)2458 709 y Ff(B)f Fb(!)2624 596 y Fn(\022)2745 642 y Fg(^)2729 659 y Ff(B)2782 670 y Fd(T)p 2685 686 V 2685 702 V 2743 748 a Fg(^)2727 765 y Ff(B)2780 776 y Fd(B)2874 596 y Fn(\023)778 855 y Fc(where)51 b Ff(L)1089 866 y Fd(T)8 b(L)1203 855 y Fg(is)24 b(0)16 b Fb(\002)f Fg(0,)24 b(and)g Ff(B)1664 866 y Fd(T)1736 855 y Fg(and)1890 838 y(^)1874 855 y Ff(B)1927 866 y Fd(T)1999 855 y Fg(ha)n(v)n(e)h(0)f(ro)n (ws)p 3545 878 4 286 v 353 882 3195 4 v 351 1081 4 200 v 454 998 a(2)p 588 1081 V 605 1081 V 656 885 a Fn(\032)q(\022)823 948 y Ff(B)876 959 y Fd(T)p 780 975 189 4 v 780 991 V 821 1047 a Ff(B)874 1058 y Fd(B)969 885 y Fn(\023)1049 998 y Fg(=)1124 885 y Fn(\022)1226 948 y Ff(L)1274 959 y Fd(T)8 b(L)1365 923 y Fa(\000)p Fh(1)1464 931 y Fg(^)1448 948 y Ff(B)1501 959 y Fd(T)p 1185 975 407 4 v 1185 991 V 1352 1038 a Fg(^)1335 1054 y Ff(B)1388 1065 y Fd(B)1591 885 y Fn(\023\033)p 3545 1081 4 200 v 353 1084 3195 4 v 351 1163 4 79 v 454 1140 a Fg(3)p 588 1163 V 605 1163 V 167 w Fc(while)49 b Ff(m)p Fg(\()p Ff(L)1027 1151 y Fd(T)8 b(L)1118 1140 y Fg(\))20 b Fb(6)p Fg(=)g Ff(m)p Fg(\()p Ff(L)p Fg(\))48 b Fc(do)p 3545 1163 V 353 1167 3195 4 v 351 1366 4 200 v 426 1283 a Fg(2,3)p 588 1366 V 605 1366 V 817 1170 a Fn(\032\022\022)1045 1233 y Ff(B)1098 1244 y Fd(T)p 1001 1260 189 4 v 1001 1276 V 1043 1331 a Ff(B)1096 1342 y Fd(B)1190 1170 y Fn(\023)1271 1283 y Fg(=)1345 1170 y Fn(\022)1448 1233 y Ff(L)1496 1244 y Fd(T)8 b(L)1587 1208 y Fa(\000)p Fh(1)1686 1216 y Fg(^)1669 1233 y Ff(B)1722 1244 y Fd(T)p 1407 1260 407 4 v 1407 1276 V 1573 1322 a Fg(^)1557 1339 y Ff(B)1610 1350 y Fd(B)1813 1170 y Fn(\023\023)1951 1283 y Fb(^)15 b Fg(\()q Ff(m)p Fg(\()p Ff(L)2178 1294 y Fd(T)8 b(L)2270 1283 y Fg(\))20 b Fb(6)p Fg(=)f Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2584 1170 y Fn(\033)p 3545 1366 4 200 v 353 1369 3195 4 v 351 2212 4 843 v 436 1424 a Fg(5a)p 588 2212 V 605 2212 V 311 w Fc(Repartition)938 1568 y Fn(\022)1043 1625 y Fq(L)1100 1637 y Fl(T)9 b(L)p 1240 1655 4 100 v 1257 1655 V 1361 1625 a Fr(0)p 999 1658 506 4 v 999 1675 V 1041 1745 a Fq(L)1098 1757 y Fl(B)s(L)p 1240 1774 4 100 v 1257 1774 V 1300 1745 a Fq(L)1357 1757 y Fl(B)s(R)1505 1568 y Fn(\023)1589 1686 y Fp(!)1696 1519 y Fn(0)1696 1668 y(@)1810 1573 y Fq(L)1867 1585 y Fm(00)p 1976 1603 V 1993 1603 V 2075 1573 a Fr(0)p 2195 1603 V 164 w(0)p 1768 1607 639 4 v 1768 1623 V 1826 1693 a Fq(l)1853 1663 y Fl(T)1851 1714 y Fm(10)p 1976 1723 4 100 v 1993 1723 V 2036 1693 a Fq(\025)2084 1705 y Fm(11)p 2195 1723 V 2281 1693 a Fr(0)p 1768 1726 639 4 v 1810 1796 a Fq(L)1867 1808 y Fm(20)p 1976 1826 4 100 v 1993 1826 V 2048 1796 a Fq(l)2073 1808 y Fm(21)p 2195 1826 V 2238 1796 a Fq(L)2295 1808 y Fm(22)2406 1519 y Fn(1)2406 1668 y(A)2479 1686 y Fr(,)2530 1568 y Fn(\022)2635 1625 y Fq(B)2698 1637 y Fl(T)p 2591 1658 204 4 v 2591 1675 V 2632 1745 a Fq(B)2695 1757 y Fl(B)2794 1568 y Fn(\023)2878 1686 y Fp(!)2984 1519 y Fn(0)2984 1668 y(@)3098 1573 y Fq(B)3161 1585 y Fm(0)p 3057 1607 184 4 v 3057 1623 V 3104 1693 a Fq(b)3140 1663 y Fl(T)3140 1714 y Fm(1)p 3057 1726 V 3098 1796 a Fq(B)3161 1808 y Fm(2)3240 1519 y Fn(1)3240 1668 y(A)3313 1686 y Fr(,)27 b(and)938 1879 y Fn(\022)1059 1925 y Fg(^)1043 1942 y Ff(B)1096 1953 y Fd(T)p 999 1969 189 4 v 999 1985 V 1057 2031 a Fg(^)1041 2048 y Ff(B)1094 2059 y Fd(B)1188 1879 y Fn(\023)1269 1992 y Fb(!)1359 1829 y Fn(0)1359 1979 y(@)1490 1880 y Fg(^)1473 1896 y Ff(B)1526 1905 y Fh(0)p 1432 1923 171 4 v 1432 1940 V 1475 1986 a Fg(^)1478 2004 y Ff(b)1508 1980 y Fd(T)1508 2026 y Fh(1)p 1432 2031 V 1490 2077 a Fg(^)1473 2094 y Ff(B)1526 2103 y Fh(2)1603 1829 y Fn(1)1603 1979 y(A)1060 2188 y Fc(where)74 b Ff(b)1376 2165 y Fd(T)1376 2210 y Fh(1)1449 2188 y Fg(and)1583 2171 y(^)1586 2188 y Ff(b)1616 2165 y Fd(T)1616 2210 y Fh(1)1688 2188 y Fg(are)24 b(ro)n(ws)f(and)h Ff(\025)2147 2197 y Fh(11)2236 2188 y Fg(is)g(a)g(scalar)p 3545 2212 4 843 v 353 2215 3195 4 v 351 2514 4 299 v 454 2381 a(6)p 588 2514 V 605 2514 V 817 2215 a Fn(8)817 2290 y(<)817 2440 y(:)891 2244 y( )998 2289 y Ff(B)1051 2298 y Fh(0)p 956 2316 171 4 v 956 2333 V 1002 2390 a Ff(b)1032 2367 y Fd(T)1032 2412 y Fh(1)p 956 2417 V 998 2472 a Ff(B)1051 2481 y Fh(2)1127 2244 y Fn(!)1213 2381 y Fg(=)1287 2219 y Fn(0)1287 2368 y(@)1401 2286 y Ff(L)1449 2256 y Fa(\000)p Fh(1)1449 2308 y(00)1548 2269 y Fg(^)1532 2286 y Ff(B)1585 2295 y Fh(0)p 1360 2313 302 4 v 1360 2330 V 1468 2376 a Fg(^)1471 2393 y Ff(b)1501 2370 y Fd(T)1501 2415 y Fh(1)p 1360 2420 V 1483 2466 a Fg(^)1466 2483 y Ff(B)1519 2492 y Fh(2)1661 2219 y Fn(1)1661 2368 y(A)1734 2215 y(9)1734 2290 y(=)1734 2440 y(;)p 3545 2514 4 299 v 353 2518 3195 4 v 351 2680 4 163 v 454 2615 a Fg(8)p 588 2680 V 605 2680 V 858 2575 a Ff(b)888 2552 y Fd(T)888 2597 y Fh(1)957 2575 y Fg(:=)19 b Ff(b)1081 2552 y Fd(T)1081 2597 y Fh(1)1145 2575 y Fb(\000)d Ff(l)1238 2552 y Fd(T)1237 2597 y Fh(10)1302 2575 y Ff(B)1355 2584 y Fh(0)858 2656 y Ff(b)888 2633 y Fd(T)888 2678 y Fh(1)957 2656 y Fg(:=)j Ff(b)1081 2633 y Fd(T)1081 2678 y Fh(1)1130 2656 y Ff(=\025)1206 2665 y Fh(11)p 3545 2680 V 353 2683 3195 4 v 351 3426 4 743 v 434 2738 a Fg(5b)p 588 3426 V 605 3426 V 309 w Fc(Con)n(tin)n(ue)27 b(with)938 2870 y Fn(\022)1043 2926 y Fq(L)1100 2938 y Fl(T)9 b(L)p 1240 2956 4 100 v 1257 2956 V 1361 2926 a Fr(0)p 999 2959 506 4 v 999 2976 V 1041 3046 a Fq(L)1098 3058 y Fl(B)s(L)p 1240 3076 4 100 v 1257 3076 V 1300 3046 a Fq(L)1357 3058 y Fl(B)s(R)1505 2870 y Fn(\023)1589 2987 y Fp( )1696 2820 y Fn(0)1696 2969 y(@)1810 2875 y Fq(L)1867 2887 y Fm(00)p 1976 2904 V 2058 2875 a Fr(0)p 2178 2904 V 2195 2904 V 181 w(0)p 1768 2908 639 4 v 1826 2978 a Fq(l)1853 2948 y Fl(T)1851 2998 y Fm(10)p 1976 3008 4 100 v 2020 2978 a Fq(\025)2068 2990 y Fm(11)p 2178 3008 V 2195 3008 V 2281 2978 a Fr(0)p 1768 3011 639 4 v 1768 3027 V 1810 3097 a Fq(L)1867 3109 y Fm(20)p 1976 3127 4 100 v 2031 3097 a Fq(l)2056 3109 y Fm(21)p 2178 3127 V 2195 3127 V 2238 3097 a Fq(L)2295 3109 y Fm(22)2406 2820 y Fn(1)2406 2969 y(A)2479 2987 y Fr(,)2530 2870 y Fn(\022)2635 2926 y Fq(B)2698 2938 y Fl(T)p 2591 2959 204 4 v 2591 2976 V 2632 3046 a Fq(B)2695 3058 y Fl(B)2794 2870 y Fn(\023)2878 2987 y Fp( )2984 2820 y Fn(0)2984 2969 y(@)3098 2875 y Fq(B)3161 2887 y Fm(0)p 3057 2908 184 4 v 3104 2978 a Fq(b)3140 2948 y Fl(T)3140 2998 y Fm(1)p 3057 3011 V 3057 3027 V 3098 3097 a Fq(B)3161 3109 y Fm(2)3240 2820 y Fn(1)3240 2969 y(A)3313 2987 y Fr(,)27 b(and)938 3180 y Fn(\022)1059 3226 y Fg(^)1043 3243 y Ff(B)1096 3254 y Fd(T)p 999 3270 189 4 v 999 3286 V 1057 3333 a Fg(^)1041 3349 y Ff(B)1094 3360 y Fd(B)1188 3180 y Fn(\023)1269 3293 y Fb( )1359 3130 y Fn(0)1359 3280 y(@)1490 3181 y Fg(^)1473 3198 y Ff(B)1526 3207 y Fh(0)p 1432 3225 171 4 v 1475 3271 a Fg(^)1478 3288 y Ff(b)1508 3265 y Fd(T)1508 3310 y Fh(1)p 1432 3315 V 1432 3332 V 1490 3378 a Fg(^)1473 3395 y Ff(B)1526 3404 y Fh(2)1603 3130 y Fn(1)1603 3280 y(A)p 3545 3426 4 743 v 353 3429 3195 4 v 351 3728 4 299 v 454 3595 a Fg(7)p 588 3728 V 605 3728 V 817 3429 a Fn(8)817 3504 y(<)817 3653 y(:)891 3458 y( )998 3503 y Ff(B)1051 3512 y Fh(0)p 956 3530 171 4 v 1002 3587 a Ff(b)1032 3564 y Fd(T)1032 3609 y Fh(1)p 956 3614 V 956 3631 V 998 3686 a Ff(B)1051 3695 y Fh(2)1127 3458 y Fn(!)1213 3595 y Fg(=)1287 3433 y Fn(0)1287 3582 y(@)1652 3500 y Ff(L)1700 3470 y Fa(\000)p Fh(1)1700 3522 y(00)1799 3483 y Fg(^)1783 3500 y Ff(B)1836 3509 y Fh(0)p 1360 3527 804 4 v 1401 3590 a Fg(\()1425 3573 y(^)1428 3590 y Ff(b)1458 3567 y Fd(T)1458 3612 y Fh(1)1523 3590 y Fb(\000)16 b Ff(l)1616 3567 y Fd(T)1615 3612 y Fh(10)1680 3590 y Fg(\()p Ff(L)1755 3561 y Fa(\000)p Fh(1)1755 3613 y(00)1854 3574 y Fg(^)1838 3590 y Ff(B)1891 3599 y Fh(0)1926 3590 y Fg(\)\))p Ff(=\025)2056 3599 y Fh(11)p 1360 3617 V 1360 3634 V 1734 3680 a Fg(^)1718 3697 y Ff(B)1771 3706 y Fh(2)2163 3433 y Fn(1)2163 3582 y(A)2236 3429 y(9)2236 3504 y(=)2236 3653 y(;)p 3545 3728 4 299 v 353 3731 3195 4 v 351 3931 4 200 v 454 3848 a Fg(2)p 588 3931 V 605 3931 V 817 3735 a Fn(\032\022)984 3798 y Ff(B)1037 3809 y Fd(T)p 940 3824 189 4 v 940 3841 V 982 3896 a Ff(B)1035 3907 y Fd(B)1129 3735 y Fn(\023)1210 3848 y Fg(=)1284 3735 y Fn(\022)1387 3798 y Ff(L)1435 3809 y Fd(T)8 b(L)1526 3773 y Fa(\000)p Fh(1)1625 3781 y Fg(^)1608 3798 y Ff(B)1661 3809 y Fd(T)p 1345 3824 407 4 v 1345 3841 V 1512 3887 a Fg(^)1496 3904 y Ff(B)1549 3915 y Fd(B)1752 3735 y Fn(\023\033)p 3545 3931 4 200 v 353 3934 3195 4 v 351 4013 4 79 v 588 4013 V 605 4013 V 656 3989 a Fc(enddo)p 3545 4013 V 353 4016 3195 4 v 351 4216 4 200 v 426 4132 a Fg(2,3)p 588 4216 V 605 4216 V 656 4020 a Fn(\032)q(\022\022)884 4082 y Ff(B)937 4093 y Fd(T)p 841 4109 189 4 v 841 4126 V 882 4181 a Ff(B)935 4192 y Fd(B)1030 4020 y Fn(\023)1110 4132 y Fg(=)1185 4020 y Fn(\022)1288 4082 y Ff(L)1336 4093 y Fd(T)g(L)1426 4057 y Fa(\000)p Fh(1)1525 4065 y Fg(^)1509 4082 y Ff(B)1562 4093 y Fd(T)p 1246 4109 407 4 v 1246 4126 V 1413 4172 a Fg(^)1396 4189 y Ff(B)1449 4200 y Fd(B)1652 4020 y Fn(\023\023)1790 4132 y Fb(^)16 b(:)c Fg(\()p Ff(m)p Fg(\()p Ff(L)2076 4143 y Fd(T)c(L)2168 4132 y Fg(\))20 b Fb(6)p Fg(=)f Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2482 4020 y Fn(\033)p 3545 4216 4 200 v 353 4219 3195 4 v 351 4318 4 100 v 434 4285 a Fg(1b)p 588 4318 V 605 4318 V 656 4222 a Fn(\010)705 4285 y Ff(B)k Fg(:=)c Ff(L)923 4262 y Fa(\000)p Fh(1)1022 4268 y Fg(^)1005 4285 y Ff(B)1062 4222 y Fn(\011)p 3545 4318 V 353 4322 3195 4 v 458 4564 a Fr(Figure)26 b(2:)37 b(W)-7 b(orksheet)27 b(for)g(deriving)d(un)n (blo)r(c)n(k)n(ed)i(algorithm)e(for)j Fq(B)g Fr(:=)c Fq(L)2832 4534 y Fj(\000)p Fm(1)2920 4564 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(1\).)1929 5356 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 536 -12 a Fk(P)m(artition)55 b Fq(L)23 b Fp(!)1153 -130 y Fn(\022)1258 -73 y Fq(L)1315 -61 y Fl(T)9 b(L)p 1455 -43 4 100 v 1471 -43 V 1576 -73 a Fr(0)p 1214 -40 506 4 v 1214 -23 V 1256 46 a Fq(L)1313 58 y Fl(B)s(L)p 1455 76 4 100 v 1471 76 V 1514 46 a Fq(L)1571 58 y Fl(B)s(R)1720 -130 y Fn(\023)1809 -12 y Fr(and)27 b Fq(B)g Fp(!)2166 -130 y Fn(\022)2272 -73 y Fq(B)2335 -61 y Fl(T)p 2228 -40 204 4 v 2228 -23 V 2269 46 a Fq(B)2332 58 y Fl(B)2431 -130 y Fn(\023)679 146 y Fk(where)59 b Fq(L)1044 158 y Fl(T)9 b(L)1169 146 y Fr(is)26 b(0)18 b Fp(\002)g Fr(0)28 b(and)f Fq(B)1689 158 y Fl(T)1769 146 y Fr(has)g(0)g(ro)n(ws)536 387 y Fk(while)54 b Fq(m)p Fr(\()p Fq(L)972 399 y Fl(T)9 b(L)1069 387 y Fr(\))24 b Fp(6)p Fr(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\))56 b Fk(do)709 487 y(Repartition)852 634 y Fn(\022)957 691 y Fq(L)1014 703 y Fl(T)9 b(L)p 1153 720 4 100 v 1170 720 V 1274 691 a Fr(0)p 913 724 506 4 v 913 740 V 954 810 a Fq(L)1011 822 y Fl(B)s(L)p 1153 840 4 100 v 1170 840 V 1213 810 a Fq(L)1270 822 y Fl(B)s(R)1419 634 y Fn(\023)1503 751 y Fp(!)1609 584 y Fn(0)1609 734 y(@)1723 639 y Fq(L)1780 651 y Fm(00)p 1890 669 V 1906 669 V 1988 639 a Fr(0)p 2108 669 V 164 w(0)p 1681 672 639 4 v 1681 689 V 1739 759 a Fq(l)1766 729 y Fl(T)1764 779 y Fm(10)p 1890 789 4 100 v 1906 789 V 1949 759 a Fq(\025)1997 771 y Fm(11)p 2108 789 V 2194 759 a Fr(0)p 1681 792 639 4 v 1723 862 a Fq(L)1780 874 y Fm(20)p 1890 892 4 100 v 1906 892 V 1961 862 a Fq(l)1986 874 y Fm(21)p 2108 892 V 2151 862 a Fq(L)2208 874 y Fm(22)2320 584 y Fn(1)2320 734 y(A)2420 751 y Fr(and)2581 634 y Fn(\022)2686 691 y Fq(B)2749 703 y Fl(T)p 2642 724 204 4 v 2642 740 V 2684 810 a Fq(B)2747 822 y Fl(B)2846 634 y Fn(\023)2930 751 y Fp(!)3036 584 y Fn(0)3036 734 y(@)3150 639 y Fq(B)3213 651 y Fm(0)p 3108 672 184 4 v 3108 689 V 3156 759 a Fq(b)3192 729 y Fl(T)3192 779 y Fm(1)p 3108 792 V 3150 862 a Fq(B)3213 874 y Fm(2)3292 584 y Fn(1)3292 734 y(A)995 961 y Fk(where)87 b Fq(b)1367 931 y Fl(T)1367 982 y Fm(1)1446 961 y Fr(is)27 b(a)g(ro)n(w)f(and)i Fq(\025)1968 973 y Fm(11)2066 961 y Fr(is)f(a)g(scalar)p 709 1063 2341 4 v 709 1055 V 750 1161 a Fq(b)786 1131 y Fl(T)786 1181 y Fm(1)861 1161 y Fr(:=)c Fq(b)1008 1131 y Fl(T)1008 1181 y Fm(1)1078 1161 y Fp(\000)18 b Fq(l)1188 1131 y Fl(T)1186 1181 y Fm(10)1256 1161 y Fq(B)1319 1173 y Fm(0)750 1260 y Fq(b)786 1230 y Fl(T)786 1281 y Fm(1)861 1260 y Fr(:=)23 b Fq(b)1008 1230 y Fl(T)1008 1281 y Fm(1)1060 1260 y Fq(=\025)1150 1272 y Fm(11)p 709 1362 V 709 1354 V 709 1460 a Fk(Con)m(tin)m(ue)31 b(with)852 1591 y Fn(\022)957 1647 y Fq(L)1014 1659 y Fl(T)9 b(L)p 1153 1677 4 100 v 1170 1677 V 1274 1647 a Fr(0)p 913 1681 506 4 v 913 1697 V 954 1767 a Fq(L)1011 1779 y Fl(B)s(L)p 1153 1797 4 100 v 1170 1797 V 1213 1767 a Fq(L)1270 1779 y Fl(B)s(R)1419 1591 y Fn(\023)1503 1708 y Fp( )1609 1541 y Fn(0)1609 1691 y(@)1723 1596 y Fq(L)1780 1608 y Fm(00)p 1890 1626 V 1971 1596 a Fr(0)p 2091 1626 V 2108 1626 V 181 w(0)p 1681 1629 639 4 v 1739 1699 a Fq(l)1766 1669 y Fl(T)1764 1720 y Fm(10)p 1890 1729 4 100 v 1933 1699 a Fq(\025)1981 1711 y Fm(11)p 2091 1729 V 2108 1729 V 2194 1699 a Fr(0)p 1681 1732 639 4 v 1681 1749 V 1723 1819 a Fq(L)1780 1831 y Fm(20)p 1890 1848 4 100 v 1945 1819 a Fq(l)1970 1831 y Fm(21)p 2091 1848 V 2108 1848 V 2151 1819 a Fq(L)2208 1831 y Fm(22)2320 1541 y Fn(1)2320 1691 y(A)2420 1708 y Fr(and)2581 1591 y Fn(\022)2686 1647 y Fq(B)2749 1659 y Fl(T)p 2642 1681 204 4 v 2642 1697 V 2684 1767 a Fq(B)2747 1779 y Fl(B)2846 1591 y Fn(\023)2930 1708 y Fp( )3036 1541 y Fn(0)3036 1691 y(@)3150 1596 y Fq(B)3213 1608 y Fm(0)p 3108 1629 184 4 v 3156 1699 a Fq(b)3192 1669 y Fl(T)3192 1720 y Fm(1)p 3108 1732 V 3108 1749 V 3150 1819 a Fq(B)3213 1831 y Fm(2)3292 1541 y Fn(1)3292 1691 y(A)536 1918 y Fk(enddo)878 2190 y Fr(Figure)26 b(3:)37 b(Un)n(blo)r(c)n(k)n(ed)26 b(algorithm)d(for)k Fq(B)h Fr(:=)22 b Fq(L)2411 2160 y Fj(\000)p Fm(1)2500 2190 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(1\).)0 2460 y Fk(Step)32 b(6:)41 b(State)33 b(after)f(repartitioning) 0 2613 y Fr(The)20 b(next)f(question)f(b)r(ecomes)g(what)i(the)g(state) f(is)f(of)i(the)g(op)r(erands,)g(in)e(terms)h(of)g(the)h(submatrices)d (that)i(w)n(ere)g(exp)r(osed)g(as)0 2713 y(part)k(of)h(the)g (repartitioning.)32 b(Notice)22 b(that)j(b)n(y)e(substituting)g(the)h (exp)r(osed)f(submatrices)e(in)i(\(8\))h(in)n(to)f(the)h(lo)r(op-in)n (v)-5 b(arian)n(t)0 2812 y(w)n(e)27 b(\014nd)h(that)653 2874 y Fn( )719 2899 y(\022)824 2955 y Fq(B)887 2967 y Fl(T)p 780 2988 204 4 v 780 3005 V 822 3075 a Fq(B)885 3087 y Fl(B)983 2899 y Fn(\023)1067 3016 y Fr(=)1155 2874 y Fn( )1262 2955 y Fq(L)1319 2967 y Fl(T)9 b(L)1417 2919 y Fj(\000)p Fm(1)1526 2934 y Fr(^)1506 2955 y Fq(B)1569 2967 y Fl(T)p 1221 2988 442 4 v 1221 3005 V 1401 3063 a Fr(^)1382 3084 y Fq(B)1445 3096 y Fl(B)1663 2874 y Fn(!)o(!)1817 3016 y Fp(\))1923 2824 y Fn(0)1923 2970 y(B)1923 3023 y(@)1996 2849 y(0)1996 2998 y(@)2213 2904 y Fq(B)2276 2916 y Fm(0)p 2068 2937 389 4 v 2068 2953 V 2110 2958 a Fn(\022)2213 3023 y Fq(B)2276 3035 y Fm(1)p 2171 3056 184 4 v 2213 3126 a Fq(B)2276 3138 y Fm(2)2354 2958 y Fn(\023)2457 2849 y(1)2457 2998 y(A)2553 3016 y Fr(=)2640 2824 y Fn(0)2640 2970 y(B)2640 3023 y(@)2784 2899 y Fq(L)2841 2864 y Fj(\000)p Fm(1)2841 2921 y(00)2950 2878 y Fr(^)2930 2899 y Fq(B)2993 2911 y Fm(0)p 2713 2932 389 4 v 2713 2949 V 2754 2963 a Fn(\022)2877 3007 y Fr(^)2857 3028 y Fq(B)2920 3040 y Fm(1)p 2816 3061 184 4 v 2877 3119 a Fr(^)2857 3140 y Fq(B)2920 3152 y Fm(2)2999 2963 y Fn(\023)3101 2824 y(1)3101 2970 y(C)3101 3023 y(A)3174 2824 y(1)3174 2970 y(C)3174 3023 y(A)0 3285 y Fr(whic)n(h,)27 b(up)r(on)g(simpli\014cation,)c(giv)n(es)i(the)j (state)g(of)f(the)h(exp)r(osed)f(submatrices)e(after)i(the)h (repartitioning:)1475 3421 y Fn(0)1475 3570 y(@)1589 3476 y Fq(B)1652 3488 y Fm(0)p 1548 3509 V 1548 3526 V 1589 3595 a Fq(B)1652 3607 y Fm(1)p 1548 3628 V 1589 3698 a Fq(B)1652 3710 y Fm(2)1731 3421 y Fn(1)1731 3570 y(A)1826 3588 y Fr(=)1914 3396 y Fn(0)1914 3542 y(B)1914 3595 y(@)2028 3471 y Fq(L)2085 3436 y Fj(\000)p Fm(1)2085 3493 y(00)2194 3450 y Fr(^)2174 3471 y Fq(B)2237 3483 y Fm(0)p 1987 3504 329 4 v 1987 3521 V 2121 3579 a Fr(^)2101 3600 y Fq(B)2164 3612 y Fm(1)p 1987 3633 V 2121 3691 a Fr(^)2101 3712 y Fq(B)2164 3724 y Fm(2)2316 3396 y Fn(1)2316 3542 y(C)2316 3595 y(A)2402 3588 y Fq(:)1327 b Fr(\(10\))0 3924 y Fk(Step)32 b(7:)41 b(State)33 b(after)f(mo)m(ving) f(the)g(double)g(lines)0 4077 y Fr(After)23 b(the)g(double)e(lines)f (are)i(mo)n(v)n(ed,)f(w)n(e)h(are)f(at)i(the)f(b)r(ottom)g(of)h(the)f (lo)r(op)f(and)h(the)h(lo)r(op-in)n(v)-5 b(arian)n(t)18 b(m)n(ust)j(again)f(b)r(e)j(true.)0 4176 y(Notice)h(that)i(the)f (rede\014nition)e(of)j(the)f(partitionings)c(giv)n(en)j(in)g(\(4\))i (means)d(that)j(in)e(terms)g(of)i(the)f(exp)r(osed)g(submatrices)0 4276 y(the)j(con)n(ten)n(ts)f(of)g(the)h(op)r(erands)f(m)n(ust)g(b)r(e) 312 4441 y Fn( )378 4466 y(\022)483 4522 y Fq(B)546 4534 y Fl(T)p 439 4555 204 4 v 439 4572 V 481 4642 a Fq(B)544 4654 y Fl(B)642 4466 y Fn(\023)727 4583 y Fr(=)814 4441 y Fn( )921 4522 y Fq(L)978 4534 y Fl(T)9 b(L)1076 4486 y Fj(\000)p Fm(1)1185 4501 y Fr(^)1165 4522 y Fq(B)1228 4534 y Fl(T)p 880 4555 442 4 v 880 4572 V 1060 4630 a Fr(^)1041 4651 y Fq(B)1104 4663 y Fl(B)1322 4441 y Fn(!)o(!)1476 4583 y Fp(\))1582 4391 y Fn(0)1582 4537 y(B)1582 4590 y(@)1655 4416 y(0)1655 4565 y(@)1769 4406 y(\022)1872 4471 y Fq(B)1935 4483 y Fm(0)p 1830 4504 184 4 v 1872 4574 a Fq(B)1935 4586 y Fm(1)2013 4406 y Fn(\023)p 1728 4607 389 4 v 1728 4623 V 1872 4693 a Fq(B)1935 4705 y Fm(2)2116 4416 y Fn(1)2116 4565 y(A)2212 4583 y Fr(=)2299 4391 y Fn(0)2299 4537 y(B)2299 4590 y(@)2414 4405 y(\022)2516 4470 y Fq(L)2573 4482 y Fm(00)p 2683 4500 4 100 v 2769 4470 a Fr(0)p 2475 4503 420 4 v 2516 4573 a Fq(L)2573 4585 y Fm(10)p 2683 4603 4 100 v 2726 4573 a Fq(L)2783 4585 y Fm(11)2894 4405 y Fn(\023)2955 4421 y Fj(\000)p Fm(1)3058 4405 y Fn(\022)3181 4449 y Fr(^)3161 4470 y Fq(B)3224 4482 y Fm(0)p 3120 4503 184 4 v 3181 4561 a Fr(^)3161 4582 y Fq(B)3224 4594 y Fm(1)3303 4405 y Fn(\023)p 2372 4615 1034 4 v 2372 4632 V 2858 4690 a Fr(^)2839 4711 y Fq(B)2902 4723 y Fm(2)3405 4391 y Fn(1)3405 4537 y(C)3405 4590 y(A)3478 4391 y(1)3478 4537 y(C)3478 4590 y(A)3565 4583 y Fq(:)0 4888 y Fr(Since)1121 4919 y Fn(\022)1223 4983 y Fq(L)1280 4995 y Fm(00)p 1390 5013 4 100 v 1476 4983 a Fr(0)p 1182 5017 420 4 v 1223 5086 a Fq(L)1280 5098 y Fm(10)p 1390 5116 4 100 v 1433 5086 a Fq(L)1490 5098 y Fm(11)1602 4919 y Fn(\023)1663 4934 y Fj(\000)p Fm(1)1775 5036 y Fr(=)1863 4919 y Fn(\022)2134 4983 y Fq(L)2191 4948 y Fj(\000)p Fm(1)2191 5006 y(00)p 2488 5013 4 103 v 2583 4983 a Fr(0)p 1924 5017 795 4 v 1965 5090 a Fp(\000)p Fq(L)2087 5054 y Fj(\000)p Fm(1)2087 5112 y(11)2175 5090 y Fq(L)2232 5059 y Fl(T)2232 5110 y Fm(10)2302 5090 y Fq(L)2359 5054 y Fj(\000)p Fm(1)2359 5112 y(00)p 2488 5120 4 103 v 2531 5090 a Fq(L)2588 5054 y Fj(\000)p Fm(1)2588 5112 y(11)2718 4919 y Fn(\023)1929 5356 y Fr(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 0 -60 a Fr(the)28 b(state)f(of)h(the)g(exp)r (osed)f(submatrices)e(after)i(mo)n(ving)e(the)j(double)e(lines)g(m)n (ust)g(b)r(e)896 49 y Fn(0)896 199 y(@)1010 39 y(\022)1112 104 y Fq(B)1175 116 y Fm(0)p 1071 137 184 4 v 1112 207 a Fq(B)1175 219 y Fm(1)1254 39 y Fn(\023)p 968 240 389 4 v 968 257 V 1112 327 a Fq(B)1175 339 y Fm(2)1357 49 y Fn(1)1357 199 y(A)1453 216 y Fr(=)1540 24 y Fn(0)1540 171 y(B)1540 224 y(@)1654 35 y(\022)1925 100 y Fq(L)1982 64 y Fj(\000)p Fm(1)1982 122 y(00)p 2279 130 4 103 v 2375 100 a Fr(0)p 1715 133 795 4 v 1757 206 a Fp(\000)p Fq(L)1879 170 y Fj(\000)p Fm(1)1879 228 y(11)1967 206 y Fq(L)2024 218 y Fm(10)2094 206 y Fq(L)2151 170 y Fj(\000)p Fm(1)2151 228 y(00)p 2279 236 4 103 v 2323 206 a Fq(L)2380 170 y Fj(\000)p Fm(1)2380 228 y(11)2510 35 y Fn(\023)14 b(\022)2707 79 y Fr(^)2687 100 y Fq(B)2750 112 y Fm(0)p 2646 133 184 4 v 2707 191 a Fr(^)2687 212 y Fq(B)2750 224 y Fm(1)2829 35 y Fn(\023)p 1613 245 1319 4 v 1613 261 V 2242 319 a Fr(^)2222 340 y Fq(B)2285 352 y Fm(2)2932 24 y Fn(1)2932 171 y(C)2932 224 y(A)0 492 y Fr(or,)27 b(simplifying,)1009 545 y Fn(0)1009 694 y(@)1123 535 y(\022)1226 600 y Fq(B)1289 612 y Fm(0)p 1184 633 184 4 v 1226 703 a Fq(B)1289 715 y Fm(1)1368 535 y Fn(\023)p 1082 736 389 4 v 1082 753 V 1226 822 a Fq(B)1289 834 y Fm(2)1470 545 y Fn(1)1470 694 y(A)1566 712 y Fr(=)1654 520 y Fn(0)1654 666 y(B)1654 719 y(@)1768 531 y(\022)2149 595 y Fq(L)2206 560 y Fj(\000)p Fm(1)2206 617 y(00)2315 574 y Fr(^)2295 595 y Fq(B)2358 607 y Fm(0)p 1829 629 887 4 v 1870 707 a Fq(L)1927 672 y Fj(\000)p Fm(1)1927 729 y(11)2016 707 y Fr(\()2068 686 y(^)2048 707 y Fq(B)2115 677 y Fl(T)2111 728 y Fm(1)2186 707 y Fp(\000)18 b Fq(L)2326 719 y Fm(10)2396 707 y Fq(L)2453 672 y Fj(\000)p Fm(1)2453 729 y(00)2561 686 y Fr(^)2542 707 y Fq(B)2605 719 y Fm(0)2642 707 y Fr(\))2716 531 y Fn(\023)p 1726 740 1093 4 v 1726 757 V 2242 815 a Fr(^)2222 836 y Fq(B)2285 848 y Fm(2)2818 520 y Fn(1)2818 666 y(C)2818 719 y(A)0 965 y Fr(whic)n(h)27 b(means)1196 1002 y Fn(0)1196 1151 y(@)1310 1057 y Fq(B)1373 1069 y Fm(0)p 1269 1090 184 4 v 1310 1160 a Fq(B)1373 1172 y Fm(1)p 1269 1193 V 1269 1209 V 1310 1279 a Fq(B)1373 1291 y Fm(2)1452 1002 y Fn(1)1452 1151 y(A)1547 1169 y Fr(=)1635 977 y Fn(0)1635 1123 y(B)1635 1176 y(@)2028 1052 y Fq(L)2085 1017 y Fj(\000)p Fm(1)2085 1074 y(00)2194 1031 y Fr(^)2174 1052 y Fq(B)2237 1064 y Fm(0)p 1708 1085 887 4 v 1749 1164 a Fq(L)1806 1129 y Fj(\000)p Fm(1)1806 1186 y(11)1895 1164 y Fr(\()1947 1143 y(^)1927 1164 y Fq(B)1994 1134 y Fl(T)1990 1185 y Fm(1)2065 1164 y Fp(\000)18 b Fq(L)2205 1176 y Fm(10)2275 1164 y Fq(L)2332 1129 y Fj(\000)p Fm(1)2332 1186 y(00)2440 1143 y Fr(^)2421 1164 y Fq(B)2484 1176 y Fm(0)2521 1164 y Fr(\))p 1708 1197 V 1708 1214 V 2121 1272 a(^)2101 1293 y Fq(B)2164 1305 y Fm(2)2595 977 y Fn(1)2595 1123 y(C)2595 1176 y(A)2681 1169 y Fq(:)1048 b Fr(\(11\))0 1500 y Fk(Step)32 b(8:)41 b(Determining)30 b(the)i(up)s(date)g(to)f(the)h(exp)s(osed)f (submatrices)0 1653 y Fr(Notice)23 b(that)i(after)f(repartitioning,)c (the)25 b(state)f(of)g(the)h(exp)r(osed)f(submatrices)d(is)i(giv)n(en)g (b)n(y)g(the)i(predicate)e(in)g(\(10\).)36 b(After)0 1753 y(mo)n(ving)30 b(the)k(double)f(lines,)g(the)h(state)f(of)g(the)h (exp)r(osed)f(submatrices)e(m)n(ust)h(b)r(e)i(as)f(giv)n(en)f(b)n(y)h (the)h(predicate)e(in)g(\(11\).)0 1852 y(Since)25 b(no)h(computation)d (happ)r(ens)j(in)f(mo)n(ving)e(the)j(double)f(lines,)f(b)r(efore)h(mo)n (ving)e(the)j(double)f(lines)f(the)i(state)g(m)n(ust)f(b)r(e)1205 1962 y Fn(0)1205 2111 y(@)1320 2016 y Fq(B)1383 2028 y Fm(0)p 1278 2050 184 4 v 1278 2066 V 1320 2136 a Fq(B)1383 2148 y Fm(1)p 1278 2169 V 1320 2239 a Fq(B)1383 2251 y Fm(2)1461 1962 y Fn(1)1461 2111 y(A)1557 2128 y Fr(=)1645 1937 y Fn(0)1645 2083 y(B)1645 2136 y(@)2028 2012 y Fq(L)2085 1976 y Fj(\000)p Fm(1)2085 2034 y(00)2194 1991 y Fr(^)2174 2012 y Fq(B)2237 2024 y Fm(0)p 1717 2045 868 4 v 1717 2062 V 1759 2140 a Fq(L)1816 2105 y Fj(\000)p Fm(1)1816 2163 y(11)1905 2140 y Fr(\()p Fq(B)2000 2152 y Fm(1)2055 2140 y Fp(\000)18 b Fq(L)2195 2152 y Fm(10)2265 2140 y Fq(L)2322 2105 y Fj(\000)p Fm(1)2322 2163 y(00)2431 2119 y Fr(^)2411 2140 y Fq(B)2474 2152 y Fm(0)2511 2140 y Fr(\))p 1717 2174 V 2121 2231 a(^)2101 2252 y Fq(B)2164 2264 y Fm(2)2585 1937 y Fn(1)2585 2083 y(C)2585 2136 y(A)2671 2128 y Fq(:)1058 b Fr(\(12\))0 2404 y(Th)n(us,)27 b(the)h(up)r(date)g(m)n(ust)f(c)n(hange)g(the)h(state)f(of)g(the)h (submatrices)d(lik)n(e)514 2489 y Fn(0)514 2635 y(B)514 2688 y(@)586 2514 y(0)586 2663 y(@)700 2568 y Fq(B)763 2580 y Fm(0)p 659 2602 184 4 v 659 2618 V 700 2688 a Fq(B)763 2700 y Fm(1)p 659 2721 V 700 2791 a Fq(B)763 2803 y Fm(2)842 2514 y Fn(1)842 2663 y(A)938 2680 y Fr(=)1025 2489 y Fn(0)1025 2635 y(B)1025 2688 y(@)1140 2564 y Fq(L)1197 2528 y Fj(\000)p Fm(1)1197 2586 y(00)1305 2543 y Fr(^)1285 2564 y Fq(B)1348 2576 y Fm(0)p 1098 2597 329 4 v 1098 2614 V 1232 2671 a Fr(^)1212 2692 y Fq(B)1275 2704 y Fm(1)p 1098 2726 V 1232 2783 a Fr(^)1212 2804 y Fq(B)1275 2816 y Fm(2)1427 2489 y Fn(1)1427 2635 y(C)1427 2688 y(A)1500 2489 y(1)1500 2635 y(C)1500 2688 y(A)1595 2680 y Fp(\000)-14 b(!)1752 2489 y Fn(0)1752 2635 y(B)1752 2688 y(@)1825 2514 y(0)1825 2663 y(@)1939 2568 y Fq(B)2002 2580 y Fm(0)p 1897 2602 184 4 v 1897 2618 V 1939 2688 a Fq(B)2002 2700 y Fm(1)p 1897 2721 V 1939 2791 a Fq(B)2002 2803 y Fm(2)2081 2514 y Fn(1)2081 2663 y(A)2176 2680 y Fr(=)2264 2489 y Fn(0)2264 2635 y(B)2264 2688 y(@)2648 2564 y Fq(L)2705 2528 y Fj(\000)p Fm(1)2705 2586 y(00)2813 2543 y Fr(^)2793 2564 y Fq(B)2856 2576 y Fm(0)p 2337 2597 868 4 v 2337 2614 V 2378 2692 a Fq(L)2435 2657 y Fj(\000)p Fm(1)2435 2715 y(11)2524 2692 y Fr(\()2576 2671 y(^)2556 2692 y Fq(B)2619 2704 y Fm(1)2675 2692 y Fp(\000)18 b Fq(L)2815 2704 y Fm(10)2885 2692 y Fq(L)2942 2657 y Fj(\000)p Fm(1)2942 2715 y(00)3050 2671 y Fr(^)3030 2692 y Fq(B)3093 2704 y Fm(0)3130 2692 y Fr(\))p 2337 2726 V 2740 2783 a(^)2720 2804 y Fq(B)2783 2816 y Fm(2)3204 2489 y Fn(1)3204 2635 y(C)3204 2688 y(A)3277 2489 y(1)3277 2635 y(C)3277 2688 y(A)3363 2680 y Fq(:)0 2956 y Fr(w)n(e)27 b(conclude)g(that)g(the)h(follo)n(wing)c(up)r(dates)k(m)n(ust)e(b)r(e)i (made)f(to)g(submatrices)e(of)j Fq(B)t Fr(:)1441 3113 y Fq(B)1504 3125 y Fm(1)1564 3113 y Fr(:=)23 b Fq(L)1732 3077 y Fj(\000)p Fm(1)1732 3135 y(11)1820 3113 y Fr(\()1872 3092 y(^)1852 3113 y Fq(B)1915 3125 y Fm(1)1971 3113 y Fp(\000)18 b Fq(L)2111 3125 y Fm(10)2181 3113 y Fq(L)2238 3077 y Fj(\000)p Fm(1)2238 3135 y(00)2347 3092 y Fr(^)2327 3113 y Fq(B)2390 3125 y Fm(0)2427 3113 y Fr(\))0 3278 y(Again,)25 b(realize)d(that)717 3257 y(^)698 3278 y Fq(B)761 3290 y Fm(1)824 3278 y Fr(and)1003 3257 y(^)983 3278 y Fq(B)1046 3290 y Fm(0)1109 3278 y Fr(refer)i(to)i(the)g (original)21 b(con)n(ten)n(ts)j(of)i(matrix)d Fq(B)t Fr(.)36 b(As)26 b(part)f(of)g(the)h(computation,)e Fq(B)30 b Fr(has)0 3378 y(b)r(een)g(partially)d(up)r(dated,)k(and)f(th)n(us)g (all)e(or)h(part)h(of)g(the)g(con)n(ten)n(ts)f(of)2330 3357 y(^)2310 3378 y Fq(B)35 b Fr(are)29 b(no)h(longer)d(a)n(v)-5 b(ailable.)40 b(Notice,)30 b(ho)n(w)n(ev)n(er,)0 3477 y(that)e Fq(B)243 3489 y Fm(0)308 3477 y Fr(curren)n(tly)d(con)n(tains) h Fq(L)1046 3442 y Fj(\000)p Fm(1)1046 3499 y(00)1154 3456 y Fr(^)1135 3477 y Fq(B)1198 3489 y Fm(0)1263 3477 y Fr(while)f Fq(B)1542 3489 y Fm(1)1607 3477 y Fr(con)n(tains)h(the)i (original)23 b(con)n(ten)n(ts)2724 3456 y(^)2704 3477 y Fq(B)2767 3489 y Fm(1)2804 3477 y Fr(.)37 b(Th)n(us,)27 b(the)h(up)r(date)1514 3633 y Fq(B)1577 3645 y Fm(1)1637 3633 y Fr(:=)23 b Fq(L)1805 3598 y Fj(\000)p Fm(1)1805 3655 y(11)1893 3633 y Fr(\()p Fq(B)1988 3645 y Fm(1)2044 3633 y Fp(\000)18 b Fq(L)2184 3645 y Fm(10)2254 3633 y Fq(B)2317 3645 y Fm(0)2354 3633 y Fr(\))0 3789 y(will)25 b(lea)n(v)n(e)g(the)j(submatrices)d(of)i Fq(B)32 b Fr(in)27 b(the)h(desired)e(state.)0 4000 y Fk(Algorithm)0 4154 y Fr(An)31 b(annotated)f(algorithm,)e(in)i(whic)n(h)g(the)h(states)f (of)h(the)g(v)-5 b(ariables)28 b(are)h(presen)n(ted)i(as)f(w)n(ell)e (as)i(the)i(op)r(erations)c(to)j(b)r(e)0 4253 y(executed,)37 b(is)d(presen)n(ted)h(in)f(Fig.)h(4.)59 b(By)35 b(remo)n(ving)d(all)g (annotations)i(and)h(remo)n(ving)c(an)n(y)k(op)r(eration)e(in)n(v)n (olving)e(the)0 4353 y(\014cticious)366 4332 y(^)346 4353 y Fq(B)5 b Fr(,)27 b(the)h(\014nal)f(algorithm)c(is)k(giv)n(en)e (Fig.)i(5.)0 4581 y Fe(2.2)112 b(V)-9 b(arian)m(t)38 b(2)0 4734 y Fr(Next,)26 b(let)f(us)g(deriv)n(e)e(an)i(algorithm,)d(V) -7 b(arian)n(t)23 b(2,)i(b)n(y)g(pic)n(king)e(the)j(lo)r(op-in)n(v)-5 b(arian)n(t)20 b(to)25 b(equal)f(that)h(giv)n(en)f(as)g(In)n(v)-5 b(arian)n(t)23 b([2])0 4834 y(in)k(Fig.)f(1:)1427 4886 y Fn(\022)1532 4943 y Fq(B)1595 4955 y Fl(T)p 1488 4976 204 4 v 1488 4993 V 1530 5062 a Fq(B)1593 5074 y Fl(B)1691 4886 y Fn(\023)1775 5003 y Fr(=)1863 4861 y Fn( )1970 4943 y Fq(L)2027 4955 y Fl(T)9 b(L)2125 4907 y Fj(\000)p Fm(1)2233 4922 y Fr(^)2214 4943 y Fq(B)2277 4955 y Fl(T)p 1929 4976 442 4 v 1929 4993 V 2109 5050 a Fr(^)2090 5071 y Fq(B)2153 5083 y Fl(B)2370 4861 y Fn(!)2450 5003 y Fq(:)1929 5356 y Fr(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 349 342 3203 4 v 347 429 4 87 v 399 405 a Fg(Step)p 584 429 V 601 429 V 117 w(Annotated)26 b(Algorithm:)64 b Ff(B)23 b Fg(:=)c Ff(L)1619 382 y Fa(\000)p Fh(1)1718 388 y Fg(^)1702 405 y Ff(B)p 3549 429 V 349 432 3203 4 v 349 449 V 347 548 4 100 v 432 515 a Fg(1a)p 584 548 V 601 548 V 652 452 a Fn(\010)701 515 y Ff(B)k Fg(=)868 498 y(^)851 515 y Ff(B)908 452 y Fn(\011)p 3549 548 V 349 551 3203 4 v 347 837 4 286 v 450 668 a Fg(4)p 584 837 V 601 837 V 167 w Fc(P)n(artition)47 b Ff(L)20 b Fb(!)1178 555 y Fn(\022)1282 617 y Ff(L)1330 628 y Fd(T)8 b(L)p 1463 641 4 79 v 1480 641 V 1579 617 a Fg(0)p 1239 644 473 4 v 1239 661 V 1280 716 a Ff(L)1328 727 y Fd(B)r(L)p 1463 740 4 79 v 1480 740 V 1523 716 a Ff(L)1571 727 y Fd(B)r(R)1711 555 y Fn(\023)1772 668 y Fg(,)23 b Ff(B)h Fb(!)1982 555 y Fn(\022)2086 617 y Ff(B)2139 628 y Fd(T)p 2043 644 189 4 v 2043 661 V 2084 716 a Ff(B)2137 727 y Fd(B)2232 555 y Fn(\023)2316 668 y Fg(and)2470 651 y(^)2454 668 y Ff(B)f Fb(!)2620 555 y Fn(\022)2741 601 y Fg(^)2725 617 y Ff(B)2778 628 y Fd(T)p 2681 644 V 2681 661 V 2739 707 a Fg(^)2723 724 y Ff(B)2776 735 y Fd(B)2870 555 y Fn(\023)774 814 y Fc(where)51 b Ff(L)1085 825 y Fd(T)8 b(L)1199 814 y Fg(is)24 b(0)16 b Fb(\002)f Fg(0,)24 b(and)g Ff(B)1660 825 y Fd(T)1732 814 y Fg(and)1886 797 y(^)1870 814 y Ff(B)1923 825 y Fd(T)1995 814 y Fg(ha)n(v)n(e)h(0)e (ro)n(ws)p 3549 837 4 286 v 349 841 3203 4 v 347 1040 4 200 v 450 957 a(2)p 584 1040 V 601 1040 V 652 844 a Fn(\032\022)819 907 y Ff(B)872 918 y Fd(T)p 776 934 189 4 v 776 950 V 817 1005 a Ff(B)870 1016 y Fd(B)965 844 y Fn(\023)1045 957 y Fg(=)1120 844 y Fn(\022)1222 907 y Ff(L)1270 918 y Fd(T)8 b(L)1361 882 y Fa(\000)p Fh(1)1460 890 y Fg(^)1444 907 y Ff(B)1497 918 y Fd(T)p 1181 934 407 4 v 1181 950 V 1348 996 a Fg(^)1331 1013 y Ff(B)1384 1024 y Fd(B)1587 844 y Fn(\023\033)p 3549 1040 4 200 v 349 1043 3203 4 v 347 1122 4 79 v 450 1098 a Fg(3)p 584 1122 V 601 1122 V 167 w Fc(while)49 b Ff(m)p Fg(\()p Ff(L)1023 1109 y Fd(T)8 b(L)1114 1098 y Fg(\))20 b Fb(6)p Fg(=)g Ff(m)p Fg(\()p Ff(L)p Fg(\))48 b Fc(do)p 3549 1122 V 349 1125 3203 4 v 347 1325 4 200 v 422 1242 a Fg(2,3)p 584 1325 V 601 1325 V 813 1129 a Fn(\032\022\022)1041 1191 y Ff(B)1094 1202 y Fd(T)p 997 1218 189 4 v 997 1235 V 1039 1290 a Ff(B)1092 1301 y Fd(B)1186 1129 y Fn(\023)1267 1242 y Fg(=)1341 1129 y Fn(\022)1444 1191 y Ff(L)1492 1202 y Fd(T)8 b(L)1583 1167 y Fa(\000)p Fh(1)1682 1175 y Fg(^)1665 1191 y Ff(B)1718 1202 y Fd(T)p 1403 1218 407 4 v 1403 1235 V 1569 1281 a Fg(^)1553 1298 y Ff(B)1606 1309 y Fd(B)1809 1129 y Fn(\023\023)1947 1242 y Fb(^)15 b Fg(\()q Ff(m)p Fg(\()p Ff(L)2174 1253 y Fd(T)8 b(L)2266 1242 y Fg(\))20 b Fb(6)p Fg(=)f Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2580 1129 y Fn(\033)p 3549 1325 4 200 v 349 1328 3203 4 v 347 2249 4 921 v 432 1383 a Fg(5a)p 584 2249 V 601 2249 V 311 w Fc(Determine)28 b(blo)r(c)n(k)g(size)g Ff(b)813 1462 y Fc(Repartition)934 1606 y Fn(\022)1039 1663 y Fq(L)1096 1675 y Fl(T)9 b(L)p 1236 1692 4 100 v 1253 1692 V 1357 1663 a Fr(0)p 995 1696 506 4 v 995 1712 V 1037 1782 a Fq(L)1094 1794 y Fl(B)s(L)p 1236 1812 4 100 v 1253 1812 V 1296 1782 a Fq(L)1353 1794 y Fl(B)s(R)1501 1606 y Fn(\023)1585 1723 y Fp(!)1691 1556 y Fn(0)1691 1706 y(@)1806 1611 y Fq(L)1863 1623 y Fm(00)p 1972 1641 V 1989 1641 V 2075 1611 a Fr(0)p 2199 1641 V 168 w(0)p 1764 1644 647 4 v 1764 1661 V 1806 1731 a Fq(L)1863 1743 y Fm(10)p 1972 1761 4 100 v 1989 1761 V 2032 1731 a Fq(L)2089 1743 y Fm(11)p 2199 1761 V 2285 1731 a Fr(0)p 1764 1764 647 4 v 1806 1834 a Fq(L)1863 1846 y Fm(20)p 1972 1864 4 100 v 1989 1864 V 2048 1834 a Fq(l)2073 1846 y Fm(21)p 2199 1864 V 2242 1834 a Fq(L)2299 1846 y Fm(22)2410 1556 y Fn(1)2410 1706 y(A)2483 1723 y Fr(,)2534 1606 y Fn(\022)2639 1663 y Fq(B)2702 1675 y Fl(T)p 2595 1696 204 4 v 2595 1712 V 2636 1782 a Fq(B)2699 1794 y Fl(B)2798 1606 y Fn(\023)2882 1723 y Fp(!)2988 1556 y Fn(0)2988 1706 y(@)3102 1611 y Fq(B)3165 1623 y Fm(0)p 3061 1644 184 4 v 3061 1661 V 3102 1731 a Fq(B)3165 1743 y Fm(1)p 3061 1764 V 3102 1834 a Fq(B)3165 1846 y Fm(2)3244 1556 y Fn(1)3244 1706 y(A)3317 1723 y Fr(,)27 b(and)934 1917 y Fn(\022)1055 1963 y Fg(^)1039 1979 y Ff(B)1092 1990 y Fd(T)p 995 2006 189 4 v 995 2023 V 1053 2069 a Fg(^)1037 2086 y Ff(B)1090 2097 y Fd(B)1184 1917 y Fn(\023)1265 2030 y Fb(!)1355 1867 y Fn(0)1355 2016 y(@)1486 1918 y Fg(^)1469 1934 y Ff(B)1522 1943 y Fh(0)p 1428 1961 171 4 v 1428 1978 V 1486 2024 a Fg(^)1469 2041 y Ff(B)1522 2050 y Fh(1)p 1428 2068 V 1486 2114 a Fg(^)1469 2131 y Ff(B)1522 2140 y Fh(2)1599 1867 y Fn(1)1599 2016 y(A)1056 2225 y Fc(where)74 b Ff(B)1395 2234 y Fh(1)1453 2225 y Fg(and)1607 2209 y(^)1591 2225 y Ff(B)1644 2234 y Fh(1)1702 2225 y Fg(ha)n(v)n(e)25 b Ff(b)e Fg(ro)n(ws)h(and)g Ff(L)2267 2234 y Fh(11)2355 2225 y Fg(is)g Ff(b)16 b Fb(\002)g Ff(b)p 3549 2249 4 921 v 349 2252 3203 4 v 347 2551 4 299 v 450 2418 a Fg(6)p 584 2551 V 601 2551 V 813 2252 a Fn(8)813 2327 y(<)813 2476 y(:)887 2281 y( )994 2327 y Ff(B)1047 2336 y Fh(0)p 952 2354 171 4 v 952 2371 V 994 2426 a Ff(B)1047 2435 y Fh(1)p 952 2453 V 994 2508 a Ff(B)1047 2517 y Fh(2)1123 2281 y Fn(!)1208 2418 y Fg(=)1283 2256 y Fn(0)1283 2405 y(@)1397 2323 y Ff(L)1445 2293 y Fa(\000)p Fh(1)1445 2346 y(00)1544 2306 y Fg(^)1528 2323 y Ff(B)1581 2332 y Fh(0)p 1356 2350 302 4 v 1356 2367 V 1479 2413 a Fg(^)1462 2430 y Ff(B)1515 2439 y Fh(1)p 1356 2457 V 1479 2503 a Fg(^)1462 2520 y Ff(B)1515 2529 y Fh(2)1657 2256 y Fn(1)1657 2405 y(A)1730 2252 y(9)1730 2327 y(=)1730 2476 y(;)p 3549 2551 4 299 v 349 2554 3203 4 v 347 2721 4 167 v 450 2654 a Fg(8)p 584 2721 V 601 2721 V 854 2612 a Ff(B)907 2621 y Fh(1)962 2612 y Fg(:=)j Ff(B)1109 2621 y Fh(1)1159 2612 y Fb(\000)d Ff(L)1278 2589 y Fd(T)1278 2634 y Fh(10)1343 2612 y Ff(B)1396 2621 y Fh(0)854 2698 y Ff(B)907 2707 y Fh(1)962 2698 y Fg(:=)j Ff(L)1104 2668 y Fa(\000)p Fh(1)1104 2720 y(11)1186 2698 y Ff(B)1239 2707 y Fh(1)p 3549 2721 V 349 2725 3203 4 v 347 3467 4 743 v 430 2780 a Fg(5b)p 584 3467 V 601 3467 V 309 w Fc(Con)n(tin)n(ue)27 b(with)934 2911 y Fn(\022)1039 2967 y Fq(L)1096 2979 y Fl(T)9 b(L)p 1236 2997 4 100 v 1253 2997 V 1357 2967 a Fr(0)p 995 3001 506 4 v 995 3017 V 1037 3087 a Fq(L)1094 3099 y Fl(B)s(L)p 1236 3117 4 100 v 1253 3117 V 1296 3087 a Fq(L)1353 3099 y Fl(B)s(R)1501 2911 y Fn(\023)1585 3028 y Fp( )1691 2861 y Fn(0)1691 3011 y(@)1806 2916 y Fq(L)1863 2928 y Fm(00)p 1972 2946 V 2058 2916 a Fr(0)p 2182 2946 V 2199 2946 V 185 w(0)p 1764 2949 647 4 v 1806 3019 a Fq(L)1863 3031 y Fm(10)p 1972 3049 4 100 v 2016 3019 a Fq(L)2073 3031 y Fm(11)p 2182 3049 V 2199 3049 V 2285 3019 a Fr(0)p 1764 3052 647 4 v 1764 3069 V 1806 3138 a Fq(L)1863 3150 y Fm(20)p 1972 3168 4 100 v 2016 3138 a Fq(L)2073 3150 y Fm(21)p 2182 3168 V 2199 3168 V 2242 3138 a Fq(L)2299 3150 y Fm(22)2410 2861 y Fn(1)2410 3011 y(A)2483 3028 y Fr(,)2534 2911 y Fn(\022)2639 2967 y Fq(B)2702 2979 y Fl(T)p 2595 3001 204 4 v 2595 3017 V 2636 3087 a Fq(B)2699 3099 y Fl(B)2798 2911 y Fn(\023)2882 3028 y Fp( )2988 2861 y Fn(0)2988 3011 y(@)3102 2916 y Fq(B)3165 2928 y Fm(0)p 3061 2949 184 4 v 3102 3019 a Fq(B)3165 3031 y Fm(1)p 3061 3052 V 3061 3069 V 3102 3138 a Fq(B)3165 3150 y Fm(2)3244 2861 y Fn(1)3244 3011 y(A)3317 3028 y Fr(,)27 b(and)934 3221 y Fn(\022)1055 3267 y Fg(^)1039 3284 y Ff(B)1092 3295 y Fd(T)p 995 3311 189 4 v 995 3328 V 1053 3374 a Fg(^)1037 3391 y Ff(B)1090 3402 y Fd(B)1184 3221 y Fn(\023)1265 3334 y Fb( )1355 3172 y Fn(0)1355 3321 y(@)1486 3222 y Fg(^)1469 3239 y Ff(B)1522 3248 y Fh(0)p 1428 3266 171 4 v 1486 3312 a Fg(^)1469 3329 y Ff(B)1522 3338 y Fh(1)p 1428 3356 V 1428 3373 V 1486 3419 a Fg(^)1469 3436 y Ff(B)1522 3445 y Fh(2)1599 3172 y Fn(1)1599 3321 y(A)p 3549 3467 4 743 v 349 3470 3203 4 v 347 3769 4 299 v 450 3637 a Fg(7)p 584 3769 V 601 3769 V 813 3470 a Fn(8)813 3545 y(<)813 3695 y(:)887 3499 y( )994 3545 y Ff(B)1047 3554 y Fh(0)p 952 3572 171 4 v 994 3627 a Ff(B)1047 3636 y Fh(1)p 952 3654 V 952 3671 V 994 3726 a Ff(B)1047 3735 y Fh(2)1123 3499 y Fn(!)1208 3637 y Fg(=)1283 3474 y Fn(0)1283 3623 y(@)1661 3541 y Ff(L)1709 3512 y Fa(\000)p Fh(1)1709 3564 y(00)1808 3525 y Fg(^)1791 3541 y Ff(B)1844 3550 y Fh(0)p 1356 3568 829 4 v 1397 3631 a Ff(L)1445 3601 y Fa(\000)p Fh(1)1445 3654 y(11)1528 3631 y Fg(\()1572 3614 y(^)1555 3631 y Ff(B)1608 3640 y Fh(1)1659 3631 y Fb(\000)15 b Ff(L)1777 3640 y Fh(10)1842 3631 y Fg(\()p Ff(L)1917 3601 y Fa(\000)p Fh(1)1917 3654 y(00)2017 3614 y Fg(^)2000 3631 y Ff(B)2053 3640 y Fh(0)2088 3631 y Fg(\)\))p 1356 3658 V 1356 3675 V 1743 3721 a(^)1726 3738 y Ff(B)1779 3747 y Fh(2)2184 3474 y Fn(1)2184 3623 y(A)2257 3470 y(9)2257 3545 y(=)2257 3695 y(;)p 3549 3769 4 299 v 349 3773 3203 4 v 347 3972 4 200 v 450 3889 a Fg(2)p 584 3972 V 601 3972 V 813 3776 a Fn(\032\022)980 3839 y Ff(B)1033 3850 y Fd(T)p 936 3866 189 4 v 936 3882 V 978 3937 a Ff(B)1031 3948 y Fd(B)1125 3776 y Fn(\023)1206 3889 y Fg(=)1280 3776 y Fn(\022)1383 3839 y Ff(L)1431 3850 y Fd(T)8 b(L)1522 3814 y Fa(\000)p Fh(1)1621 3822 y Fg(^)1604 3839 y Ff(B)1657 3850 y Fd(T)p 1341 3866 407 4 v 1341 3882 V 1508 3928 a Fg(^)1492 3945 y Ff(B)1545 3956 y Fd(B)1748 3776 y Fn(\023\033)p 3549 3972 4 200 v 349 3975 3203 4 v 347 4054 4 79 v 584 4054 V 601 4054 V 652 4030 a Fc(enddo)p 3549 4054 V 349 4057 3203 4 v 347 4257 4 200 v 422 4174 a Fg(2,3)p 584 4257 V 601 4257 V 652 4061 a Fn(\032\022)q(\022)880 4123 y Ff(B)933 4134 y Fd(T)p 837 4150 189 4 v 837 4167 V 878 4222 a Ff(B)931 4233 y Fd(B)1026 4061 y Fn(\023)1106 4174 y Fg(=)1181 4061 y Fn(\022)1283 4123 y Ff(L)1331 4134 y Fd(T)g(L)1422 4099 y Fa(\000)p Fh(1)1521 4107 y Fg(^)1505 4123 y Ff(B)1558 4134 y Fd(T)p 1242 4150 407 4 v 1242 4167 V 1409 4213 a Fg(^)1392 4230 y Ff(B)1445 4241 y Fd(B)1648 4061 y Fn(\023\023)1786 4174 y Fb(^)16 b(:)c Fg(\()p Ff(m)p Fg(\()p Ff(L)2072 4185 y Fd(T)c(L)2164 4174 y Fg(\))20 b Fb(6)p Fg(=)f Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2478 4061 y Fn(\033)p 3549 4257 4 200 v 349 4260 3203 4 v 347 4360 4 100 v 430 4326 a Fg(1b)p 584 4360 V 601 4360 V 652 4263 a Fn(\010)701 4326 y Ff(B)k Fg(:=)c Ff(L)919 4303 y Fa(\000)p Fh(1)1018 4310 y Fg(^)1001 4326 y Ff(B)1058 4263 y Fn(\011)p 3549 4360 V 349 4363 3203 4 v 469 4605 a Fr(Figure)26 b(4:)36 b(W)-7 b(orksheet)27 b(for)g(a)g(deriving)e(blo) r(c)n(k)n(ed)h(algorithm)d(for)k Fq(B)g Fr(:=)c Fq(L)2821 4575 y Fj(\000)p Fm(1)2910 4605 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(1\).)1908 5356 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 532 -12 a Fk(P)m(artition)55 b Fq(L)23 b Fp(!)1149 -130 y Fn(\022)1254 -73 y Fq(L)1311 -61 y Fl(T)9 b(L)p 1451 -43 4 100 v 1467 -43 V 1572 -73 a Fr(0)p 1210 -40 506 4 v 1210 -23 V 1252 46 a Fq(L)1309 58 y Fl(B)s(L)p 1451 76 4 100 v 1467 76 V 1510 46 a Fq(L)1567 58 y Fl(B)s(R)1716 -130 y Fn(\023)1805 -12 y Fr(and)27 b Fq(B)g Fp(!)2162 -130 y Fn(\022)2267 -73 y Fq(B)2330 -61 y Fl(T)p 2224 -40 204 4 v 2224 -23 V 2265 46 a Fq(B)2328 58 y Fl(B)2427 -130 y Fn(\023)675 146 y Fk(where)59 b Fq(L)1040 158 y Fl(T)9 b(L)1165 146 y Fr(is)26 b(0)18 b Fp(\002)g Fr(0)28 b(and)f Fq(B)1685 158 y Fl(T)1765 146 y Fr(has)g(0)g(ro)n(ws)532 387 y Fk(while)54 b Fq(m)p Fr(\()p Fq(L)968 399 y Fl(T)9 b(L)1065 387 y Fr(\))24 b Fp(6)p Fr(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\))56 b Fk(do)705 487 y(Determine)30 b(blo)s(c)m(k)h(size)h Fq(b)705 586 y Fk(Repartition)848 734 y Fn(\022)953 790 y Fq(L)1010 802 y Fl(T)9 b(L)p 1149 820 4 100 v 1166 820 V 1270 790 a Fr(0)p 909 823 506 4 v 909 840 V 950 910 a Fq(L)1007 922 y Fl(B)s(L)p 1149 940 4 100 v 1166 940 V 1209 910 a Fq(L)1266 922 y Fl(B)s(R)1415 734 y Fn(\023)1499 851 y Fp(!)1605 684 y Fn(0)1605 833 y(@)1719 739 y Fq(L)1776 751 y Fm(00)p 1886 769 V 1902 769 V 1988 739 a Fr(0)p 2112 769 V 168 w(0)p 1677 772 647 4 v 1677 789 V 1719 858 a Fq(L)1776 870 y Fm(10)p 1886 888 4 100 v 1902 888 V 1945 858 a Fq(L)2002 870 y Fm(11)p 2112 888 V 2198 858 a Fr(0)p 1677 891 647 4 v 1719 961 a Fq(L)1776 973 y Fm(20)p 1886 991 4 100 v 1902 991 V 1961 961 a Fq(l)1986 973 y Fm(21)p 2112 991 V 2155 961 a Fq(L)2212 973 y Fm(22)2324 684 y Fn(1)2324 833 y(A)2424 851 y Fr(and)2585 734 y Fn(\022)2690 790 y Fq(B)2753 802 y Fl(T)p 2646 823 204 4 v 2646 840 V 2688 910 a Fq(B)2751 922 y Fl(B)2850 734 y Fn(\023)2934 851 y Fp(!)3040 684 y Fn(0)3040 833 y(@)3154 739 y Fq(B)3217 751 y Fm(0)p 3113 772 184 4 v 3113 789 V 3154 858 a Fq(B)3217 870 y Fm(1)p 3113 891 V 3154 961 a Fq(B)3217 973 y Fm(2)3296 684 y Fn(1)3296 833 y(A)991 1061 y Fk(where)87 b Fq(B)1390 1073 y Fm(1)1454 1061 y Fr(has)28 b Fq(b)f Fr(ro)n(ws)f(and)h Fq(L)2076 1073 y Fm(11)2174 1061 y Fr(is)f Fq(b)18 b Fp(\002)g Fq(b)p 705 1162 2341 4 v 705 1154 V 746 1260 a(B)809 1272 y Fm(1)869 1260 y Fr(:=)23 b Fq(B)1043 1272 y Fm(1)1099 1260 y Fp(\000)18 b Fq(L)1239 1230 y Fl(T)1239 1281 y Fm(10)1309 1260 y Fq(B)1372 1272 y Fm(0)746 1363 y Fq(B)809 1375 y Fm(1)869 1363 y Fr(:=)23 b Fq(L)1037 1328 y Fj(\000)p Fm(1)1037 1385 y(11)1126 1363 y Fq(B)1189 1375 y Fm(1)p 705 1465 V 705 1457 V 705 1562 a Fk(Con)m(tin)m(ue)31 b(with)848 1693 y Fn(\022)953 1750 y Fq(L)1010 1762 y Fl(T)9 b(L)p 1149 1780 4 100 v 1166 1780 V 1270 1750 a Fr(0)p 909 1783 506 4 v 909 1800 V 950 1869 a Fq(L)1007 1881 y Fl(B)s(L)p 1149 1899 4 100 v 1166 1899 V 1209 1869 a Fq(L)1266 1881 y Fl(B)s(R)1415 1693 y Fn(\023)1499 1811 y Fp( )1605 1644 y Fn(0)1605 1793 y(@)1719 1698 y Fq(L)1776 1710 y Fm(00)p 1886 1728 V 1971 1698 a Fr(0)p 2095 1728 V 2112 1728 V 185 w(0)p 1677 1732 647 4 v 1719 1801 a Fq(L)1776 1813 y Fm(10)p 1886 1831 4 100 v 1929 1801 a Fq(L)1986 1813 y Fm(11)p 2095 1831 V 2112 1831 V 2198 1801 a Fr(0)p 1677 1835 647 4 v 1677 1851 V 1719 1921 a Fq(L)1776 1933 y Fm(20)p 1886 1951 4 100 v 1929 1921 a Fq(L)1986 1933 y Fm(21)p 2095 1951 V 2112 1951 V 2155 1921 a Fq(L)2212 1933 y Fm(22)2324 1644 y Fn(1)2324 1793 y(A)2424 1811 y Fr(and)2585 1693 y Fn(\022)2690 1750 y Fq(B)2753 1762 y Fl(T)p 2646 1783 204 4 v 2646 1800 V 2688 1869 a Fq(B)2751 1881 y Fl(B)2850 1693 y Fn(\023)2934 1811 y Fp( )3040 1644 y Fn(0)3040 1793 y(@)3154 1698 y Fq(B)3217 1710 y Fm(0)p 3113 1732 184 4 v 3154 1801 a Fq(B)3217 1813 y Fm(1)p 3113 1835 V 3113 1851 V 3154 1921 a Fq(B)3217 1933 y Fm(2)3296 1644 y Fn(1)3296 1793 y(A)532 2021 y Fk(enddo)925 2292 y Fr(Figure)26 b(5:)37 b(Blo)r(c)n(k)n(ed)25 b(algorithm)f(for)j Fq(B)g Fr(:=)c Fq(L)2365 2262 y Fj(\000)p Fm(1)2453 2292 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(1\).)0 2543 y Fk(Step)32 b(3:)41 b(Determination)30 b(of)i(the)g(Lo)s(op-Guard)0 2696 y Fr(Again,)d(w)n(e)h(need)g(to)h(determine)d(the)i(condition)e (under)i(whic)n(h)f(the)i(\014nal)e(result)g(has)g(not)i(y)n(et)e(b)r (een)i(computed,)f(whic)n(h)0 2796 y(means)c(that)i(the)g(lo)r(op)e (cannot)h(y)n(et)h(b)r(e)g(terminated,)d(lo)r(op-guard)g Fq(G)p Fr(.)125 2896 y(Notice)h(that)i(the)g(lo)r(op)e(is)h(exited)f (when)i(this)f(condition)e(b)r(ecomes)i Fo(false)p Fr(.)38 b(Th)n(us,)27 b Fq(G)h Fr(needs)g(to)f(b)r(e)h(c)n(hoses)e(so)h(that) 1581 3054 y(\()p Fq(P)1666 3066 y Fm(pre)1782 3054 y Fp(^)18 b(:)p Fq(G)p Fr(\))24 b Fp(\))f Fq(P)2190 3066 y Fm(p)r(ost)0 3213 y Fr(or,)k(for)g(our)g(sp)r(eci\014c)f(case,)645 3300 y Fn( )777 3325 y(\022)882 3381 y Fq(B)945 3393 y Fl(T)p 838 3415 204 4 v 838 3431 V 879 3501 a Fq(B)942 3513 y Fl(B)1041 3325 y Fn(\023)1125 3442 y Fr(=)1213 3300 y Fn( )1543 3381 y Fq(L)1600 3393 y Fl(T)9 b(L)1697 3345 y Fj(\000)p Fm(1)1806 3360 y Fr(^)1786 3381 y Fq(B)1849 3393 y Fl(T)p 1279 3415 888 4 v 1279 3431 V 1340 3489 a Fr(^)1320 3510 y Fq(B)1383 3522 y Fl(B)1459 3510 y Fp(\000)18 b Fq(L)1599 3522 y Fl(B)s(L)1701 3510 y Fr(\()p Fq(L)1790 3522 y Fl(T)9 b(L)1888 3474 y Fj(\000)p Fm(1)1996 3489 y Fr(^)1977 3510 y Fq(B)2040 3522 y Fl(T)2092 3510 y Fr(\))2166 3300 y Fn(!)o(!)2316 3442 y Fp(^)18 b(:)p Fq(G)2509 3300 y Fn(!)2599 3442 y Fp(\))2705 3350 y Fn(\020)2754 3442 y Fq(B)28 b Fr(:=)22 b Fq(L)3012 3408 y Fj(\000)p Fm(1)3121 3421 y Fr(^)3101 3442 y Fq(B)3168 3350 y Fn(\021)3232 3442 y Fq(:)0 3692 y Fr(Recalling)e(that)j(the)h(partitionings)19 b(of)k Fq(L)p Fr(,)h Fq(B)t Fr(,)g(and)1641 3671 y(^)1621 3692 y Fq(B)k Fr(need)23 b(to)g(b)r(e)h(suc)n(h)f(that)g(the)h(v)-5 b(arious)21 b(matrix)g(m)n(ultiplications)c(mak)n(e)0 3791 y(sense,)34 b(w)n(e)g(\014nd)f(that)h(when)g Fq(L)1009 3803 y Fl(T)9 b(L)1140 3791 y Fr(equals)31 b(all)g(of)j Fq(L)p Fr(,)g(then)g Fq(B)1991 3803 y Fl(T)2077 3791 y Fr(m)n(ust)f(equal)f(all)f(of)i Fq(B)38 b Fr(and)3023 3770 y(^)3003 3791 y Fq(B)3066 3803 y Fl(T)3152 3791 y Fr(m)n(ust)32 b(equal)g(all)f(of)3830 3770 y(^)3810 3791 y Fq(B)t Fr(.)0 3891 y(Letting)c Fq(m)p Fr(\()p Fq(X)7 b Fr(\))27 b(denote)h(the)g(ro)n(w)e(dimension)f(of)i(a)g (matrix,)e(and)j(noticing)d(that)j Fq(L)2668 3903 y Fl(T)9 b(L)2793 3891 y Fr(is)26 b(square,)h(w)n(e)g(can)g(argue)f(that)845 4075 y(\()p Fq(m)p Fr(\()p Fq(L)1039 4087 y Fl(T)9 b(L)1137 4075 y Fr(\))23 b(=)g Fq(m)p Fr(\()p Fq(L)p Fr(\)\))g Fp(\))1635 3983 y Fn(\020)1685 4075 y Fr(\()p Fq(L)1774 4087 y Fl(T)9 b(L)1895 4075 y Fr(=)22 b Fq(L)p Fr(\))c Fp(^)h Fr(\()p Fq(B)2258 4087 y Fl(T)2334 4075 y Fr(=)j Fq(B)t Fr(\))d Fp(^)2613 3983 y Fn(\020)2682 4054 y Fr(^)2663 4075 y Fq(B)2726 4087 y Fl(T)2801 4075 y Fr(=)2908 4054 y(^)2889 4075 y Fq(B)2956 3983 y Fn(\021)o(\021)0 4254 y Fr(so)27 b(that)375 4325 y Fn( )507 4349 y(\022)612 4406 y Fq(B)675 4418 y Fl(T)p 568 4439 204 4 v 568 4456 V 609 4526 a Fq(B)672 4538 y Fl(B)771 4349 y Fn(\023)855 4467 y Fr(=)943 4325 y Fn( )1273 4406 y Fq(L)1330 4418 y Fl(T)9 b(L)1427 4370 y Fj(\000)p Fm(1)1536 4385 y Fr(^)1516 4406 y Fq(B)1579 4418 y Fl(T)p 1008 4439 888 4 v 1008 4456 V 1070 4513 a Fr(^)1050 4534 y Fq(B)1113 4546 y Fl(B)1188 4534 y Fp(\000)18 b Fq(L)1328 4546 y Fl(B)s(L)1431 4534 y Fr(\()p Fq(L)1520 4546 y Fl(T)9 b(L)1617 4498 y Fj(\000)p Fm(1)1726 4513 y Fr(^)1706 4534 y Fq(B)1769 4546 y Fl(T)1822 4534 y Fr(\))1896 4325 y Fn(!)o(!)2045 4467 y Fp(^)19 b Fr(\()p Fq(m)p Fr(\()p Fq(L)2313 4479 y Fl(T)9 b(L)2411 4467 y Fr(\))23 b(=)g Fq(m)p Fr(\()p Fq(L)p Fr(\)\))2780 4325 y Fn(!)2869 4467 y Fp(\))2975 4374 y Fn(\020)3025 4467 y Fq(B)k Fr(:=)c Fq(L)3283 4432 y Fj(\000)p Fm(1)3391 4446 y Fr(^)3371 4467 y Fq(B)3438 4374 y Fn(\021)3502 4467 y Fq(:)0 4716 y Fr(The)30 b(fact)f(that)h Fq(B)583 4728 y Fl(B)670 4716 y Fr(and)853 4695 y(^)833 4716 y Fq(B)896 4728 y Fl(B)983 4716 y Fr(con)n(tain)e(no)h(ro)n(ws)g (can)g(b)r(e)h(depicted)f(b)n(y)g(the)h(double)f(lines)e(reac)n(hing)g (the)j(b)r(ottom)f(of)h(the)0 4816 y(resp)r(ectiv)n(e)c(partitioned)f (matrices)g(in)i(the)h(lo)r(op-in)n(v)-5 b(arian)n(t:)853 4903 y Fn( )984 4928 y(\022)1089 4984 y Fq(B)1152 4996 y Fl(T)p 1045 5018 204 4 v 1045 5034 V 1087 5104 a Fq(B)1150 5116 y Fl(B)1248 4928 y Fn(\023)1333 5045 y Fr(=)1420 4903 y Fn( )1528 4984 y Fq(L)1585 4996 y Fl(T)9 b(L)1682 4948 y Fj(\000)p Fm(1)1791 4963 y Fr(^)1771 4984 y Fq(B)1834 4996 y Fl(T)p 1486 5018 442 4 v 1486 5034 V 1667 5092 a Fr(^)1647 5113 y Fq(B)1710 5125 y Fl(B)1928 4903 y Fn(!!)2078 5045 y Fp(^)19 b Fr(\()p Fq(m)p Fr(\()p Fq(L)2346 5057 y Fl(T)9 b(L)2443 5045 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))2813 4903 y Fn(!)1908 5356 y Fr(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 936 2 a Fp(\))1102 -139 y Fn( )1233 -115 y(\022)1336 -58 y Fq(B)1399 -46 y Fl(T)p 1294 -25 199 4 v 1294 -8 V 1493 -115 a Fn(\023)1577 2 y Fr(=)1665 -139 y Fn( )1772 -54 y Fq(L)1829 -89 y Fj(\000)p Fm(1)1829 -29 y Fl(T)9 b(L)1946 -75 y Fr(^)1926 -54 y Fq(B)1989 -42 y Fl(T)p 1730 -20 353 4 v 1730 -4 V 2083 -139 a Fn(!!)2233 2 y Fp(^)19 b Fr(\()p Fq(m)p Fr(\()p Fq(L)2501 14 y Fl(T)9 b(L)2598 2 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))2968 -139 y Fn(!)936 285 y Fp(\))1102 143 y Fn( )1168 168 y(\022)1270 224 y Fq(B)p 1229 257 151 4 v 1229 274 V 1379 168 a Fn(\023)1463 285 y Fr(=)1551 143 y Fn( )1658 229 y Fq(L)1715 198 y Fj(\000)p Fm(1)1823 208 y Fr(^)1804 229 y Fq(B)p 1616 262 296 4 v 1616 278 V 1912 143 a Fn(!!)936 517 y Fp(\))1102 425 y Fn(\020)1151 517 y Fq(B)28 b Fr(=)22 b Fq(L)1386 483 y Fj(\000)p Fm(1)1495 496 y Fr(^)1475 517 y Fq(B)1542 425 y Fn(\021)1605 517 y Fq(:)0 696 y Fr(Th)n(us,)27 b(the)h(lo)r(op-guard)d(can)i(b)r(e)h(c)n(hosen)f(as)g Fq(G)c Fr(:)g Fp(:)14 b Fr(\()q Fq(m)p Fr(\()p Fq(L)1825 708 y Fl(T)9 b(L)1922 696 y Fr(\))24 b(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\)\))28 b(or,)f(simpli\014ed,)d(as)1573 850 y Fq(G)f Fr(:)g Fq(m)p Fr(\()p Fq(L)1869 862 y Fl(T)9 b(L)1967 850 y Fr(\))23 b Fp(6)p Fr(=)g Fq(m)p Fr(\()p Fq(L)p Fr(\))p Fq(:)125 1004 y Fr(Notice)j(that)i(this)f(lo)r(op-guard) e(is)h(iden)n(tical)e(to)k(the)g(one)f(deriv)n(ed)f(from)g(In)n(v)-5 b(arian)n(t)26 b([1].)0 1215 y Fk(Step)32 b(4:)41 b(Initialization)0 1368 y Fr(Next,)28 b(w)n(e)e(m)n(ust)h(initialize)22 b(the)27 b(partitionings)c(of)k(the)h(matrices)c(so)j(that)g Fq(B)k Fr(con)n(tains)25 b(the)j(correct)d(con)n(ten)n(ts)i(b)r(efore)g (the)0 1468 y(lo)r(op)f(commenses.)34 b(Notice)27 b(that)h(the)g (partitionings)775 1692 y Fq(L)22 b Fp(!)960 1575 y Fn(\022)1065 1632 y Fq(L)1122 1644 y Fl(T)9 b(L)p 1262 1662 4 100 v 1279 1662 V 1383 1632 a Fr(0)p 1022 1665 506 4 v 1022 1682 V 1063 1751 a Fq(L)1120 1763 y Fl(B)s(L)p 1262 1781 4 100 v 1279 1781 V 1322 1751 a Fq(L)1379 1763 y Fl(B)s(R)1527 1575 y Fn(\023)1602 1692 y Fq(;)97 b(B)27 b Fp(!)1918 1575 y Fn(\022)2024 1632 y Fq(B)2087 1644 y Fl(T)p 1980 1665 204 4 v 1980 1682 V 2021 1751 a Fq(B)2084 1763 y Fl(B)2183 1575 y Fn(\023)2258 1692 y Fq(;)97 b Fr(and)2614 1671 y(^)2594 1692 y Fq(B)28 b Fp(!)2791 1550 y Fn( )2920 1611 y Fr(^)2900 1632 y Fq(B)2963 1644 y Fl(T)p 2856 1665 V 2856 1682 V 2918 1739 a Fr(^)2898 1760 y Fq(B)2961 1772 y Fl(B)3060 1550 y Fn(!)0 1937 y Fr(where)k Fq(L)302 1949 y Fl(T)9 b(L)431 1937 y Fr(is)32 b(0)21 b Fp(\002)g Fr(0,)33 b(and)f Fq(B)996 1949 y Fl(T)1081 1937 y Fr(and)1266 1916 y(^)1247 1937 y Fq(B)1310 1949 y Fl(T)1394 1937 y Fr(con)n(tain)f(0)h(ro)n(ws,)g(ha)n(v)n(e)f(the)h(prop)r(ert)n(y)f (that)i(they)f(lea)n(v)n(e)e(the)j(v)-5 b(ariables)29 b(in)i(a)0 2037 y(state)c(where)g(the)h(lo)r(op-in)n(v)-5 b(arian)n(t)1223 2144 y Fn(\022)1328 2201 y Fq(B)1391 2213 y Fl(T)p 1284 2234 V 1284 2251 V 1325 2320 a Fq(B)1388 2332 y Fl(B)1487 2144 y Fn(\023)1571 2261 y Fr(=)1659 2119 y Fn( )1989 2201 y Fq(L)2046 2213 y Fl(T)9 b(L)2143 2164 y Fj(\000)p Fm(1)2252 2180 y Fr(^)2232 2201 y Fq(B)2295 2213 y Fl(T)p 1724 2234 888 4 v 1724 2251 V 1786 2308 a Fr(^)1766 2329 y Fq(B)1829 2341 y Fl(B)1905 2329 y Fp(\000)18 b Fq(L)2045 2341 y Fl(B)s(L)2147 2329 y Fr(\()p Fq(L)2236 2341 y Fl(T)9 b(L)2333 2293 y Fj(\000)p Fm(1)2442 2308 y Fr(^)2423 2329 y Fq(B)2486 2341 y Fl(T)2538 2329 y Fr(\))2612 2119 y Fn(!)0 2506 y Fr(is)36 b Fo(true)p Fr(.)64 b(This)36 b(is)f(b)r(ecause)i Fq(B)997 2518 y Fl(T)1086 2506 y Fr(and)g Fq(L)1314 2471 y Fj(\000)p Fm(1)1314 2531 y Fl(T)9 b(L)1431 2485 y Fr(^)1411 2506 y Fq(B)1474 2518 y Fl(T)1563 2506 y Fr(con)n(tain)36 b(no)n(w)g(ro)n(ws)f(and)i Fq(L)2477 2518 y Fl(B)s(L)2579 2506 y Fq(L)2636 2471 y Fj(\000)p Fm(1)2636 2531 y Fl(T)9 b(L)2753 2485 y Fr(^)2733 2506 y Fq(B)2796 2518 y Fl(T)2887 2506 y Fr(=)38 b(0)f(since)e Fq(L)3338 2518 y Fl(B)s(L)3477 2506 y Fr(con)n(tains)g(no)0 2618 y(columns,)366 2597 y(^)346 2618 y Fq(B)409 2630 y Fl(T)489 2618 y Fr(con)n(tains)26 b(no)h(ro)n(ws,)f(and)i Fq(L)1364 2630 y Fl(T)9 b(L)1489 2618 y Fr(is)26 b(0)18 b Fp(\002)g Fr(0.)37 b(Therefore,)26 b(the)i(lo)r(op-in)n(v)-5 b(arian)n(t)23 b(reduces)k(to)1546 2725 y Fn(\022)p 1607 2815 204 4 v 1607 2831 V 1649 2901 a Fq(B)1712 2913 y Fl(B)1810 2725 y Fn(\023)1895 2842 y Fr(=)1982 2700 y Fn( )p 2048 2810 V 2048 2827 V 2109 2884 a Fr(^)2090 2905 y Fq(B)2153 2917 y Fl(B)2251 2700 y Fn(!)2331 2842 y Fq(:)0 3087 y Fr(whic)n(h)22 b(is)g(equiv)-5 b(alen)n(t)21 b(to)i Fq(B)k Fr(=)997 3066 y(^)977 3087 y Fq(B)t Fr(,)d(the)g(precondition,)d(giv)n(en)h(ho)n(w)g(the)i (matrices)c(w)n(ere)j(partitioned)d(in)j(the)g(initialization.)125 3187 y(Notice)j(that)i(the)g(initializatio)o(n)22 b(is)27 b(iden)n(tical)d(to)k(the)g(one)f(deriv)n(ed)f(from)g(In)n(v)-5 b(arian)n(t)25 b([1].)0 3398 y Fk(Step)32 b(5:)41 b(Mo)m(ving)32 b(through)g(the)f(op)s(erands)0 3551 y Fr(The)d(next)f(question)g(is)f (ho)n(w)h(to)h(mo)n(v)n(e)d(through)i(the)h(op)r(erands)e(so)h(that)h (w)n(e)g(progress)d(from)h(the)i(original)23 b(state)k(where)1300 3658 y Fn(\022)p 1361 3748 V 1361 3764 V 1402 3834 a Fq(B)1465 3846 y Fl(B)1564 3658 y Fn(\023)1648 3775 y Fr(=)1736 3633 y Fn( )p 1801 3743 734 4 v 1801 3760 V 1863 3818 a Fr(^)1843 3839 y Fq(B)1906 3851 y Fl(B)1981 3839 y Fp(\000)18 b Fq(L)2121 3851 y Fl(B)s(L)2224 3839 y Fq(L)2281 3803 y Fj(\000)p Fm(1)2281 3863 y Fl(T)9 b(L)2398 3818 y Fr(^)2378 3839 y Fq(B)2441 3851 y Fl(T)2535 3633 y Fn(!)0 3999 y Fr(to)27 b(the)h(\014nal)f(state)g(where)1474 4036 y Fn(\022)1577 4092 y Fq(B)1640 4104 y Fl(T)p 1535 4126 199 4 v 1535 4142 V 1733 4036 a Fn(\023)1817 4153 y Fr(=)1905 4011 y Fn( )2012 4097 y Fq(L)2069 4061 y Fj(\000)p Fm(1)2069 4121 y Fl(T)9 b(L)2186 4076 y Fr(^)2167 4097 y Fq(B)2230 4109 y Fl(T)p 1971 4130 353 4 v 1971 4147 V 2323 4011 a Fn(!)2403 4153 y Fq(:)0 4355 y Fr(In)28 b(considering)c(the)k(partitioned)d(matrices)1057 4462 y Fn(\022)1162 4519 y Fq(L)1219 4531 y Fl(T)9 b(L)p 1359 4549 4 100 v 1375 4549 V 1480 4519 a Fr(0)p 1118 4552 506 4 v 1118 4569 V 1160 4638 a Fq(L)1217 4650 y Fl(B)s(L)p 1359 4668 4 100 v 1375 4668 V 1418 4638 a Fq(L)1475 4650 y Fl(B)s(R)1624 4462 y Fn(\023)1699 4579 y Fq(;)1819 4462 y Fn(\022)1924 4519 y Fq(B)1987 4531 y Fl(T)p 1880 4552 204 4 v 1880 4569 V 1921 4638 a Fq(B)1984 4650 y Fl(B)2083 4462 y Fn(\023)2158 4579 y Fq(;)97 b Fr(and)2508 4438 y Fn( )2638 4498 y Fr(^)2618 4519 y Fq(B)2681 4531 y Fl(T)p 2574 4552 V 2574 4569 V 2635 4626 a Fr(^)2616 4647 y Fq(B)2679 4659 y Fl(B)2777 4438 y Fn(!)0 4808 y Fr(w)n(e)27 b(notice)g(that,)h(as)f(for)h(the)g(algorithms)23 b(deriv)n(ed)j(from)h(In)n(v)-5 b(arian)n(t)25 b([1],)j(w)n(e)f(need)h (to)g(gro)n(w)e(the)i(submatrices)d Fq(L)3613 4820 y Fl(T)9 b(L)3711 4808 y Fr(,)28 b Fq(B)3825 4820 y Fl(T)3877 4808 y Fr(,)0 4908 y(and)f Fq(B)224 4920 y Fl(B)282 4908 y Fr(,)g(while)f(shrinking)f(the)j(other)f(submatrices)e(in)h(these)i (partitionings.)125 5008 y(T)-7 b(o)28 b(do)h(so,)f(w)n(e)h(will)d (repartition)f(the)30 b(matrices,)c(exp)r(osing)h(submatrices)f(that)j (can)f(then)i(b)r(e)f(mo)n(v)n(ed)e(to)i(other)f(parts)0 5107 y(of)g(the)g(partitionings:)1908 5356 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 0 -60 a Fk(2.2.1)94 b(V)-8 b(arian)m(t)33 b(2:)42 b(un)m(blo)s(c)m(k)m(ed)32 b(algorithm)0 93 y Fr(First,)c(w)n(e)g(will)d(repartition)h(the)j(matrices)c(so)j(that)h (the)g(submatrices)c(that)k(are)e(mo)n(v)n(ed)g(b)r(et)n(w)n(een)h(the) h(v)-5 b(arious)26 b(parts)i(of)0 193 y(the)f(matrix)e(are)h (individual)d(elemen)n(ts,)i(ro)n(ws,)g(and/or)h(columns.)34 b(This)26 b(will)e(lead)i(to)h(so-called)c(un)n(blo)r(c)n(k)n(ed)i (algorithms.)143 362 y Fn(\022)248 418 y Fq(L)305 430 y Fl(T)9 b(L)p 445 448 4 100 v 461 448 V 566 418 a Fr(0)p 204 451 506 4 v 204 468 V 246 538 a Fq(L)303 550 y Fl(B)s(L)p 445 568 4 100 v 461 568 V 504 538 a Fq(L)561 550 y Fl(B)s(R)710 362 y Fn(\023)794 479 y Fp(!)900 312 y Fn(0)900 461 y(@)1014 367 y Fq(L)1071 379 y Fm(00)p 1181 397 V 1198 397 V 1279 367 a Fr(0)p 1399 397 V 164 w(0)p 973 400 639 4 v 973 416 V 1030 486 a Fq(l)1057 456 y Fl(T)1055 507 y Fm(10)p 1181 516 4 100 v 1198 516 V 1241 486 a Fq(\025)1289 498 y Fm(11)p 1399 516 V 1485 486 a Fr(0)p 973 519 639 4 v 1014 589 a Fq(L)1071 601 y Fm(20)p 1181 619 4 100 v 1198 619 V 1253 589 a Fq(l)1278 601 y Fm(21)p 1399 619 V 1443 589 a Fq(L)1500 601 y Fm(22)1611 312 y Fn(1)1611 461 y(A)1697 479 y Fq(;)1817 362 y Fn(\022)1922 418 y Fq(B)1985 430 y Fl(T)p 1878 451 204 4 v 1878 468 V 1920 538 a Fq(B)1983 550 y Fl(B)2082 362 y Fn(\023)2166 479 y Fp(!)2272 312 y Fn(0)2272 461 y(@)2386 367 y Fq(B)2449 379 y Fm(0)p 2344 400 184 4 v 2344 416 V 2392 486 a Fq(b)2428 456 y Fl(T)2428 507 y Fm(1)p 2344 519 V 2386 589 a Fq(B)2449 601 y Fm(2)2528 312 y Fn(1)2528 461 y(A)2614 479 y Fq(;)97 b Fr(and)2965 337 y Fn( )3094 397 y Fr(^)3074 418 y Fq(B)3137 430 y Fl(T)p 3030 451 204 4 v 3030 468 V 3092 526 a Fr(^)3072 547 y Fq(B)3135 559 y Fl(B)3234 337 y Fn(!)3322 479 y Fp(!)3428 287 y Fn(0)3428 433 y(B)3428 486 y(@)3562 341 y Fr(^)3543 362 y Fq(B)3606 374 y Fm(0)p 3501 395 184 4 v 3501 412 V 3546 469 a Fr(^)3549 491 y Fq(b)3585 461 y Fl(T)3585 512 y Fm(1)p 3501 524 V 3562 582 a Fr(^)3543 603 y Fq(B)3606 615 y Fm(2)3684 287 y Fn(1)3684 433 y(C)3684 486 y(A)0 760 y Fr(to)n(w)n(ards)26 b(the)i(top)f(of)h(the)g(lo)r(op)e (and)143 924 y Fn(\022)248 980 y Fq(L)305 992 y Fl(T)9 b(L)p 445 1010 4 100 v 461 1010 V 566 980 a Fr(0)p 204 1014 506 4 v 204 1030 V 246 1100 a Fq(L)303 1112 y Fl(B)s(L)p 445 1130 4 100 v 461 1130 V 504 1100 a Fq(L)561 1112 y Fl(B)s(R)710 924 y Fn(\023)794 1041 y Fp( )900 874 y Fn(0)900 1023 y(@)1014 929 y Fq(L)1071 941 y Fm(00)p 1181 959 V 1263 929 a Fr(0)p 1383 959 V 1400 959 V 180 w(0)p 973 962 639 4 v 1030 1032 a Fq(l)1057 1002 y Fl(T)1055 1052 y Fm(10)p 1181 1062 4 100 v 1224 1032 a Fq(\025)1272 1044 y Fm(11)p 1383 1062 V 1399 1062 V 1485 1032 a Fr(0)p 973 1065 639 4 v 973 1082 V 1014 1151 a Fq(L)1071 1163 y Fm(20)p 1181 1181 4 100 v 1236 1151 a Fq(l)1261 1163 y Fm(21)p 1383 1181 V 1399 1181 V 1443 1151 a Fq(L)1500 1163 y Fm(22)1611 874 y Fn(1)1611 1023 y(A)1697 1041 y Fq(;)1817 924 y Fn(\022)1922 980 y Fq(B)1985 992 y Fl(T)p 1878 1014 204 4 v 1878 1030 V 1920 1100 a Fq(B)1983 1112 y Fl(B)2082 924 y Fn(\023)2166 1041 y Fp( )2272 874 y Fn(0)2272 1023 y(@)2386 929 y Fq(B)2449 941 y Fm(0)p 2344 962 184 4 v 2392 1032 a Fq(b)2428 1002 y Fl(T)2428 1052 y Fm(1)p 2344 1065 V 2344 1082 V 2386 1151 a Fq(B)2449 1163 y Fm(2)2528 874 y Fn(1)2528 1023 y(A)2614 1041 y Fq(;)97 b Fr(and)2965 899 y Fn( )3094 959 y Fr(^)3074 980 y Fq(B)3137 992 y Fl(T)p 3030 1014 204 4 v 3030 1030 V 3092 1088 a Fr(^)3072 1109 y Fq(B)3135 1121 y Fl(B)3234 899 y Fn(!)3322 1041 y Fp( )3428 849 y Fn(0)3428 995 y(B)3428 1048 y(@)3562 903 y Fr(^)3543 924 y Fq(B)3606 936 y Fm(0)p 3501 957 184 4 v 3546 1015 a Fr(^)3549 1037 y Fq(b)3585 1007 y Fl(T)3585 1057 y Fm(1)p 3501 1070 V 3501 1087 V 3562 1144 a Fr(^)3543 1165 y Fq(B)3606 1177 y Fm(2)3684 849 y Fn(1)3684 995 y(C)3684 1048 y(A)0 1322 y Fr(to)n(w)n(ards)27 b(the)i(b)r(ottom)e(of)i(the)g(lo)r(op.)38 b(The)28 b(idea,)g(again,)e(is)i(that)h(the)f(double)g(lines)e(ha)n(v)n (e)i(seman)n(tic)e(meaning)g(and)i(sho)n(w)0 1422 y(that)g(up)r(on)g (repartitioning)326 1605 y Fq(L)383 1617 y Fl(T)9 b(L)503 1605 y Fp(!)23 b Fq(L)666 1617 y Fm(00)p 881 1635 4 101 v 897 1635 V 1047 1605 a Fq(L)1104 1617 y Fl(T)9 b(R)1229 1605 y Fp(!)1336 1538 y Fn(\000)1415 1604 y Fr(0)p 1496 1634 4 100 v 83 w(0)1623 1538 y Fn(\001)p 179 1638 1631 4 v 179 1655 V 220 1777 a Fq(L)277 1789 y Fl(B)s(L)403 1777 y Fp(!)509 1660 y Fn(\022)627 1725 y Fq(l)654 1694 y Fl(T)652 1745 y Fm(10)p 570 1758 210 4 v 611 1827 a Fq(L)668 1839 y Fm(20)780 1660 y Fn(\023)p 881 1857 4 203 v 898 1857 V 941 1777 a Fq(L)998 1789 y Fl(B)s(R)1128 1777 y Fp(!)1234 1660 y Fn(\022)1336 1724 y Fq(\025)1384 1736 y Fm(11)p 1495 1754 4 100 v 1581 1724 a Fr(0)p 1295 1758 412 4 v 1348 1827 a Fq(l)1373 1839 y Fm(21)p 1495 1857 4 100 v 1538 1827 a Fq(L)1595 1839 y Fm(22)1706 1660 y Fn(\023)1809 1717 y Fq(;)2076 1604 y(B)2139 1616 y Fl(T)2214 1604 y Fp(!)23 b Fq(B)2383 1616 y Fm(0)p 1929 1638 638 4 v 1929 1654 V 1970 1776 a Fq(B)2033 1788 y Fl(B)2114 1776 y Fp(!)2220 1659 y Fn(\022)2328 1724 y Fq(b)2364 1694 y Fl(T)2364 1745 y Fm(1)p 2281 1757 184 4 v 2322 1827 a Fq(B)2385 1839 y Fm(2)2464 1659 y Fn(\023)2567 1717 y Fq(;)97 b Fr(and)3074 1565 y(^)3055 1586 y Fq(B)3118 1598 y Fl(T)3193 1586 y Fp(!)3319 1565 y Fr(^)3299 1586 y Fq(B)3362 1598 y Fm(0)p 2903 1619 647 4 v 2903 1636 V 2965 1760 a Fr(^)2945 1781 y Fq(B)3008 1793 y Fl(B)3088 1781 y Fp(!)3194 1639 y Fn( )3305 1707 y Fr(^)3308 1729 y Fq(b)3344 1699 y Fl(T)3344 1750 y Fm(1)p 3260 1762 184 4 v 3321 1820 a Fr(^)3301 1841 y Fq(B)3364 1853 y Fm(2)3443 1639 y Fn(!)3550 1717 y Fq(:)179 b Fr(\(13\))0 2012 y(T)-7 b(o)n(w)n(ards)25 b(the)j(end)g(of)g(the)g (lo)r(op)e(the)i(quadran)n(ts)e(of)i(the)g(partitioned)d(matrix)g(are)h (rede\014ned)i(lik)n(e)347 2246 y Fq(L)404 2258 y Fl(T)9 b(L)524 2246 y Fp( )630 2129 y Fn(\022)733 2194 y Fq(L)790 2206 y Fm(00)p 899 2224 4 100 v 981 2194 a Fr(0)p 691 2227 412 4 v 749 2297 a Fq(l)776 2267 y Fl(T)774 2318 y Fm(10)p 899 2327 4 100 v 943 2297 a Fq(\025)991 2309 y Fm(11)1103 2129 y Fn(\023)p 1204 2327 4 203 v 1221 2327 V 1264 2246 a Fq(L)1321 2258 y Fl(T)g(R)1446 2246 y Fp( )1552 2129 y Fn(\022)1655 2194 y Fr(0)p 1613 2227 125 4 v 1655 2297 a(0)1738 2129 y Fn(\023)p 305 2330 1536 4 v 305 2347 V 379 2417 a Fq(L)436 2429 y Fl(B)s(L)561 2417 y Fp( )667 2350 y Fn(\000)747 2416 y Fq(L)804 2428 y Fm(20)p 914 2446 4 100 v 957 2416 a Fq(l)982 2428 y Fm(21)1093 2350 y Fn(\001)p 1204 2447 4 101 v 1221 2447 V 1321 2417 a Fq(L)1378 2429 y Fl(B)s(R)1508 2417 y Fp( )23 b Fq(L)1671 2429 y Fm(22)1840 2307 y Fq(;)2002 2247 y(B)2065 2259 y Fl(T)2140 2247 y Fp( )2246 2130 y Fn(\022)2349 2194 y Fq(B)2412 2206 y Fm(0)p 2307 2228 184 4 v 2355 2297 a Fq(b)2391 2267 y Fl(T)2391 2318 y Fm(1)2490 2130 y Fn(\023)p 1960 2331 633 4 v 1960 2347 V 2102 2417 a Fq(B)2165 2429 y Fl(B)2245 2417 y Fp( )g Fq(B)2414 2429 y Fm(2)2593 2307 y Fq(;)97 b Fr(and)2991 2221 y(^)2971 2242 y Fq(B)3034 2254 y Fl(T)3110 2242 y Fp( )3216 2100 y Fn( )3343 2169 y Fr(^)3323 2190 y Fq(B)3386 2202 y Fm(0)p 3281 2223 184 4 v 3326 2280 a Fr(^)3329 2302 y Fq(b)3365 2272 y Fl(T)3365 2323 y Fm(1)3465 2100 y Fn(!)p 2930 2349 643 4 v 2930 2366 V 3096 2424 a Fr(^)3076 2445 y Fq(B)3139 2457 y Fl(B)3219 2445 y Fp( )3345 2424 y Fr(^)3325 2445 y Fq(B)3388 2457 y Fm(0)3572 2307 y Fq(:)157 b Fr(\(14\))0 2653 y Fk(Step)32 b(6:)41 b(State)33 b(after)f (repartitioning)0 2806 y Fr(The)25 b(next)g(question)e(b)r(ecomes)g (what)i(the)g(state)g(is)e(of)i(the)g(op)r(erands,)g(in)f(terms)f(of)i (the)g(submatrices)d(that)j(w)n(ere)f(exp)r(osed)0 2906 y(as)34 b(part)g(of)g(the)h(repartitioning.)53 b(Notice)34 b(that)g(b)n(y)g(substituting)f(the)i(exp)r(osed)f(submatrices)e(in)i (\(13\))g(in)n(to)f(the)i(lo)r(op-)0 3005 y(in)n(v)-5 b(arian)n(t)24 b(w)n(e)k(\014nd)g(that)79 3142 y Fn( )145 3167 y(\022)250 3224 y Fq(B)313 3236 y Fl(T)p 206 3257 204 4 v 206 3273 V 248 3343 a Fq(B)311 3355 y Fl(B)409 3167 y Fn(\023)494 3284 y Fr(=)581 3142 y Fn( )911 3224 y Fq(L)968 3236 y Fl(T)9 b(L)1065 3187 y Fj(\000)p Fm(1)1174 3203 y Fr(^)1155 3224 y Fq(B)1218 3236 y Fl(T)p 647 3257 888 4 v 647 3273 V 708 3331 a Fr(^)688 3352 y Fq(B)751 3364 y Fl(B)827 3352 y Fp(\000)18 b Fq(L)967 3364 y Fl(B)s(L)1069 3352 y Fr(\()p Fq(L)1158 3364 y Fl(T)9 b(L)1256 3316 y Fj(\000)p Fm(1)1365 3331 y Fr(^)1345 3352 y Fq(B)1408 3364 y Fl(T)1460 3352 y Fr(\))1534 3142 y Fn(!!)1689 3284 y Fp(\))1795 3092 y Fn(0)1795 3239 y(B)1795 3292 y(@)1867 3117 y(0)1867 3267 y(@)2084 3172 y Fq(B)2147 3184 y Fm(0)p 1940 3205 389 4 v 1940 3222 V 1981 3227 a Fn(\022)2090 3292 y Fq(b)2126 3262 y Fl(T)2126 3312 y Fm(1)p 2043 3325 184 4 v 2084 3395 a Fq(B)2147 3407 y Fm(2)2226 3227 y Fn(\023)2328 3117 y(1)2328 3267 y(A)2424 3284 y Fr(=)2512 3092 y Fn(0)2512 3239 y(B)2512 3292 y(@)3007 3153 y Fq(L)3064 3118 y Fj(\000)p Fm(1)3064 3175 y(00)3172 3132 y Fr(^)3153 3153 y Fq(B)3216 3165 y Fm(0)p 2584 3187 1091 4 v 2584 3203 V 2626 3206 a Fn( )2736 3275 y Fr(^)2739 3297 y Fq(b)2775 3266 y Fl(T)2775 3317 y Fm(1)p 2692 3330 184 4 v 2753 3387 a Fr(^)2733 3408 y Fq(B)2796 3420 y Fm(2)2875 3206 y Fn(!)2959 3348 y Fp(\000)3042 3231 y Fn(\022)3161 3296 y Fq(l)3188 3266 y Fl(T)3186 3317 y Fm(10)p 3103 3329 210 4 v 3145 3399 a Fq(L)3202 3411 y Fm(20)3313 3231 y Fn(\023)3388 3348 y Fq(L)3445 3313 y Fj(\000)p Fm(1)3445 3371 y(00)3553 3327 y Fr(^)3534 3348 y Fq(B)3597 3360 y Fm(0)3675 3092 y Fn(1)3675 3239 y(C)3675 3292 y(A)3748 3092 y(1)3748 3239 y(C)3748 3292 y(A)0 3579 y Fr(whic)n(h,)27 b(up)r(on)g (simpli\014cation,)c(giv)n(es)i(the)j(state)g(of)f(the)h(exp)r(osed)f (submatrices)e(after)i(the)h(repartitioning:)1311 3693 y Fn(0)1311 3843 y(@)1425 3748 y Fq(B)1488 3760 y Fm(0)p 1383 3781 184 4 v 1383 3798 V 1431 3868 a Fq(b)1467 3838 y Fl(T)1467 3888 y Fm(1)p 1383 3901 V 1425 3971 a Fq(B)1488 3983 y Fm(2)1566 3693 y Fn(1)1566 3843 y(A)1662 3860 y Fr(=)1750 3668 y Fn(0)1750 3815 y(B)1750 3868 y(@)2028 3743 y Fq(L)2085 3708 y Fj(\000)p Fm(1)2085 3765 y(00)2194 3722 y Fr(^)2174 3743 y Fq(B)2237 3755 y Fm(0)p 1822 3776 658 4 v 1822 3793 V 1883 3851 a Fr(^)1886 3873 y Fq(b)1922 3842 y Fl(T)1922 3893 y Fm(1)1992 3873 y Fp(\000)18 b Fq(l)2102 3842 y Fl(T)2100 3893 y Fm(10)2170 3873 y Fq(L)2227 3837 y Fj(\000)p Fm(1)2227 3895 y(00)2336 3852 y Fr(^)2316 3873 y Fq(B)2379 3885 y Fm(0)p 1822 3906 V 1884 3963 a Fr(^)1864 3984 y Fq(B)1927 3996 y Fm(2)1983 3984 y Fp(\000)g Fq(L)2123 3996 y Fm(20)2193 3984 y Fq(L)2250 3949 y Fj(\000)p Fm(1)2250 4007 y(00)2358 3963 y Fr(^)2338 3984 y Fq(B)2401 3996 y Fm(0)2480 3668 y Fn(1)2480 3815 y(C)2480 3868 y(A)2566 3860 y Fq(:)1163 b Fr(\(15\))0 4193 y Fk(Step)32 b(7:)41 b(State)33 b(after)f(mo)m(ving)f(the)g (double)g(lines)0 4346 y Fr(After)23 b(the)g(double)e(lines)f(are)i(mo) n(v)n(ed,)f(w)n(e)h(are)f(at)i(the)f(b)r(ottom)g(of)h(the)f(lo)r(op)f (and)h(the)h(lo)r(op-in)n(v)-5 b(arian)n(t)18 b(m)n(ust)j(again)f(b)r (e)j(true.)0 4446 y(Notice)f(that)h(the)g(rede\014nition)e(of)h(the)i (partitionings)18 b(giv)n(en)j(in)h(\(14\))h(means)e(that)i(in)f(terms) g(of)g(the)i(exp)r(osed)e(submatrices)0 4545 y(the)28 b(con)n(ten)n(ts)f(of)g(the)h(op)r(erands)f(m)n(ust)g(b)r(e)0 4768 y Fn( )66 4793 y(\022)171 4850 y Fq(B)234 4862 y Fl(T)p 127 4883 204 4 v 127 4900 V 168 4969 a Fq(B)231 4981 y Fl(B)330 4793 y Fn(\023)414 4910 y Fr(=)502 4768 y Fn( )832 4850 y Fq(L)889 4862 y Fl(T)9 b(L)986 4814 y Fj(\000)p Fm(1)1095 4829 y Fr(^)1075 4850 y Fq(B)1138 4862 y Fl(T)p 568 4883 888 4 v 568 4900 V 629 4957 a Fr(^)609 4978 y Fq(B)672 4990 y Fl(B)748 4978 y Fp(\000)18 b Fq(L)888 4990 y Fl(B)s(L)990 4978 y Fr(\()p Fq(L)1079 4990 y Fl(T)9 b(L)1176 4942 y Fj(\000)p Fm(1)1285 4957 y Fr(^)1266 4978 y Fq(B)1329 4990 y Fl(T)1381 4978 y Fr(\))1455 4768 y Fn(!)o(!)1609 4910 y Fp(\))1715 4644 y Fn(0)1715 4790 y(B)1715 4840 y(B)1715 4890 y(B)1715 4939 y(B)1715 4993 y(@)1788 4744 y(0)1788 4893 y(@)1902 4734 y(\022)2005 4798 y Fq(B)2068 4810 y Fm(0)p 1963 4831 184 4 v 2011 4901 a Fq(b)2047 4871 y Fl(T)2047 4922 y Fm(1)2146 4734 y Fn(\023)p 1861 4935 389 4 v 1861 4951 V 2005 5021 a Fq(B)2068 5033 y Fm(2)2249 4744 y Fn(1)2249 4893 y(A)2345 4910 y Fr(=)2432 4644 y Fn(0)2432 4790 y(B)2432 4840 y(B)2432 4890 y(B)2432 4939 y(B)2432 4993 y(@)2886 4659 y(\022)2989 4724 y Fq(L)3046 4736 y Fm(00)p 3156 4753 4 100 v 3237 4724 a Fr(0)p 2947 4757 412 4 v 3005 4827 a Fq(l)3032 4796 y Fl(T)3030 4847 y Fm(10)p 3156 4857 4 100 v 3199 4827 a Fq(\025)3247 4839 y Fm(11)3359 4659 y Fn(\023)3420 4674 y Fj(\000)p Fm(1)3523 4634 y Fn( )3650 4702 y Fr(^)3630 4723 y Fq(B)3693 4735 y Fm(0)p 3589 4756 184 4 v 3634 4814 a Fr(^)3637 4836 y Fq(b)3673 4806 y Fl(T)3673 4857 y Fm(1)3772 4634 y Fn(!)p 2505 4883 1715 4 v 2505 4900 V 2566 5024 a Fr(^)2547 5045 y Fq(B)2610 5057 y Fm(2)2665 5045 y Fp(\000)2748 4978 y Fn(\000)2828 5044 y Fq(L)2885 5056 y Fm(20)p 2994 5074 4 100 v 3038 5044 a Fq(l)3063 5056 y Fm(21)3174 4978 y Fn(\001)3226 4928 y(\022)3329 4993 y Fq(L)3386 5005 y Fm(00)p 3495 5022 V 3577 4993 a Fr(0)p 3287 5026 412 4 v 3345 5096 a Fq(l)3372 5065 y Fl(T)3370 5116 y Fm(10)p 3495 5125 4 100 v 3539 5096 a Fq(\025)3587 5108 y Fm(11)3699 4928 y Fn(\023)3760 4943 y Fj(\000)p Fm(1)3863 4903 y Fn( )3990 4971 y Fr(^)3970 4992 y Fq(B)4033 5004 y Fm(0)p 3929 5025 184 4 v 3973 5083 a Fr(^)3976 5105 y Fq(b)4012 5075 y Fl(T)4012 5126 y Fm(1)4112 4903 y Fn(!)4219 4644 y(1)4219 4790 y(C)4219 4840 y(C)4219 4890 y(C)4219 4939 y(C)4219 4993 y(A)4292 4644 y(1)4292 4790 y(C)4292 4840 y(C)4292 4890 y(C)4292 4939 y(C)4292 4993 y(A)4378 4910 y Fq(:)1908 5356 y Fr(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 -60 a Fr(Since)1149 -30 y Fn(\022)1251 35 y Fq(L)1308 47 y Fm(00)p 1418 65 4 100 v 1500 35 a Fr(0)p 1210 68 412 4 v 1267 138 a Fq(l)1294 108 y Fl(T)1292 159 y Fm(10)p 1418 168 4 100 v 1461 138 a Fq(\025)1509 150 y Fm(11)1622 -30 y Fn(\023)1683 -14 y Fj(\000)p Fm(1)1795 88 y Fr(=)1882 -30 y Fn(\022)2134 35 y Fq(L)2191 0 y Fj(\000)p Fm(1)2191 57 y(00)p 2468 65 4 103 v 2559 35 a Fr(0)p 1944 68 747 4 v 1985 141 a Fp(\000)p Fq(\025)2098 106 y Fj(\000)p Fm(1)2098 164 y(11)2187 141 y Fq(l)2214 111 y Fl(T)2212 162 y Fm(10)2282 141 y Fq(L)2339 106 y Fj(\000)p Fm(1)2339 164 y(00)p 2468 171 4 103 v 2511 141 a Fq(\025)2559 106 y Fj(\000)p Fm(1)2559 164 y(11)2690 -30 y Fn(\023)0 287 y Fr(the)28 b(state)f(of)h(the)g(exp)r(osed)f (submatrices)e(after)i(mo)n(ving)e(the)j(double)e(lines)g(m)n(ust)g(b)r (e)575 507 y Fn(0)575 657 y(@)689 497 y(\022)792 562 y Fq(B)855 574 y Fm(0)p 750 595 184 4 v 798 665 a Fq(b)834 635 y Fl(T)834 686 y Fm(1)934 497 y Fn(\023)p 648 698 389 4 v 648 715 V 792 785 a Fq(B)855 797 y Fm(2)1036 507 y Fn(1)1036 657 y(A)1132 674 y Fr(=)1220 408 y Fn(0)1220 554 y(B)1220 604 y(B)1220 654 y(B)1220 703 y(B)1220 757 y(@)1674 423 y(\022)1925 488 y Fq(L)1982 452 y Fj(\000)p Fm(1)1982 510 y(00)p 2259 517 4 103 v 2350 488 a Fr(0)p 1735 521 747 4 v 1776 594 a Fp(\000)p Fq(\025)1889 558 y Fj(\000)p Fm(1)1889 616 y(11)1978 594 y Fq(l)2005 564 y Fl(T)2003 614 y Fm(10)2073 594 y Fq(L)2130 558 y Fj(\000)p Fm(1)2130 616 y(00)p 2259 624 4 103 v 2302 594 a Fq(\025)2350 558 y Fj(\000)p Fm(1)2350 616 y(11)2481 423 y Fn(\023)2556 398 y( )2683 466 y Fr(^)2663 487 y Fq(B)2726 499 y Fm(0)p 2622 520 184 4 v 2667 578 a Fr(^)2670 600 y Fq(b)2706 570 y Fl(T)2706 620 y Fm(1)2805 398 y Fn(!)p 1292 647 1960 4 v 1292 664 V 1354 788 a Fr(^)1334 809 y Fq(B)1397 821 y Fm(2)1452 809 y Fp(\000)1535 742 y Fn(\000)1615 808 y Fq(L)1672 820 y Fm(20)p 1782 838 4 100 v 1825 808 a Fq(l)1850 820 y Fm(21)1962 742 y Fn(\001)2013 692 y(\022)2265 757 y Fq(L)2322 721 y Fj(\000)p Fm(1)2322 779 y(00)p 2599 786 4 103 v 2690 757 a Fr(0)p 2075 790 747 4 v 2116 863 a Fp(\000)p Fq(\025)2229 827 y Fj(\000)p Fm(1)2229 885 y(11)2318 863 y Fq(l)2345 833 y Fl(T)2343 883 y Fm(10)2413 863 y Fq(L)2470 827 y Fj(\000)p Fm(1)2470 885 y(00)p 2599 893 4 103 v 2642 863 a Fq(\025)2690 827 y Fj(\000)p Fm(1)2690 885 y(11)2821 692 y Fn(\023)2896 667 y( )3023 735 y Fr(^)3003 756 y Fq(B)3066 768 y Fm(0)p 2962 789 184 4 v 3006 847 a Fr(^)3009 869 y Fq(b)3045 839 y Fl(T)3045 889 y Fm(1)3145 667 y Fn(!)3252 408 y(1)3252 554 y(C)3252 604 y(C)3252 654 y(C)3252 703 y(C)3252 757 y(A)0 1062 y Fr(or,)h(simplifying,)700 1282 y Fn(0)700 1431 y(@)814 1272 y(\022)917 1337 y Fq(B)980 1349 y Fm(0)p 876 1370 V 923 1440 a Fq(b)959 1410 y Fl(T)959 1460 y Fm(1)1059 1272 y Fn(\023)p 773 1473 389 4 v 773 1490 V 917 1559 a Fq(B)980 1571 y Fm(2)1161 1282 y Fn(1)1161 1431 y(A)1257 1449 y Fr(=)1345 1182 y Fn(0)1345 1328 y(B)1345 1378 y(B)1345 1428 y(B)1345 1478 y(B)1345 1531 y(@)1799 1172 y( )2149 1262 y Fq(L)2206 1226 y Fj(\000)p Fm(1)2206 1284 y(00)2315 1241 y Fr(^)2295 1262 y Fq(B)2358 1274 y Fm(0)p 1864 1295 816 4 v 1906 1374 a Fq(\025)1954 1339 y Fj(\000)p Fm(1)1954 1396 y(11)2044 1374 y Fr(\()2073 1352 y(^)2076 1374 y Fq(b)2112 1344 y Fl(T)2112 1395 y Fm(1)2182 1374 y Fp(\000)18 b Fq(l)2292 1344 y Fl(T)2290 1395 y Fm(10)2360 1374 y Fq(L)2417 1339 y Fj(\000)p Fm(1)2417 1396 y(00)2526 1353 y Fr(^)2506 1374 y Fq(B)2569 1386 y Fm(0)2606 1374 y Fr(\))2680 1172 y Fn(!)p 1417 1421 1710 4 v 1417 1438 V 1479 1562 a Fr(^)1459 1583 y Fq(B)1522 1595 y Fm(2)1578 1583 y Fp(\000)1661 1516 y Fn(\000)1740 1582 y Fq(L)1797 1594 y Fm(20)p 1907 1612 4 100 v 1950 1582 a Fq(l)1975 1594 y Fm(21)2087 1516 y Fn(\001)2139 1441 y( )2489 1531 y Fq(L)2546 1495 y Fj(\000)p Fm(1)2546 1553 y(00)2654 1510 y Fr(^)2635 1531 y Fq(B)2698 1543 y Fm(0)p 2204 1564 816 4 v 2246 1643 a Fq(\025)2294 1608 y Fj(\000)p Fm(1)2294 1665 y(11)2383 1643 y Fr(\()2412 1621 y(^)2415 1643 y Fq(b)2451 1613 y Fl(T)2451 1664 y Fm(1)2522 1643 y Fp(\000)g Fq(l)2632 1613 y Fl(T)2630 1664 y Fm(10)2700 1643 y Fq(L)2757 1608 y Fj(\000)p Fm(1)2757 1665 y(00)2866 1622 y Fr(^)2846 1643 y Fq(B)2909 1655 y Fm(0)2946 1643 y Fr(\))3020 1441 y Fn(!)3127 1182 y(1)3127 1328 y(C)3127 1378 y(C)3127 1428 y(C)3127 1478 y(C)3127 1531 y(A)0 1836 y Fr(whic)n(h)27 b(means)846 1874 y Fn(0)846 2023 y(@)960 1928 y Fq(B)1023 1940 y Fm(0)p 919 1962 184 4 v 966 2031 a Fq(b)1002 2001 y Fl(T)1002 2052 y Fm(1)p 919 2065 V 919 2081 V 960 2151 a Fq(B)1023 2163 y Fm(2)1102 1874 y Fn(1)1102 2023 y(A)1198 2040 y Fr(=)1285 1849 y Fn(0)1285 1995 y(B)1285 2048 y(@)2028 1923 y Fq(L)2085 1887 y Fj(\000)p Fm(1)2085 1945 y(00)2194 1902 y Fr(^)2174 1923 y Fq(B)2237 1935 y Fm(0)p 1358 1956 1587 4 v 1785 2036 a Fq(\025)1833 2000 y Fj(\000)p Fm(1)1833 2058 y(11)1922 2036 y Fr(\()1951 2014 y(^)1954 2036 y Fq(b)1990 2006 y Fl(T)1990 2056 y Fm(1)2061 2036 y Fp(\000)18 b Fq(l)2171 2006 y Fl(T)2169 2056 y Fm(10)2239 2036 y Fq(L)2296 2000 y Fj(\000)p Fm(1)2296 2058 y(00)2405 2015 y Fr(^)2385 2036 y Fq(B)2448 2048 y Fm(0)2485 2036 y Fr(\))p 1358 2069 V 1358 2086 V 1419 2144 a(^)1399 2165 y Fq(B)1462 2177 y Fm(2)1518 2165 y Fp(\000)g Fq(L)1658 2177 y Fm(20)1728 2165 y Fq(L)1785 2130 y Fj(\000)p Fm(1)1785 2187 y(00)1893 2144 y Fr(^)1874 2165 y Fq(B)1937 2177 y Fm(0)1992 2165 y Fp(\000)g Fq(l)2100 2177 y Fm(21)2170 2165 y Fq(\025)2218 2130 y Fj(\000)p Fm(1)2218 2187 y(11)2308 2165 y Fr(\()2337 2143 y(^)2340 2165 y Fq(b)2376 2135 y Fl(T)2376 2186 y Fm(1)2447 2165 y Fp(\000)g Fq(l)2557 2135 y Fl(T)2555 2186 y Fm(10)2625 2165 y Fq(L)2682 2130 y Fj(\000)p Fm(1)2682 2187 y(00)2790 2144 y Fr(^)2770 2165 y Fq(B)2833 2177 y Fm(0)2871 2165 y Fr(\))2944 1849 y Fn(1)2944 1995 y(C)2944 2048 y(A)3031 2040 y Fq(:)698 b Fr(\(16\))0 2377 y Fk(Step)32 b(8:)41 b(Determining)30 b(the)i(up)s(date)g(to)f(the)h(exp)s(osed)f (submatrices)0 2530 y Fr(Notice)23 b(that)i(after)f(repartitioning,)c (the)25 b(state)f(of)g(the)h(exp)r(osed)f(submatrices)d(is)i(giv)n(en)g (b)n(y)g(the)i(predicate)e(in)g(\(15\).)36 b(After)0 2630 y(mo)n(ving)30 b(the)k(double)f(lines,)g(the)h(state)f(of)g(the)h (exp)r(osed)f(submatrices)e(m)n(ust)h(b)r(e)i(as)f(giv)n(en)f(b)n(y)h (the)h(predicate)e(in)g(\(16\).)0 2730 y(Since)25 b(no)h(computation)d (happ)r(ens)j(in)f(mo)n(ving)e(the)j(double)f(lines,)f(b)r(efore)h(mo)n (ving)e(the)j(double)f(lines)f(the)i(state)g(m)n(ust)f(b)r(e)846 2866 y Fn(0)846 3016 y(@)960 2921 y Fq(B)1023 2933 y Fm(0)p 919 2954 184 4 v 919 2971 V 966 3041 a Fq(b)1002 3011 y Fl(T)1002 3061 y Fm(1)p 919 3074 V 960 3144 a Fq(B)1023 3156 y Fm(2)1102 2866 y Fn(1)1102 3016 y(A)1198 3033 y Fr(=)1285 2841 y Fn(0)1285 2988 y(B)1285 3041 y(@)2028 2916 y Fq(L)2085 2880 y Fj(\000)p Fm(1)2085 2938 y(00)2194 2895 y Fr(^)2174 2916 y Fq(B)2237 2928 y Fm(0)p 1358 2949 1587 4 v 1358 2966 V 1785 3045 a Fq(\025)1833 3010 y Fj(\000)p Fm(1)1833 3067 y(11)1922 3045 y Fr(\()1951 3023 y(^)1954 3045 y Fq(b)1990 3015 y Fl(T)1990 3066 y Fm(1)2061 3045 y Fp(\000)18 b Fq(l)2171 3015 y Fl(T)2169 3066 y Fm(10)2239 3045 y Fq(L)2296 3010 y Fj(\000)p Fm(1)2296 3067 y(00)2405 3024 y Fr(^)2385 3045 y Fq(B)2448 3057 y Fm(0)2485 3045 y Fr(\))p 1358 3078 V 1419 3137 a(^)1399 3158 y Fq(B)1462 3170 y Fm(2)1518 3158 y Fp(\000)g Fq(L)1658 3170 y Fm(20)1728 3158 y Fq(L)1785 3122 y Fj(\000)p Fm(1)1785 3180 y(00)1893 3137 y Fr(^)1874 3158 y Fq(B)1937 3170 y Fm(0)1992 3158 y Fp(\000)g Fq(l)2100 3170 y Fm(21)2170 3158 y Fq(\025)2218 3122 y Fj(\000)p Fm(1)2218 3180 y(11)2308 3158 y Fr(\()2337 3136 y(^)2340 3158 y Fq(b)2376 3128 y Fl(T)2376 3179 y Fm(1)2447 3158 y Fp(\000)g Fq(l)2557 3128 y Fl(T)2555 3179 y Fm(10)2625 3158 y Fq(L)2682 3122 y Fj(\000)p Fm(1)2682 3180 y(00)2790 3137 y Fr(^)2770 3158 y Fq(B)2833 3170 y Fm(0)2871 3158 y Fr(\))2944 2841 y Fn(1)2944 2988 y(C)2944 3041 y(A)3031 3033 y Fq(:)698 b Fr(\(17\))0 3337 y(Th)n(us,)27 b(the)h(up)r(date)g(m)n(ust)f(c)n (hange)g(the)h(state)f(of)g(the)h(submatrices)d(lik)n(e)0 3449 y Fn(0)0 3595 y(B)0 3648 y(@)73 3473 y(0)73 3623 y(@)187 3528 y Fq(B)250 3540 y Fm(0)p 145 3561 184 4 v 145 3578 V 193 3648 a Fq(b)229 3618 y Fl(T)229 3668 y Fm(1)p 145 3681 V 187 3751 a Fq(B)250 3763 y Fm(2)329 3473 y Fn(1)329 3623 y(A)424 3640 y Fr(=)512 3449 y Fn(0)512 3595 y(B)512 3648 y(@)790 3523 y Fq(L)847 3488 y Fj(\000)p Fm(1)847 3546 y(00)956 3502 y Fr(^)936 3523 y Fq(B)999 3535 y Fm(0)p 585 3557 658 4 v 585 3573 V 645 3631 a Fr(^)648 3653 y Fq(b)684 3623 y Fl(T)684 3673 y Fm(1)754 3653 y Fp(\000)18 b Fq(l)864 3623 y Fl(T)862 3673 y Fm(10)933 3653 y Fq(L)990 3617 y Fj(\000)p Fm(1)990 3675 y(00)1098 3632 y Fr(^)1078 3653 y Fq(B)1141 3665 y Fm(0)p 585 3686 V 646 3744 a Fr(^)626 3765 y Fq(B)689 3777 y Fm(2)745 3765 y Fp(\000)g Fq(L)885 3777 y Fm(20)955 3765 y Fq(L)1012 3729 y Fj(\000)p Fm(1)1012 3787 y(00)1120 3744 y Fr(^)1100 3765 y Fq(B)1163 3777 y Fm(0)1242 3449 y Fn(1)1242 3595 y(C)1242 3648 y(A)1328 3640 y Fq(:)1351 3449 y Fn(1)1351 3595 y(C)1351 3648 y(A)1447 3640 y Fp(\000)-14 b(!)1604 3449 y Fn(0)1604 3595 y(B)1604 3648 y(@)1677 3473 y(0)1677 3623 y(@)1791 3528 y Fq(B)1854 3540 y Fm(0)p 1749 3561 184 4 v 1749 3578 V 1797 3648 a Fq(b)1833 3618 y Fl(T)1833 3668 y Fm(1)p 1749 3681 V 1791 3751 a Fq(B)1854 3763 y Fm(2)1933 3473 y Fn(1)1933 3623 y(A)2028 3640 y Fr(=)2116 3449 y Fn(0)2116 3595 y(B)2116 3648 y(@)2859 3523 y Fq(L)2916 3487 y Fj(\000)p Fm(1)2916 3545 y(00)3024 3502 y Fr(^)3004 3523 y Fq(B)3067 3535 y Fm(0)p 2188 3556 1587 4 v 2188 3573 V 2615 3652 a Fq(\025)2663 3617 y Fj(\000)p Fm(1)2663 3674 y(11)2753 3652 y Fr(\()2782 3630 y(^)2785 3652 y Fq(b)2821 3622 y Fl(T)2821 3673 y Fm(1)2892 3652 y Fp(\000)18 b Fq(l)3002 3622 y Fl(T)3000 3673 y Fm(10)3070 3652 y Fq(L)3127 3617 y Fj(\000)p Fm(1)3127 3674 y(00)3235 3631 y Fr(^)3215 3652 y Fq(B)3278 3664 y Fm(0)3316 3652 y Fr(\))p 2188 3685 V 2250 3744 a(^)2230 3765 y Fq(B)2293 3777 y Fm(2)2349 3765 y Fp(\000)g Fq(L)2489 3777 y Fm(20)2559 3765 y Fq(L)2616 3730 y Fj(\000)p Fm(1)2616 3787 y(00)2724 3744 y Fr(^)2704 3765 y Fq(B)2767 3777 y Fm(0)2823 3765 y Fp(\000)g Fq(l)2931 3777 y Fm(21)3001 3765 y Fq(\025)3049 3730 y Fj(\000)p Fm(1)3049 3787 y(11)3139 3765 y Fr(\()3168 3743 y(^)3171 3765 y Fq(b)3207 3735 y Fl(T)3207 3786 y Fm(1)3277 3765 y Fp(\000)g Fq(l)3387 3735 y Fl(T)3385 3786 y Fm(10)3455 3765 y Fq(L)3512 3730 y Fj(\000)p Fm(1)3512 3787 y(00)3621 3744 y Fr(^)3601 3765 y Fq(B)3664 3777 y Fm(0)3701 3765 y Fr(\))3775 3449 y Fn(1)3775 3595 y(C)3775 3648 y(A)3848 3449 y(1)3848 3595 y(C)3848 3648 y(A)3934 3640 y Fq(:)0 3944 y Fr(w)n(e)27 b(conclude)g(that)g(the)h(follo)n (wing)c(up)r(dates)k(m)n(ust)e(b)r(e)i(made)f(to)g(submatrices)e(of)j Fq(B)t Fr(:)1070 4131 y Fq(b)1106 4101 y Fl(T)1106 4152 y Fm(1)1181 4131 y Fr(:=)22 b Fq(\025)1339 4095 y Fj(\000)p Fm(1)1339 4153 y(11)1429 4131 y Fr(\()1458 4109 y(^)1461 4131 y Fq(b)1497 4101 y Fl(T)1497 4152 y Fm(1)1568 4131 y Fp(\000)c Fq(l)1678 4101 y Fl(T)1676 4152 y Fm(10)1746 4131 y Fq(L)1803 4095 y Fj(\000)p Fm(1)1803 4153 y(00)1911 4110 y Fr(^)1891 4131 y Fq(B)1954 4143 y Fm(0)1992 4131 y Fr(\))1070 4240 y Fq(B)1133 4252 y Fm(2)1193 4240 y Fr(:=)1323 4219 y(^)1304 4240 y Fq(B)1367 4252 y Fm(2)1422 4240 y Fp(\000)g Fq(L)1562 4252 y Fm(20)1632 4240 y Fq(L)1689 4205 y Fj(\000)p Fm(1)1689 4263 y(00)1798 4219 y Fr(^)1778 4240 y Fq(B)1841 4252 y Fm(0)1897 4240 y Fp(\000)g Fq(l)2005 4252 y Fm(21)2075 4240 y Fq(\025)2123 4205 y Fj(\000)p Fm(1)2123 4263 y(11)2212 4240 y Fr(\()2241 4218 y(^)2244 4240 y Fq(b)2280 4210 y Fl(T)2280 4261 y Fm(1)2351 4240 y Fp(\000)g Fq(l)2461 4210 y Fl(T)2459 4261 y Fm(10)2529 4240 y Fq(L)2586 4205 y Fj(\000)p Fm(1)2586 4263 y(00)2694 4219 y Fr(^)2675 4240 y Fq(B)2738 4252 y Fm(0)2775 4240 y Fr(\))p Fq(:)0 4441 y Fr(Ho)n(w)n(ev)n(er,)27 b(realize)f(that)797 4419 y(^)800 4441 y Fq(b)836 4411 y Fl(T)836 4462 y Fm(1)917 4441 y Fr(and)1100 4420 y(^)1080 4441 y Fq(B)1143 4453 y Fm(0)1209 4441 y Fr(refer)i(to)h(the)g(original)c(con)n(ten)n(ts)j (of)h(matrix)d Fq(B)t Fr(.)41 b(As)29 b(part)g(of)g(the)g(computation,) e Fq(B)0 4541 y Fr(has)j(b)r(een)h(partially)c(up)r(dated,)32 b(and)e(th)n(us)h(all)d(or)i(part)g(of)g(the)h(con)n(ten)n(ts)f(of)2488 4520 y(^)2468 4541 y Fq(B)35 b Fr(are)29 b(no)i(longer)d(a)n(v)-5 b(ailable.)41 b(Notice,)31 b(that)0 4640 y Fq(b)36 4610 y Fl(T)36 4661 y Fm(1)116 4640 y Fr(curren)n(tly)25 b(con)n(tains)h(\() 826 4619 y(^)829 4640 y Fq(b)865 4610 y Fl(T)865 4661 y Fm(1)936 4640 y Fp(\000)18 b Fq(l)1046 4610 y Fl(T)1044 4661 y Fm(10)1114 4640 y Fq(L)1171 4605 y Fj(\000)p Fm(1)1171 4663 y(00)1279 4619 y Fr(^)1259 4640 y Fq(B)1322 4652 y Fm(0)1359 4640 y Fr(\))28 b(while)e Fq(B)1699 4652 y Fm(2)1764 4640 y Fr(con)n(tains)2109 4619 y(^)2090 4640 y Fq(B)2153 4652 y Fm(2)2208 4640 y Fp(\000)18 b Fq(L)2348 4652 y Fm(20)2418 4640 y Fq(L)2475 4605 y Fj(\000)p Fm(1)2475 4663 y(00)2584 4619 y Fr(^)2564 4640 y Fq(B)2627 4652 y Fm(0)2664 4640 y Fr(.)37 b(Th)n(us,)27 b(the)h(up)r(date)1641 4824 y Fq(b)1677 4794 y Fl(T)1677 4844 y Fm(1)1752 4824 y Fr(:=)22 b Fq(b)1898 4794 y Fl(T)1898 4844 y Fm(1)1950 4824 y Fq(=\025)2040 4836 y Fm(11)1641 4923 y Fq(B)1704 4935 y Fm(2)1764 4923 y Fr(:=)h Fq(B)1938 4935 y Fm(2)1993 4923 y Fp(\000)18 b Fq(l)2101 4935 y Fm(21)2171 4923 y Fq(b)2207 4893 y Fl(T)2207 4944 y Fm(1)0 5102 y Fr(will)25 b(lea)n(v)n(e)g(the)j(submatrices)d(of)i Fq(B)32 b Fr(in)27 b(the)h(desired)e(state.)1908 5356 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 353 383 3195 4 v 351 470 4 87 v 403 446 a Fg(Step)p 588 470 V 605 470 V 117 w(Annotated)26 b(Algorithm:)64 b Ff(B)23 b Fg(:=)c Ff(L)1640 429 y Fg(^)1623 446 y Ff(B)p 3545 470 V 353 473 3195 4 v 353 490 V 351 589 4 100 v 436 556 a Fg(1a)p 588 589 V 605 589 V 656 493 a Fn(\010)705 556 y Ff(B)k Fg(=)872 539 y(^)855 556 y Ff(B)912 493 y Fn(\011)p 3545 589 V 353 593 3195 4 v 351 878 4 286 v 454 709 a Fg(4)p 588 878 V 605 878 V 167 w Fc(P)n(artition)47 b Ff(L)20 b Fb(!)1182 596 y Fn(\022)1286 659 y Ff(L)1334 670 y Fd(T)8 b(L)p 1467 682 4 79 v 1484 682 V 1583 659 a Fg(0)p 1243 686 473 4 v 1243 702 V 1284 757 a Ff(L)1332 768 y Fd(B)r(L)p 1467 781 4 79 v 1484 781 V 1527 757 a Ff(L)1575 768 y Fd(B)r(R)1715 596 y Fn(\023)1776 709 y Fg(,)23 b Ff(B)h Fb(!)1986 596 y Fn(\022)2090 659 y Ff(B)2143 670 y Fd(T)p 2047 686 189 4 v 2047 702 V 2088 757 a Ff(B)2141 768 y Fd(B)2236 596 y Fn(\023)2320 709 y Fg(and)2474 692 y(^)2458 709 y Ff(B)f Fb(!)2624 596 y Fn(\022)2745 642 y Fg(^)2729 659 y Ff(B)2782 670 y Fd(T)p 2685 686 V 2685 702 V 2743 748 a Fg(^)2727 765 y Ff(B)2780 776 y Fd(B)2874 596 y Fn(\023)778 855 y Fc(where)51 b Ff(L)1089 866 y Fd(T)8 b(L)1203 855 y Fg(is)24 b(0)16 b Fb(\002)f Fg(0,)24 b(and)g Ff(B)1664 866 y Fd(T)1736 855 y Fg(and)1890 838 y(^)1874 855 y Ff(B)1927 866 y Fd(T)1999 855 y Fg(ha)n(v)n(e)h(0)f(ro)n (ws)p 3545 878 4 286 v 353 882 3195 4 v 351 1081 4 200 v 454 998 a(2)p 588 1081 V 605 1081 V 656 885 a Fn(\032)q(\022)823 948 y Ff(B)876 959 y Fd(T)p 780 975 189 4 v 780 991 V 821 1047 a Ff(B)874 1058 y Fd(B)969 885 y Fn(\023)1049 998 y Fg(=)1124 885 y Fn(\022)1421 948 y Ff(L)1469 959 y Fd(T)8 b(L)1560 923 y Fa(\000)p Fh(1)1659 931 y Fg(^)1643 948 y Ff(B)1696 959 y Fd(T)p 1185 975 797 4 v 1185 991 V 1243 1038 a Fg(^)1226 1054 y Ff(B)1279 1065 y Fd(B)1348 1054 y Fb(\000)16 b Ff(L)1467 1065 y Fd(B)r(L)1561 1054 y Fg(\()p Ff(L)1636 1065 y Fd(T)8 b(L)1728 1029 y Fa(\000)p Fh(1)1827 1038 y Fg(^)1810 1054 y Ff(B)1863 1065 y Fd(T)1912 1054 y Fg(\))1981 885 y Fn(\023\033)p 3545 1081 4 200 v 353 1084 3195 4 v 351 1163 4 79 v 454 1140 a Fg(3)p 588 1163 V 605 1163 V 167 w Fc(while)49 b Ff(m)p Fg(\()p Ff(L)1027 1151 y Fd(T)8 b(L)1118 1140 y Fg(\))20 b Fb(6)p Fg(=)g Ff(m)p Fg(\()p Ff(L)p Fg(\))48 b Fc(do)p 3545 1163 V 353 1167 3195 4 v 351 1366 4 200 v 426 1283 a Fg(2,3)p 588 1366 V 605 1366 V 817 1170 a Fn(\032\022\022)1045 1233 y Ff(B)1098 1244 y Fd(T)p 1001 1260 189 4 v 1001 1276 V 1043 1331 a Ff(B)1096 1342 y Fd(B)1190 1170 y Fn(\023)1271 1283 y Fg(=)1345 1170 y Fn(\022)1643 1233 y Ff(L)1691 1244 y Fd(T)8 b(L)1782 1208 y Fa(\000)p Fh(1)1881 1216 y Fg(^)1864 1233 y Ff(B)1917 1244 y Fd(T)p 1407 1260 797 4 v 1407 1276 V 1465 1322 a Fg(^)1448 1339 y Ff(B)1501 1350 y Fd(B)1570 1339 y Fb(\000)15 b Ff(L)1688 1350 y Fd(B)r(L)1783 1339 y Fg(\()p Ff(L)1858 1350 y Fd(T)8 b(L)1949 1314 y Fa(\000)p Fh(1)2048 1322 y Fg(^)2032 1339 y Ff(B)2085 1350 y Fd(T)2134 1339 y Fg(\))2203 1170 y Fn(\023\023)2341 1283 y Fb(^)15 b Fg(\()q Ff(m)p Fg(\()p Ff(L)2568 1294 y Fd(T)8 b(L)2659 1283 y Fg(\))20 b Fb(6)p Fg(=)g Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2974 1170 y Fn(\033)p 3545 1366 4 200 v 353 1369 3195 4 v 351 2212 4 843 v 436 1424 a Fg(5a)p 588 2212 V 605 2212 V 311 w Fc(Repartition)938 1568 y Fn(\022)1043 1625 y Fq(L)1100 1637 y Fl(T)9 b(L)p 1240 1655 4 100 v 1257 1655 V 1361 1625 a Fr(0)p 999 1658 506 4 v 999 1675 V 1041 1745 a Fq(L)1098 1757 y Fl(B)s(L)p 1240 1774 4 100 v 1257 1774 V 1300 1745 a Fq(L)1357 1757 y Fl(B)s(R)1505 1568 y Fn(\023)1589 1686 y Fp(!)1696 1519 y Fn(0)1696 1668 y(@)1810 1573 y Fq(L)1867 1585 y Fm(00)p 1976 1603 V 1993 1603 V 2075 1573 a Fr(0)p 2195 1603 V 164 w(0)p 1768 1607 639 4 v 1768 1623 V 1826 1693 a Fq(l)1853 1663 y Fl(T)1851 1714 y Fm(10)p 1976 1723 4 100 v 1993 1723 V 2036 1693 a Fq(\025)2084 1705 y Fm(11)p 2195 1723 V 2281 1693 a Fr(0)p 1768 1726 639 4 v 1810 1796 a Fq(L)1867 1808 y Fm(20)p 1976 1826 4 100 v 1993 1826 V 2048 1796 a Fq(l)2073 1808 y Fm(21)p 2195 1826 V 2238 1796 a Fq(L)2295 1808 y Fm(22)2406 1519 y Fn(1)2406 1668 y(A)2479 1686 y Fr(,)2530 1568 y Fn(\022)2635 1625 y Fq(B)2698 1637 y Fl(T)p 2591 1658 204 4 v 2591 1675 V 2632 1745 a Fq(B)2695 1757 y Fl(B)2794 1568 y Fn(\023)2878 1686 y Fp(!)2984 1519 y Fn(0)2984 1668 y(@)3098 1573 y Fq(B)3161 1585 y Fm(0)p 3057 1607 184 4 v 3057 1623 V 3104 1693 a Fq(b)3140 1663 y Fl(T)3140 1714 y Fm(1)p 3057 1726 V 3098 1796 a Fq(B)3161 1808 y Fm(2)3240 1519 y Fn(1)3240 1668 y(A)3313 1686 y Fr(,)27 b(and)938 1879 y Fn(\022)1059 1925 y Fg(^)1043 1942 y Ff(B)1096 1953 y Fd(T)p 999 1969 189 4 v 999 1985 V 1057 2031 a Fg(^)1041 2048 y Ff(B)1094 2059 y Fd(B)1188 1879 y Fn(\023)1269 1992 y Fb(!)1359 1829 y Fn(0)1359 1979 y(@)1490 1880 y Fg(^)1473 1896 y Ff(B)1526 1905 y Fh(0)p 1432 1923 171 4 v 1432 1940 V 1475 1986 a Fg(^)1478 2004 y Ff(b)1508 1980 y Fd(T)1508 2026 y Fh(1)p 1432 2031 V 1490 2077 a Fg(^)1473 2094 y Ff(B)1526 2103 y Fh(2)1603 1829 y Fn(1)1603 1979 y(A)1060 2188 y Fc(where)74 b Ff(b)1376 2165 y Fd(T)1376 2210 y Fh(1)1449 2188 y Fg(and)1583 2171 y(^)1586 2188 y Ff(b)1616 2165 y Fd(T)1616 2210 y Fh(1)1688 2188 y Fg(are)24 b(ro)n(ws)f(and)h Ff(\025)2147 2197 y Fh(11)2236 2188 y Fg(is)g(a)g(scalar)p 3545 2212 4 843 v 353 2215 3195 4 v 351 2514 4 299 v 454 2381 a(6)p 588 2514 V 605 2514 V 817 2215 a Fn(8)817 2290 y(<)817 2440 y(:)891 2244 y( )998 2289 y Ff(B)1051 2298 y Fh(0)p 956 2316 171 4 v 956 2333 V 1002 2390 a Ff(b)1032 2367 y Fd(T)1032 2412 y Fh(1)p 956 2417 V 998 2472 a Ff(B)1051 2481 y Fh(2)1127 2244 y Fn(!)1213 2381 y Fg(=)1287 2219 y Fn(0)1287 2368 y(@)1545 2286 y Ff(L)1593 2256 y Fa(\000)p Fh(1)1593 2308 y(00)1692 2269 y Fg(^)1675 2286 y Ff(B)1728 2295 y Fh(0)p 1360 2313 589 4 v 1360 2330 V 1417 2376 a Fg(^)1419 2393 y Ff(b)1449 2370 y Fd(T)1449 2415 y Fh(1)1514 2393 y Fb(\000)1583 2376 y Fg(^)1584 2393 y Ff(l)1606 2370 y Fd(T)1605 2415 y Fh(10)1670 2393 y Ff(L)1718 2363 y Fa(\000)p Fh(1)1718 2416 y(00)1817 2376 y Fg(^)1801 2393 y Ff(B)1854 2402 y Fh(0)p 1360 2420 V 1418 2466 a Fg(^)1401 2483 y Ff(B)1454 2492 y Fh(2)1505 2483 y Fb(\000)1584 2466 y Fg(^)1575 2483 y Ff(L)1623 2492 y Fh(20)1688 2483 y Ff(L)1736 2453 y Fa(\000)p Fh(1)1736 2506 y(00)1835 2466 y Fg(^)1819 2483 y Ff(B)1872 2492 y Fh(0)1948 2219 y Fn(1)1948 2368 y(A)2033 2381 y Ff(:)2053 2215 y Fn(9)2053 2290 y(=)2053 2440 y(;)p 3545 2514 4 299 v 353 2518 3195 4 v 351 2680 4 163 v 454 2615 a Fg(8)p 588 2680 V 605 2680 V 858 2575 a Ff(b)888 2552 y Fd(T)888 2597 y Fh(1)957 2575 y Fg(:=)19 b Ff(b)1081 2552 y Fd(T)1081 2597 y Fh(1)1130 2575 y Ff(=\025)1206 2584 y Fh(11)858 2656 y Ff(B)911 2665 y Fh(2)966 2656 y Fg(:=)g Ff(B)1113 2665 y Fh(2)1163 2656 y Fb(\000)d Ff(l)1255 2665 y Fh(21)1320 2656 y Ff(b)1350 2633 y Fd(T)1350 2678 y Fh(1)p 3545 2680 V 353 2683 3195 4 v 351 3426 4 743 v 434 2738 a Fg(5b)p 588 3426 V 605 3426 V 309 w Fc(Con)n(tin)n(ue)27 b(with)938 2870 y Fn(\022)1043 2926 y Fq(L)1100 2938 y Fl(T)9 b(L)p 1240 2956 4 100 v 1257 2956 V 1361 2926 a Fr(0)p 999 2959 506 4 v 999 2976 V 1041 3046 a Fq(L)1098 3058 y Fl(B)s(L)p 1240 3076 4 100 v 1257 3076 V 1300 3046 a Fq(L)1357 3058 y Fl(B)s(R)1505 2870 y Fn(\023)1589 2987 y Fp( )1696 2820 y Fn(0)1696 2969 y(@)1810 2875 y Fq(L)1867 2887 y Fm(00)p 1976 2904 V 2058 2875 a Fr(0)p 2178 2904 V 2195 2904 V 181 w(0)p 1768 2908 639 4 v 1826 2978 a Fq(l)1853 2948 y Fl(T)1851 2998 y Fm(10)p 1976 3008 4 100 v 2020 2978 a Fq(\025)2068 2990 y Fm(11)p 2178 3008 V 2195 3008 V 2281 2978 a Fr(0)p 1768 3011 639 4 v 1768 3027 V 1810 3097 a Fq(L)1867 3109 y Fm(20)p 1976 3127 4 100 v 2031 3097 a Fq(l)2056 3109 y Fm(21)p 2178 3127 V 2195 3127 V 2238 3097 a Fq(L)2295 3109 y Fm(22)2406 2820 y Fn(1)2406 2969 y(A)2479 2987 y Fr(,)2530 2870 y Fn(\022)2635 2926 y Fq(B)2698 2938 y Fl(T)p 2591 2959 204 4 v 2591 2976 V 2632 3046 a Fq(B)2695 3058 y Fl(B)2794 2870 y Fn(\023)2878 2987 y Fp( )2984 2820 y Fn(0)2984 2969 y(@)3098 2875 y Fq(B)3161 2887 y Fm(0)p 3057 2908 184 4 v 3104 2978 a Fq(b)3140 2948 y Fl(T)3140 2998 y Fm(1)p 3057 3011 V 3057 3027 V 3098 3097 a Fq(B)3161 3109 y Fm(2)3240 2820 y Fn(1)3240 2969 y(A)3313 2987 y Fr(,)27 b(and)938 3180 y Fn(\022)1059 3226 y Fg(^)1043 3243 y Ff(B)1096 3254 y Fd(T)p 999 3270 189 4 v 999 3286 V 1057 3333 a Fg(^)1041 3349 y Ff(B)1094 3360 y Fd(B)1188 3180 y Fn(\023)1269 3293 y Fb( )1359 3130 y Fn(0)1359 3280 y(@)1490 3181 y Fg(^)1473 3198 y Ff(B)1526 3207 y Fh(0)p 1432 3225 171 4 v 1475 3271 a Fg(^)1478 3288 y Ff(b)1508 3265 y Fd(T)1508 3310 y Fh(1)p 1432 3315 V 1432 3332 V 1490 3378 a Fg(^)1473 3395 y Ff(B)1526 3404 y Fh(2)1603 3130 y Fn(1)1603 3280 y(A)p 3545 3426 4 743 v 353 3429 3195 4 v 351 3728 4 299 v 454 3595 a Fg(7)p 588 3728 V 605 3728 V 817 3429 a Fn(8)817 3504 y(<)817 3653 y(:)891 3458 y( )998 3503 y Ff(B)1051 3512 y Fh(0)p 956 3530 171 4 v 1002 3587 a Ff(b)1032 3564 y Fd(T)1032 3609 y Fh(1)p 956 3614 V 956 3631 V 998 3686 a Ff(B)1051 3695 y Fh(2)1127 3458 y Fn(!)1213 3595 y Fg(=)1287 3433 y Fn(0)1287 3582 y(@)1955 3499 y Ff(L)2003 3470 y Fa(\000)p Fh(1)2003 3522 y(00)2102 3483 y Fg(^)2086 3499 y Ff(B)2139 3508 y Fh(0)p 1360 3526 1409 4 v 1740 3590 a Ff(\025)1781 3560 y Fa(\000)p Fh(1)1781 3613 y(11)1864 3590 y Fg(\()1888 3573 y(^)1891 3590 y Ff(b)1921 3567 y Fd(T)1921 3612 y Fh(1)1986 3590 y Fb(\000)15 b Ff(l)2078 3567 y Fd(T)2077 3612 y Fh(10)2142 3590 y Ff(L)2190 3560 y Fa(\000)p Fh(1)2190 3613 y(00)2289 3573 y Fg(^)2273 3590 y Ff(B)2326 3599 y Fh(0)2361 3590 y Fg(\))p 1360 3617 V 1360 3634 V 1418 3681 a(^)1401 3697 y Ff(B)1454 3706 y Fh(2)1505 3697 y Fb(\000)g Ff(L)1623 3706 y Fh(20)1688 3697 y Ff(L)1736 3668 y Fa(\000)p Fh(1)1736 3720 y(00)1835 3681 y Fg(^)1819 3697 y Ff(B)1872 3706 y Fh(0)1922 3697 y Fb(\000)h Ff(l)2014 3706 y Fh(21)2079 3697 y Ff(\025)2120 3668 y Fa(\000)p Fh(1)2120 3720 y(11)2203 3697 y Fg(\()2227 3680 y(^)2230 3697 y Ff(b)2260 3674 y Fd(T)2260 3719 y Fh(1)2325 3697 y Fb(\000)f Ff(l)2417 3674 y Fd(T)2416 3719 y Fh(10)2481 3697 y Ff(L)2529 3668 y Fa(\000)p Fh(1)2529 3720 y(00)2628 3681 y Fg(^)2612 3697 y Ff(B)2665 3706 y Fh(0)2700 3697 y Fg(\))2769 3433 y Fn(1)2769 3582 y(A)2853 3595 y Ff(:)2873 3429 y Fn(9)2873 3504 y(=)2873 3653 y(;)p 3545 3728 4 299 v 353 3731 3195 4 v 351 3931 4 200 v 454 3848 a Fg(2)p 588 3931 V 605 3931 V 817 3735 a Fn(\032\022)984 3798 y Ff(B)1037 3809 y Fd(T)p 940 3824 189 4 v 940 3841 V 982 3896 a Ff(B)1035 3907 y Fd(B)1129 3735 y Fn(\023)1210 3848 y Fg(=)1284 3735 y Fn(\022)1582 3798 y Ff(L)1630 3809 y Fd(T)8 b(L)1721 3773 y Fa(\000)p Fh(1)1820 3781 y Fg(^)1803 3798 y Ff(B)1856 3809 y Fd(T)p 1345 3824 797 4 v 1345 3841 V 1403 3887 a Fg(^)1387 3904 y Ff(B)1440 3915 y Fd(B)1509 3904 y Fb(\000)15 b Ff(L)1627 3915 y Fd(B)r(L)1722 3904 y Fg(\()p Ff(L)1797 3915 y Fd(T)8 b(L)1888 3879 y Fa(\000)p Fh(1)1987 3887 y Fg(^)1971 3904 y Ff(B)2024 3915 y Fd(T)2073 3904 y Fg(\))2142 3735 y Fn(\023\033)p 3545 3931 4 200 v 353 3934 3195 4 v 351 4013 4 79 v 588 4013 V 605 4013 V 656 3989 a Fc(enddo)p 3545 4013 V 353 4016 3195 4 v 351 4216 4 200 v 426 4132 a Fg(2,3)p 588 4216 V 605 4216 V 656 4020 a Fn(\032)q(\022\022)884 4082 y Ff(B)937 4093 y Fd(T)p 841 4109 189 4 v 841 4126 V 882 4181 a Ff(B)935 4192 y Fd(B)1030 4020 y Fn(\023)1110 4132 y Fg(=)1185 4020 y Fn(\022)1482 4082 y Ff(L)1530 4093 y Fd(T)g(L)1621 4057 y Fa(\000)p Fh(1)1720 4065 y Fg(^)1704 4082 y Ff(B)1757 4093 y Fd(T)p 1246 4109 797 4 v 1246 4126 V 1304 4172 a Fg(^)1288 4189 y Ff(B)1341 4200 y Fd(B)1409 4189 y Fb(\000)16 b Ff(L)1528 4200 y Fd(B)r(L)1623 4189 y Fg(\()p Ff(L)1698 4200 y Fd(T)8 b(L)1789 4164 y Fa(\000)p Fh(1)1888 4172 y Fg(^)1871 4189 y Ff(B)1924 4200 y Fd(T)1973 4189 y Fg(\))2042 4020 y Fn(\023\023)2180 4132 y Fb(^)16 b(:)c Fg(\()p Ff(m)p Fg(\()p Ff(L)2466 4143 y Fd(T)c(L)2558 4132 y Fg(\))20 b Fb(6)p Fg(=)f Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2872 4020 y Fn(\033)p 3545 4216 4 200 v 353 4219 3195 4 v 351 4318 4 100 v 434 4285 a Fg(1b)p 588 4318 V 605 4318 V 656 4222 a Fn(\010)705 4285 y Ff(B)k Fg(:=)c Ff(L)923 4262 y Fa(\000)p Fh(1)1022 4268 y Fg(^)1005 4285 y Ff(B)1062 4222 y Fn(\011)p 3545 4318 V 353 4322 3195 4 v 458 4564 a Fr(Figure)26 b(6:)37 b(W)-7 b(orksheet)27 b(for)g(deriving)d(un)n (blo)r(c)n(k)n(ed)i(algorithm)e(for)j Fq(B)g Fr(:=)c Fq(L)2832 4534 y Fj(\000)p Fm(1)2920 4564 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(2\).)1908 5356 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 536 -12 a Fk(P)m(artition)55 b Fq(L)23 b Fp(!)1153 -130 y Fn(\022)1258 -73 y Fq(L)1315 -61 y Fl(T)9 b(L)p 1455 -43 4 100 v 1471 -43 V 1576 -73 a Fr(0)p 1214 -40 506 4 v 1214 -23 V 1256 46 a Fq(L)1313 58 y Fl(B)s(L)p 1455 76 4 100 v 1471 76 V 1514 46 a Fq(L)1571 58 y Fl(B)s(R)1720 -130 y Fn(\023)1809 -12 y Fr(and)27 b Fq(B)g Fp(!)2166 -130 y Fn(\022)2272 -73 y Fq(B)2335 -61 y Fl(T)p 2228 -40 204 4 v 2228 -23 V 2269 46 a Fq(B)2332 58 y Fl(B)2431 -130 y Fn(\023)679 146 y Fk(where)59 b Fq(L)1044 158 y Fl(T)9 b(L)1169 146 y Fr(is)26 b(0)18 b Fp(\002)g Fr(0)28 b(and)f Fq(B)1689 158 y Fl(T)1769 146 y Fr(has)g(0)g(ro)n(ws)536 387 y Fk(while)54 b Fq(m)p Fr(\()p Fq(L)972 399 y Fl(T)9 b(L)1069 387 y Fr(\))24 b Fp(6)p Fr(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\))56 b Fk(do)709 487 y(Repartition)852 634 y Fn(\022)957 691 y Fq(L)1014 703 y Fl(T)9 b(L)p 1153 720 4 100 v 1170 720 V 1274 691 a Fr(0)p 913 724 506 4 v 913 740 V 954 810 a Fq(L)1011 822 y Fl(B)s(L)p 1153 840 4 100 v 1170 840 V 1213 810 a Fq(L)1270 822 y Fl(B)s(R)1419 634 y Fn(\023)1503 751 y Fp(!)1609 584 y Fn(0)1609 734 y(@)1723 639 y Fq(L)1780 651 y Fm(00)p 1890 669 V 1906 669 V 1988 639 a Fr(0)p 2108 669 V 164 w(0)p 1681 672 639 4 v 1681 689 V 1739 759 a Fq(l)1766 729 y Fl(T)1764 779 y Fm(10)p 1890 789 4 100 v 1906 789 V 1949 759 a Fq(\025)1997 771 y Fm(11)p 2108 789 V 2194 759 a Fr(0)p 1681 792 639 4 v 1723 862 a Fq(L)1780 874 y Fm(20)p 1890 892 4 100 v 1906 892 V 1961 862 a Fq(l)1986 874 y Fm(21)p 2108 892 V 2151 862 a Fq(L)2208 874 y Fm(22)2320 584 y Fn(1)2320 734 y(A)2420 751 y Fr(and)2581 634 y Fn(\022)2686 691 y Fq(B)2749 703 y Fl(T)p 2642 724 204 4 v 2642 740 V 2684 810 a Fq(B)2747 822 y Fl(B)2846 634 y Fn(\023)2930 751 y Fp(!)3036 584 y Fn(0)3036 734 y(@)3150 639 y Fq(B)3213 651 y Fm(0)p 3108 672 184 4 v 3108 689 V 3156 759 a Fq(b)3192 729 y Fl(T)3192 779 y Fm(1)p 3108 792 V 3150 862 a Fq(B)3213 874 y Fm(2)3292 584 y Fn(1)3292 734 y(A)995 961 y Fk(where)87 b Fq(b)1367 931 y Fl(T)1367 982 y Fm(1)1446 961 y Fr(is)27 b(a)g(ro)n(w)f(and)i Fq(\025)1968 973 y Fm(11)2066 961 y Fr(is)f(a)g(scalar)p 709 1063 2341 4 v 709 1055 V 750 1161 a Fq(b)786 1131 y Fl(T)786 1181 y Fm(1)861 1161 y Fr(:=)c Fq(b)1008 1131 y Fl(T)1008 1181 y Fm(1)1060 1161 y Fq(=\025)1150 1173 y Fm(11)750 1260 y Fq(B)813 1272 y Fm(2)873 1260 y Fr(:=)g Fq(B)1047 1272 y Fm(2)1103 1260 y Fp(\000)18 b Fq(l)1211 1272 y Fm(21)1281 1260 y Fq(b)1317 1230 y Fl(T)1317 1281 y Fm(1)p 709 1362 V 709 1354 V 709 1460 a Fk(Con)m(tin)m(ue)31 b(with)852 1591 y Fn(\022)957 1647 y Fq(L)1014 1659 y Fl(T)9 b(L)p 1153 1677 4 100 v 1170 1677 V 1274 1647 a Fr(0)p 913 1681 506 4 v 913 1697 V 954 1767 a Fq(L)1011 1779 y Fl(B)s(L)p 1153 1797 4 100 v 1170 1797 V 1213 1767 a Fq(L)1270 1779 y Fl(B)s(R)1419 1591 y Fn(\023)1503 1708 y Fp( )1609 1541 y Fn(0)1609 1691 y(@)1723 1596 y Fq(L)1780 1608 y Fm(00)p 1890 1626 V 1971 1596 a Fr(0)p 2091 1626 V 2108 1626 V 181 w(0)p 1681 1629 639 4 v 1739 1699 a Fq(l)1766 1669 y Fl(T)1764 1720 y Fm(10)p 1890 1729 4 100 v 1933 1699 a Fq(\025)1981 1711 y Fm(11)p 2091 1729 V 2108 1729 V 2194 1699 a Fr(0)p 1681 1732 639 4 v 1681 1749 V 1723 1819 a Fq(L)1780 1831 y Fm(20)p 1890 1848 4 100 v 1945 1819 a Fq(l)1970 1831 y Fm(21)p 2091 1848 V 2108 1848 V 2151 1819 a Fq(L)2208 1831 y Fm(22)2320 1541 y Fn(1)2320 1691 y(A)2420 1708 y Fr(and)2581 1591 y Fn(\022)2686 1647 y Fq(B)2749 1659 y Fl(T)p 2642 1681 204 4 v 2642 1697 V 2684 1767 a Fq(B)2747 1779 y Fl(B)2846 1591 y Fn(\023)2930 1708 y Fp( )3036 1541 y Fn(0)3036 1691 y(@)3150 1596 y Fq(B)3213 1608 y Fm(0)p 3108 1629 184 4 v 3156 1699 a Fq(b)3192 1669 y Fl(T)3192 1720 y Fm(1)p 3108 1732 V 3108 1749 V 3150 1819 a Fq(B)3213 1831 y Fm(2)3292 1541 y Fn(1)3292 1691 y(A)536 1918 y Fk(enddo)878 2190 y Fr(Figure)26 b(7:)37 b(Un)n(blo)r(c)n(k)n(ed)26 b(algorithm)d(for)k Fq(B)h Fr(:=)22 b Fq(L)2411 2160 y Fj(\000)p Fm(1)2500 2190 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(2\).)0 2457 y Fk(Algorithm)0 2610 y Fr(An)31 b(annotated)f(algorithm,)e(in)i(whic)n (h)g(the)h(states)f(of)h(the)g(v)-5 b(ariables)28 b(are)h(presen)n(ted) i(as)f(w)n(ell)e(as)i(the)i(op)r(erations)c(to)j(b)r(e)0 2710 y(executed,)37 b(is)d(presen)n(ted)h(in)f(Fig.)h(6.)59 b(By)35 b(remo)n(ving)d(all)g(annotations)i(and)h(remo)n(ving)c(an)n(y) k(op)r(eration)e(in)n(v)n(olving)e(the)0 2810 y(\014cticious)366 2789 y(^)346 2810 y Fq(B)5 b Fr(,)27 b(the)h(\014nal)f(algorithm)c(is)k (giv)n(en)e(Fig.)i(7.)0 3025 y Fk(2.2.2)94 b(V)-8 b(arian)m(t)33 b(2:)42 b(blo)s(c)m(k)m(ed)32 b(algorithm)0 3178 y Fr(Next,)d(w)n(e)f (will)e(repartition)f(the)k(matrices)d(so)h(that)i(the)g(submatrices)c (that)k(are)f(mo)n(v)n(ed)e(b)r(et)n(w)n(een)i(the)h(v)-5 b(arious)26 b(parts)i(of)0 3278 y(the)g(matrix)d(are)i(submatrices.)34 b(This)26 b(will)f(lead)h(to)i(so-called)c(blo)r(c)n(k)n(ed)i (algorithms.)125 3377 y(Repartition)121 3560 y Fn(\022)226 3616 y Fq(L)283 3628 y Fl(T)9 b(L)p 422 3646 4 100 v 439 3646 V 543 3616 a Fr(0)p 182 3649 506 4 v 182 3666 V 223 3736 a Fq(L)280 3748 y Fl(B)s(L)p 422 3766 4 100 v 439 3766 V 482 3736 a Fq(L)539 3748 y Fl(B)s(R)687 3560 y Fn(\023)772 3677 y Fp(!)878 3510 y Fn(0)878 3659 y(@)992 3565 y Fq(L)1049 3577 y Fm(00)p 1159 3595 V 1175 3595 V 1261 3565 a Fr(0)p 1385 3595 V 168 w(0)p 950 3598 647 4 v 950 3615 V 992 3684 a Fq(L)1049 3696 y Fm(10)p 1159 3714 4 100 v 1175 3714 V 1218 3684 a Fq(L)1275 3696 y Fm(11)p 1385 3714 V 1471 3684 a Fr(0)p 950 3717 647 4 v 992 3787 a Fq(L)1049 3799 y Fm(20)p 1159 3817 4 100 v 1175 3817 V 1234 3787 a Fq(l)1259 3799 y Fm(21)p 1385 3817 V 1428 3787 a Fq(L)1485 3799 y Fm(22)1597 3510 y Fn(1)1597 3659 y(A)1683 3677 y Fq(;)1803 3560 y Fn(\022)1908 3616 y Fq(B)1971 3628 y Fl(T)p 1864 3649 204 4 v 1864 3666 V 1906 3736 a Fq(B)1969 3748 y Fl(B)2067 3560 y Fn(\023)2151 3677 y Fp(!)2257 3510 y Fn(0)2257 3659 y(@)2372 3565 y Fq(B)2435 3577 y Fm(0)p 2330 3598 184 4 v 2330 3615 V 2372 3684 a Fq(B)2435 3696 y Fm(1)p 2330 3717 V 2372 3787 a Fq(B)2435 3799 y Fm(2)2513 3510 y Fn(1)2513 3659 y(A)2600 3677 y Fq(;)97 b Fr(and)2950 3535 y Fn( )3080 3595 y Fr(^)3060 3616 y Fq(B)3123 3628 y Fl(T)p 3016 3649 204 4 v 3016 3666 V 3077 3724 a Fr(^)3058 3745 y Fq(B)3121 3757 y Fl(B)3219 3535 y Fn(!)3308 3677 y Fp(!)3414 3485 y Fn(0)3414 3631 y(B)3414 3684 y(@)3548 3539 y Fr(^)3528 3560 y Fq(B)3591 3572 y Fm(0)p 3487 3593 184 4 v 3487 3610 V 3548 3668 a Fr(^)3528 3689 y Fq(B)3591 3701 y Fm(1)p 3487 3722 V 3548 3780 a Fr(^)3528 3801 y Fq(B)3591 3813 y Fm(2)3670 3485 y Fn(1)3670 3631 y(C)3670 3684 y(A)3756 3677 y Fq(;)0 3997 y Fr(where)27 b Fq(L)297 4009 y Fm(11)395 3997 y Fr(is)f(a)h Fq(b)18 b Fp(\002)g Fq(b)28 b Fr(matrix,)d(and)i Fq(B)1265 4009 y Fm(1)1330 3997 y Fr(and)1511 3976 y(^)1492 3997 y Fq(B)1555 4009 y Fm(1)1619 3997 y Fr(ha)n(v)n(e)g Fq(b)g Fr(ro)n(ws.)36 b(\\Mo)n(ving)25 b(the)j(double)e(lines")f(is)i(represen)n(ted)f(b)n(y) 139 4180 y Fn(\022)244 4236 y Fq(L)301 4248 y Fl(T)9 b(L)p 441 4266 4 100 v 457 4266 V 562 4236 a Fr(0)p 200 4269 506 4 v 200 4286 V 242 4356 a Fq(L)299 4368 y Fl(B)s(L)p 441 4385 4 100 v 457 4385 V 500 4356 a Fq(L)557 4368 y Fl(B)s(R)706 4180 y Fn(\023)790 4297 y Fp( )896 4130 y Fn(0)896 4279 y(@)1010 4185 y Fq(L)1067 4197 y Fm(00)p 1177 4214 V 1263 4185 a Fr(0)p 1387 4214 V 1404 4214 V 184 w(0)p 969 4218 647 4 v 1010 4287 a Fq(L)1067 4299 y Fm(10)p 1177 4317 4 100 v 1220 4287 a Fq(L)1277 4299 y Fm(11)p 1387 4317 V 1403 4317 V 1489 4287 a Fr(0)p 969 4321 647 4 v 969 4337 V 1010 4407 a Fq(L)1067 4419 y Fm(20)p 1177 4437 4 100 v 1220 4407 a Fq(L)1277 4419 y Fm(21)p 1387 4437 V 1403 4437 V 1447 4407 a Fq(L)1504 4419 y Fm(22)1615 4130 y Fn(1)1615 4279 y(A)1701 4297 y Fq(;)1821 4180 y Fn(\022)1926 4236 y Fq(B)1989 4248 y Fl(T)p 1882 4269 204 4 v 1882 4286 V 1924 4356 a Fq(B)1987 4368 y Fl(B)2086 4180 y Fn(\023)2170 4297 y Fp( )2276 4130 y Fn(0)2276 4279 y(@)2390 4185 y Fq(B)2453 4197 y Fm(0)p 2349 4218 184 4 v 2390 4287 a Fq(B)2453 4299 y Fm(1)p 2349 4321 V 2349 4337 V 2390 4407 a Fq(B)2453 4419 y Fm(2)2532 4130 y Fn(1)2532 4279 y(A)2618 4297 y Fq(;)97 b Fr(and)2969 4155 y Fn( )3098 4215 y Fr(^)3078 4236 y Fq(B)3141 4248 y Fl(T)p 3035 4269 204 4 v 3035 4286 V 3096 4343 a Fr(^)3076 4364 y Fq(B)3139 4376 y Fl(B)3238 4155 y Fn(!)3326 4297 y Fp( )3433 4105 y Fn(0)3433 4251 y(B)3433 4304 y(@)3566 4159 y Fr(^)3547 4180 y Fq(B)3610 4192 y Fm(0)p 3505 4213 184 4 v 3566 4271 a Fr(^)3547 4292 y Fq(B)3610 4304 y Fm(1)p 3505 4325 V 3505 4342 V 3566 4399 a Fr(^)3547 4420 y Fq(B)3610 4432 y Fm(2)3688 4105 y Fn(1)3688 4251 y(C)3688 4304 y(A)0 4596 y Fr(to)n(w)n(ards)31 b(the)i(b)r(ottom)f(of)h(the)h(lo)r(op.)51 b(The)33 b(idea)e(here)i(is) f(that)h(the)g(double)f(lines)f(ha)n(v)n(e)g(seman)n(tic)g(meaning)f (and)j(sho)n(w)0 4696 y(that)28 b(up)r(on)g(repartitioning)326 4883 y Fq(L)383 4895 y Fl(T)9 b(L)504 4883 y Fp(!)23 b Fq(L)667 4895 y Fm(00)p 881 4913 4 101 v 898 4913 V 1052 4883 a Fq(L)1109 4895 y Fl(T)9 b(R)1234 4883 y Fp(!)1340 4816 y Fn(\000)1420 4882 y Fr(0)p 1501 4912 4 100 v 82 w(0)1627 4816 y Fn(\001)p 180 4917 1639 4 v 180 4933 V 221 5055 a Fq(L)278 5067 y Fl(B)s(L)403 5055 y Fp(!)509 4938 y Fn(\022)612 5003 y Fq(L)669 5015 y Fm(10)p 571 5036 210 4 v 612 5106 a Fq(L)669 5118 y Fm(20)780 4938 y Fn(\023)p 881 5136 4 203 v 898 5136 V 941 5055 a Fq(L)998 5067 y Fl(B)s(R)1128 5055 y Fp(!)1234 4938 y Fn(\022)1337 5003 y Fq(L)1394 5015 y Fm(11)p 1504 5033 4 100 v 1589 5003 a Fr(0)p 1295 5036 420 4 v 1337 5106 a Fq(L)1394 5118 y Fm(21)p 1504 5136 4 100 v 1547 5106 a Fq(L)1604 5118 y Fm(22)1715 4938 y Fn(\023)1818 4995 y Fq(;)2084 4883 y(B)2147 4895 y Fl(T)2222 4883 y Fp(!)24 b Fq(B)2392 4895 y Fm(0)p 1938 4916 638 4 v 1938 4933 V 1979 5055 a Fq(B)2042 5067 y Fl(B)2122 5055 y Fp(!)2228 4938 y Fn(\022)2331 5003 y Fq(B)2394 5015 y Fm(1)p 2289 5036 184 4 v 2331 5105 a Fq(B)2394 5117 y Fm(2)2473 4938 y Fn(\023)2575 4995 y Fq(;)97 b Fr(and)3078 4858 y(^)3059 4879 y Fq(B)3122 4891 y Fl(T)3197 4879 y Fp(!)3323 4858 y Fr(^)3303 4879 y Fq(B)3366 4891 y Fm(0)p 2912 4912 638 4 v 2912 4928 V 2973 5038 a Fr(^)2954 5059 y Fq(B)3017 5071 y Fl(B)3097 5059 y Fp(!)3203 4942 y Fn(\022)3325 4986 y Fr(^)3305 5007 y Fq(B)3368 5019 y Fm(1)p 3264 5040 184 4 v 3325 5098 a Fr(^)3305 5119 y Fq(B)3368 5131 y Fm(2)3447 4942 y Fn(\023)3550 4995 y Fq(:)179 b Fr(\(18\))1908 5356 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 0 -60 a Fr(T)-7 b(o)n(w)n(ards)25 b(the)j(end)g(of)g(the)g(lo)r(op)e(the)i(quadran)n(ts)e(of)i(the)g (partitioned)d(matrix)g(are)h(rede\014ned)i(lik)n(e)347 183 y Fq(L)404 195 y Fl(T)9 b(L)525 183 y Fp( )631 66 y Fn(\022)733 130 y Fq(L)790 142 y Fm(00)p 900 160 4 100 v 986 130 a Fr(0)p 692 164 420 4 v 733 233 a Fq(L)790 245 y Fm(01)p 900 263 4 100 v 943 233 a Fq(L)1000 245 y Fm(11)1112 66 y Fn(\023)p 1213 263 4 203 v 1230 263 V 1272 183 a Fq(L)1329 195 y Fl(T)g(R)1454 183 y Fp( )1561 66 y Fn(\022)1663 130 y Fr(0)p 1622 164 125 4 v 1663 233 a(0)1746 66 y Fn(\023)p 306 267 1544 4 v 306 283 V 368 354 a Fq(L)425 366 y Fl(B)s(L)550 354 y Fp( )656 286 y Fn(\000)736 353 y Fq(L)793 365 y Fm(20)p 902 383 4 100 v 946 353 a Fq(L)1003 365 y Fm(21)1114 286 y Fn(\001)p 1213 384 4 101 v 1230 384 V 1330 354 a Fq(L)1387 366 y Fl(B)s(R)1517 354 y Fp( )23 b Fq(L)1680 366 y Fm(22)1849 243 y Fq(;)2010 183 y(B)2073 195 y Fl(T)2149 183 y Fp( )2255 66 y Fn(\022)2357 131 y Fq(B)2420 143 y Fm(0)p 2316 164 184 4 v 2357 234 a Fq(B)2420 246 y Fm(1)2499 66 y Fn(\023)p 1969 267 633 4 v 1969 284 V 2110 353 a Fq(B)2173 365 y Fl(B)2254 353 y Fp( )g Fq(B)2423 365 y Fm(2)2602 243 y Fq(;)97 b Fr(and)3000 158 y(^)2980 179 y Fq(B)3043 191 y Fl(T)3118 179 y Fp( )3224 62 y Fn(\022)3347 105 y Fr(^)3327 126 y Fq(B)3390 138 y Fm(0)p 3285 160 184 4 v 3347 217 a Fr(^)3327 238 y Fq(B)3390 250 y Fm(1)3469 62 y Fn(\023)p 2938 271 633 4 v 2938 288 V 3100 346 a Fr(^)3080 367 y Fq(B)3143 379 y Fl(B)3223 367 y Fp( )3349 346 y Fr(^)3329 367 y Fq(B)3392 379 y Fm(0)3571 243 y Fq(:)158 b Fr(\(19\))0 579 y Fk(Step)32 b(6:)41 b(State)33 b(after)f(repartitioning)0 732 y(Step)g(7:)41 b(State)33 b(after)f(mo)m(ving)f(the)g(double)g(lines)0 885 y(Step)h(8:)41 b(Determining)30 b(the)i(up)s(date)g(to)f(the)h(exp)s(osed)f (submatrices)0 1038 y(Algorithm)0 1192 y Fr(An)g(annotated)f (algorithm,)e(in)i(whic)n(h)g(the)h(states)f(of)h(the)g(v)-5 b(ariables)28 b(are)h(presen)n(ted)i(as)f(w)n(ell)e(as)i(the)i(op)r (erations)c(to)j(b)r(e)0 1291 y(executed,)37 b(is)d(presen)n(ted)h(in)f (Fig.)h(8.)59 b(By)35 b(remo)n(ving)d(all)g(annotations)i(and)h(remo)n (ving)c(an)n(y)k(op)r(eration)e(in)n(v)n(olving)e(the)0 1391 y(\014cticious)366 1370 y(^)346 1391 y Fq(B)5 b Fr(,)27 b(the)h(\014nal)f(algorithm)c(is)k(giv)n(en)e(Fig.)i(9.)0 1665 y Fs(3)135 b(Algorithms)42 b(Deriv)l(ed)j(from)g(a)g(P)l (artitioning)f(of)h Fi(B)1908 5356 y Fr(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 349 343 3203 4 v 347 430 4 87 v 399 406 a Fg(Step)p 584 430 V 601 430 V 117 w(Annotated)26 b(Algorithm:)64 b Ff(B)23 b Fg(:=)c Ff(L)1619 383 y Fa(\000)p Fh(1)1718 389 y Fg(^)1702 406 y Ff(B)p 3549 430 V 349 433 3203 4 v 349 450 V 347 549 4 100 v 432 516 a Fg(1a)p 584 549 V 601 549 V 652 453 a Fn(\010)701 516 y Ff(B)k Fg(=)868 499 y(^)851 516 y Ff(B)908 453 y Fn(\011)p 3549 549 V 349 553 3203 4 v 347 838 4 286 v 450 669 a Fg(4)p 584 838 V 601 838 V 167 w Fc(P)n(artition)47 b Ff(L)20 b Fb(!)1178 556 y Fn(\022)1282 619 y Ff(L)1330 630 y Fd(T)8 b(L)p 1463 642 4 79 v 1480 642 V 1579 619 a Fg(0)p 1239 646 473 4 v 1239 662 V 1280 717 a Ff(L)1328 728 y Fd(B)r(L)p 1463 741 4 79 v 1480 741 V 1523 717 a Ff(L)1571 728 y Fd(B)r(R)1711 556 y Fn(\023)1772 669 y Fg(,)23 b Ff(B)h Fb(!)1982 556 y Fn(\022)2086 619 y Ff(B)2139 630 y Fd(T)p 2043 646 189 4 v 2043 662 V 2084 717 a Ff(B)2137 728 y Fd(B)2232 556 y Fn(\023)2316 669 y Fg(and)2470 652 y(^)2454 669 y Ff(B)f Fb(!)2620 556 y Fn(\022)2741 602 y Fg(^)2725 619 y Ff(B)2778 630 y Fd(T)p 2681 646 V 2681 662 V 2739 708 a Fg(^)2723 725 y Ff(B)2776 736 y Fd(B)2870 556 y Fn(\023)774 815 y Fc(where)51 b Ff(L)1085 826 y Fd(T)8 b(L)1199 815 y Fg(is)24 b(0)16 b Fb(\002)f Fg(0,)24 b(and)g Ff(B)1660 826 y Fd(T)1732 815 y Fg(and)1886 798 y(^)1870 815 y Ff(B)1923 826 y Fd(T)1995 815 y Fg(ha)n(v)n(e)h(0)e (ro)n(ws)p 3549 838 4 286 v 349 842 3203 4 v 347 1041 4 200 v 450 958 a(2)p 584 1041 V 601 1041 V 652 845 a Fn(\032\022)819 908 y Ff(B)872 919 y Fd(T)p 776 935 189 4 v 776 951 V 817 1007 a Ff(B)870 1018 y Fd(B)965 845 y Fn(\023)1045 958 y Fg(=)1120 845 y Fn(\022)1417 908 y Ff(L)1465 919 y Fd(T)8 b(L)1556 883 y Fa(\000)p Fh(1)1655 891 y Fg(^)1639 908 y Ff(B)1692 919 y Fd(T)p 1181 935 797 4 v 1181 951 V 1239 997 a Fg(^)1222 1014 y Ff(B)1275 1025 y Fd(B)1344 1014 y Fb(\000)16 b Ff(L)1463 1025 y Fd(B)r(L)1557 1014 y Fg(\()p Ff(L)1632 1025 y Fd(T)8 b(L)1724 989 y Fa(\000)p Fh(1)1823 997 y Fg(^)1806 1014 y Ff(B)1859 1025 y Fd(T)1908 1014 y Fg(\))1977 845 y Fn(\023\033)p 3549 1041 4 200 v 349 1044 3203 4 v 347 1123 4 79 v 450 1100 a Fg(3)p 584 1123 V 601 1123 V 167 w Fc(while)49 b Ff(m)p Fg(\()p Ff(L)1023 1111 y Fd(T)8 b(L)1114 1100 y Fg(\))20 b Fb(6)p Fg(=)g Ff(m)p Fg(\()p Ff(L)p Fg(\))48 b Fc(do)p 3549 1123 V 349 1127 3203 4 v 347 1326 4 200 v 422 1243 a Fg(2,3)p 584 1326 V 601 1326 V 813 1130 a Fn(\032\022\022)1041 1193 y Ff(B)1094 1204 y Fd(T)p 997 1220 189 4 v 997 1236 V 1039 1291 a Ff(B)1092 1302 y Fd(B)1186 1130 y Fn(\023)1267 1243 y Fg(=)1341 1130 y Fn(\022)1639 1193 y Ff(L)1687 1204 y Fd(T)8 b(L)1778 1168 y Fa(\000)p Fh(1)1877 1176 y Fg(^)1860 1193 y Ff(B)1913 1204 y Fd(T)p 1403 1220 797 4 v 1403 1236 V 1461 1282 a Fg(^)1444 1299 y Ff(B)1497 1310 y Fd(B)1566 1299 y Fb(\000)15 b Ff(L)1684 1310 y Fd(B)r(L)1779 1299 y Fg(\()p Ff(L)1854 1310 y Fd(T)8 b(L)1945 1274 y Fa(\000)p Fh(1)2044 1282 y Fg(^)2028 1299 y Ff(B)2081 1310 y Fd(T)2130 1299 y Fg(\))2199 1130 y Fn(\023\023)2337 1243 y Fb(^)15 b Fg(\()q Ff(m)p Fg(\()p Ff(L)2564 1254 y Fd(T)8 b(L)2655 1243 y Fg(\))20 b Fb(6)p Fg(=)g Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2970 1130 y Fn(\033)p 3549 1326 4 200 v 349 1329 3203 4 v 347 2250 4 921 v 432 1384 a Fg(5a)p 584 2250 V 601 2250 V 311 w Fc(Determine)28 b(blo)r(c)n(k)g (size)g Ff(b)813 1463 y Fc(Repartition)934 1607 y Fn(\022)1039 1664 y Fq(L)1096 1676 y Fl(T)9 b(L)p 1236 1694 4 100 v 1253 1694 V 1357 1664 a Fr(0)p 995 1697 506 4 v 995 1714 V 1037 1783 a Fq(L)1094 1795 y Fl(B)s(L)p 1236 1813 4 100 v 1253 1813 V 1296 1783 a Fq(L)1353 1795 y Fl(B)s(R)1501 1607 y Fn(\023)1585 1724 y Fp(!)1691 1557 y Fn(0)1691 1707 y(@)1806 1612 y Fq(L)1863 1624 y Fm(00)p 1972 1642 V 1989 1642 V 2075 1612 a Fr(0)p 2199 1642 V 168 w(0)p 1764 1645 647 4 v 1764 1662 V 1806 1732 a Fq(L)1863 1744 y Fm(10)p 1972 1762 4 100 v 1989 1762 V 2032 1732 a Fq(L)2089 1744 y Fm(11)p 2199 1762 V 2285 1732 a Fr(0)p 1764 1765 647 4 v 1806 1835 a Fq(L)1863 1847 y Fm(20)p 1972 1865 4 100 v 1989 1865 V 2048 1835 a Fq(l)2073 1847 y Fm(21)p 2199 1865 V 2242 1835 a Fq(L)2299 1847 y Fm(22)2410 1557 y Fn(1)2410 1707 y(A)2483 1724 y Fr(,)2534 1607 y Fn(\022)2639 1664 y Fq(B)2702 1676 y Fl(T)p 2595 1697 204 4 v 2595 1714 V 2636 1783 a Fq(B)2699 1795 y Fl(B)2798 1607 y Fn(\023)2882 1724 y Fp(!)2988 1557 y Fn(0)2988 1707 y(@)3102 1612 y Fq(B)3165 1624 y Fm(0)p 3061 1645 184 4 v 3061 1662 V 3102 1732 a Fq(B)3165 1744 y Fm(1)p 3061 1765 V 3102 1835 a Fq(B)3165 1847 y Fm(2)3244 1557 y Fn(1)3244 1707 y(A)3317 1724 y Fr(,)27 b(and)934 1918 y Fn(\022)1055 1964 y Fg(^)1039 1980 y Ff(B)1092 1991 y Fd(T)p 995 2007 189 4 v 995 2024 V 1053 2070 a Fg(^)1037 2087 y Ff(B)1090 2098 y Fd(B)1184 1918 y Fn(\023)1265 2031 y Fb(!)1355 1868 y Fn(0)1355 2017 y(@)1486 1919 y Fg(^)1469 1936 y Ff(B)1522 1945 y Fh(0)p 1428 1963 171 4 v 1428 1979 V 1486 2025 a Fg(^)1469 2042 y Ff(B)1522 2051 y Fh(1)p 1428 2069 V 1486 2115 a Fg(^)1469 2132 y Ff(B)1522 2141 y Fh(2)1599 1868 y Fn(1)1599 2017 y(A)1056 2226 y Fc(where)74 b Ff(B)1395 2235 y Fh(1)1453 2226 y Fg(and)1607 2210 y(^)1591 2226 y Ff(B)1644 2235 y Fh(1)1702 2226 y Fg(ha)n(v)n(e)25 b Ff(b)e Fg(ro)n(ws)h(and)g Ff(L)2267 2235 y Fh(11)2355 2226 y Fg(is)g Ff(b)16 b Fb(\002)g Ff(b)p 3549 2250 4 921 v 349 2253 3203 4 v 347 2552 4 299 v 450 2419 a Fg(6)p 584 2552 V 601 2552 V 813 2253 a Fn(8)813 2328 y(<)813 2478 y(:)887 2282 y( )994 2328 y Ff(B)1047 2337 y Fh(0)p 952 2355 171 4 v 952 2372 V 994 2427 a Ff(B)1047 2436 y Fh(1)p 952 2454 V 994 2509 a Ff(B)1047 2518 y Fh(2)1123 2282 y Fn(!)1208 2419 y Fg(=)1283 2257 y Fn(0)1283 2406 y(@)1541 2324 y Ff(L)1589 2295 y Fa(\000)p Fh(1)1589 2347 y(00)1688 2308 y Fg(^)1671 2324 y Ff(B)1724 2333 y Fh(0)p 1356 2351 589 4 v 1356 2368 V 1414 2414 a Fg(^)1397 2431 y Ff(B)1450 2440 y Fh(1)1501 2431 y Fb(\000)1580 2414 y Fg(^)1571 2431 y Ff(L)1619 2407 y Fd(T)1619 2453 y Fh(10)1684 2431 y Ff(L)1732 2401 y Fa(\000)p Fh(1)1732 2453 y(00)1831 2414 y Fg(^)1815 2431 y Ff(B)1868 2440 y Fh(0)p 1356 2458 V 1414 2504 a Fg(^)1397 2521 y Ff(B)1450 2530 y Fh(2)1501 2521 y Fb(\000)1580 2504 y Fg(^)1571 2521 y Ff(L)1619 2530 y Fh(20)1684 2521 y Ff(L)1732 2491 y Fa(\000)p Fh(1)1732 2543 y(00)1831 2504 y Fg(^)1815 2521 y Ff(B)1868 2530 y Fh(0)1944 2257 y Fn(1)1944 2406 y(A)2029 2419 y Ff(:)2049 2253 y Fn(9)2049 2328 y(=)2049 2478 y(;)p 3549 2552 4 299 v 349 2556 3203 4 v 347 2720 4 165 v 450 2654 a Fg(8)p 584 2720 V 601 2720 V 854 2618 a Ff(B)907 2627 y Fh(1)962 2618 y Fg(:=)j Ff(L)1104 2588 y Fa(\000)p Fh(1)1104 2640 y(11)1186 2618 y Ff(B)1239 2627 y Fh(1)854 2696 y Ff(B)907 2705 y Fh(2)962 2696 y Fg(:=)g Ff(B)1109 2705 y Fh(2)1159 2696 y Fb(\000)d Ff(L)1278 2705 y Fh(21)1343 2696 y Ff(B)1396 2705 y Fh(1)p 3549 2720 V 349 2723 3203 4 v 347 3466 4 743 v 430 2779 a Fg(5b)p 584 3466 V 601 3466 V 309 w Fc(Con)n(tin)n(ue)27 b(with)934 2910 y Fn(\022)1039 2966 y Fq(L)1096 2978 y Fl(T)9 b(L)p 1236 2996 4 100 v 1253 2996 V 1357 2966 a Fr(0)p 995 2999 506 4 v 995 3016 V 1037 3086 a Fq(L)1094 3098 y Fl(B)s(L)p 1236 3116 4 100 v 1253 3116 V 1296 3086 a Fq(L)1353 3098 y Fl(B)s(R)1501 2910 y Fn(\023)1585 3027 y Fp( )1691 2860 y Fn(0)1691 3009 y(@)1806 2915 y Fq(L)1863 2927 y Fm(00)p 1972 2945 V 2058 2915 a Fr(0)p 2182 2945 V 2199 2945 V 185 w(0)p 1764 2948 647 4 v 1806 3018 a Fq(L)1863 3030 y Fm(10)p 1972 3048 4 100 v 2016 3018 a Fq(L)2073 3030 y Fm(11)p 2182 3048 V 2199 3048 V 2285 3018 a Fr(0)p 1764 3051 647 4 v 1764 3068 V 1806 3137 a Fq(L)1863 3149 y Fm(20)p 1972 3167 4 100 v 2016 3137 a Fq(L)2073 3149 y Fm(21)p 2182 3167 V 2199 3167 V 2242 3137 a Fq(L)2299 3149 y Fm(22)2410 2860 y Fn(1)2410 3009 y(A)2483 3027 y Fr(,)2534 2910 y Fn(\022)2639 2966 y Fq(B)2702 2978 y Fl(T)p 2595 2999 204 4 v 2595 3016 V 2636 3086 a Fq(B)2699 3098 y Fl(B)2798 2910 y Fn(\023)2882 3027 y Fp( )2988 2860 y Fn(0)2988 3009 y(@)3102 2915 y Fq(B)3165 2927 y Fm(0)p 3061 2948 184 4 v 3102 3018 a Fq(B)3165 3030 y Fm(1)p 3061 3051 V 3061 3068 V 3102 3137 a Fq(B)3165 3149 y Fm(2)3244 2860 y Fn(1)3244 3009 y(A)3317 3027 y Fr(,)27 b(and)934 3220 y Fn(\022)1055 3266 y Fg(^)1039 3283 y Ff(B)1092 3294 y Fd(T)p 995 3310 189 4 v 995 3327 V 1053 3373 a Fg(^)1037 3389 y Ff(B)1090 3400 y Fd(B)1184 3220 y Fn(\023)1265 3333 y Fb( )1355 3170 y Fn(0)1355 3320 y(@)1486 3221 y Fg(^)1469 3238 y Ff(B)1522 3247 y Fh(0)p 1428 3265 171 4 v 1486 3311 a Fg(^)1469 3328 y Ff(B)1522 3337 y Fh(1)p 1428 3355 V 1428 3371 V 1486 3418 a Fg(^)1469 3434 y Ff(B)1522 3443 y Fh(2)1599 3170 y Fn(1)1599 3320 y(A)p 3549 3466 4 743 v 349 3469 3203 4 v 347 3768 4 299 v 450 3635 a Fg(7)p 584 3768 V 601 3768 V 813 3469 a Fn(8)813 3544 y(<)813 3693 y(:)887 3498 y( )994 3544 y Ff(B)1047 3553 y Fh(0)p 952 3571 171 4 v 994 3626 a Ff(B)1047 3635 y Fh(1)p 952 3653 V 952 3670 V 994 3725 a Ff(B)1047 3734 y Fh(2)1123 3498 y Fn(!)1208 3635 y Fg(=)1283 3473 y Fn(0)1283 3622 y(@)1986 3540 y Ff(L)2034 3510 y Fa(\000)p Fh(1)2034 3563 y(00)2133 3523 y Fg(^)2116 3540 y Ff(B)2169 3549 y Fh(0)p 1356 3567 1479 4 v 1749 3630 a Ff(L)1797 3600 y Fa(\000)p Fh(1)1797 3653 y(11)1880 3630 y Fg(\()1924 3613 y(^)1907 3630 y Ff(B)1960 3639 y Fh(1)2011 3630 y Fb(\000)16 b Ff(L)2130 3639 y Fh(10)2195 3630 y Ff(L)2243 3600 y Fa(\000)p Fh(1)2243 3653 y(00)2342 3613 y Fg(^)2325 3630 y Ff(B)2378 3639 y Fh(0)2413 3630 y Fg(\))p 1356 3657 V 1356 3674 V 1414 3720 a(^)1397 3737 y Ff(B)1450 3746 y Fh(2)1501 3737 y Fb(\000)f Ff(L)1619 3746 y Fh(20)1684 3737 y Ff(L)1732 3707 y Fa(\000)p Fh(1)1732 3759 y(00)1831 3720 y Fg(^)1815 3737 y Ff(B)1868 3746 y Fh(0)1918 3737 y Fb(\000)h Ff(L)2037 3746 y Fh(21)2102 3737 y Ff(L)2150 3707 y Fa(\000)p Fh(1)2150 3759 y(11)2232 3737 y Fg(\()2276 3720 y(^)2259 3737 y Ff(B)2312 3746 y Fh(1)2363 3737 y Fb(\000)g Ff(L)2482 3746 y Fh(10)2547 3737 y Ff(L)2595 3707 y Fa(\000)p Fh(1)2595 3759 y(00)2694 3720 y Fg(^)2678 3737 y Ff(B)2731 3746 y Fh(0)2765 3737 y Fg(\))2834 3473 y Fn(1)2834 3622 y(A)2919 3635 y Ff(:)2939 3469 y Fn(9)2939 3544 y(=)2939 3693 y(;)p 3549 3768 4 299 v 349 3772 3203 4 v 347 3971 4 200 v 450 3888 a Fg(2)p 584 3971 V 601 3971 V 813 3775 a Fn(\032\022)980 3838 y Ff(B)1033 3849 y Fd(T)p 936 3865 189 4 v 936 3881 V 978 3936 a Ff(B)1031 3947 y Fd(B)1125 3775 y Fn(\023)1206 3888 y Fg(=)1280 3775 y Fn(\022)1578 3838 y Ff(L)1626 3849 y Fd(T)8 b(L)1717 3813 y Fa(\000)p Fh(1)1816 3821 y Fg(^)1799 3838 y Ff(B)1852 3849 y Fd(T)p 1341 3865 797 4 v 1341 3881 V 1399 3927 a Fg(^)1383 3944 y Ff(B)1436 3955 y Fd(B)1505 3944 y Fb(\000)15 b Ff(L)1623 3955 y Fd(B)r(L)1718 3944 y Fg(\()p Ff(L)1793 3955 y Fd(T)8 b(L)1884 3919 y Fa(\000)p Fh(1)1983 3927 y Fg(^)1967 3944 y Ff(B)2020 3955 y Fd(T)2069 3944 y Fg(\))2138 3775 y Fn(\023\033)p 3549 3971 4 200 v 349 3974 3203 4 v 347 4053 4 79 v 584 4053 V 601 4053 V 652 4029 a Fc(enddo)p 3549 4053 V 349 4056 3203 4 v 347 4256 4 200 v 422 4173 a Fg(2,3)p 584 4256 V 601 4256 V 652 4060 a Fn(\032\022)q(\022)880 4122 y Ff(B)933 4133 y Fd(T)p 837 4149 189 4 v 837 4166 V 878 4221 a Ff(B)931 4232 y Fd(B)1026 4060 y Fn(\023)1106 4173 y Fg(=)1181 4060 y Fn(\022)1478 4122 y Ff(L)1526 4133 y Fd(T)g(L)1617 4097 y Fa(\000)p Fh(1)1716 4106 y Fg(^)1700 4122 y Ff(B)1753 4133 y Fd(T)p 1242 4149 797 4 v 1242 4166 V 1300 4212 a Fg(^)1283 4229 y Ff(B)1336 4240 y Fd(B)1405 4229 y Fb(\000)16 b Ff(L)1524 4240 y Fd(B)r(L)1618 4229 y Fg(\()p Ff(L)1693 4240 y Fd(T)8 b(L)1785 4204 y Fa(\000)p Fh(1)1884 4212 y Fg(^)1867 4229 y Ff(B)1920 4240 y Fd(T)1969 4229 y Fg(\))2038 4060 y Fn(\023\023)2176 4173 y Fb(^)16 b(:)c Fg(\()p Ff(m)p Fg(\()p Ff(L)2462 4184 y Fd(T)c(L)2554 4173 y Fg(\))20 b Fb(6)p Fg(=)f Ff(m)p Fg(\()p Ff(L)p Fg(\)\))2868 4060 y Fn(\033)p 3549 4256 4 200 v 349 4259 3203 4 v 347 4359 4 100 v 430 4325 a Fg(1b)p 584 4359 V 601 4359 V 652 4262 a Fn(\010)701 4325 y Ff(B)k Fg(:=)c Ff(L)919 4302 y Fa(\000)p Fh(1)1018 4309 y Fg(^)1001 4325 y Ff(B)1058 4262 y Fn(\011)p 3549 4359 V 349 4362 3203 4 v 469 4604 a Fr(Figure)26 b(8:)36 b(W)-7 b(orksheet)27 b(for)g(a)g(deriving)e(blo)r(c)n(k)n(ed)h(algorithm)d(for)k Fq(B)g Fr(:=)c Fq(L)2821 4574 y Fj(\000)p Fm(1)2910 4604 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(2\).)1908 5356 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 532 1385 a Fk(P)m(artition)55 b Fq(L)23 b Fp(!)1149 1268 y Fn(\022)1254 1324 y Fq(L)1311 1336 y Fl(T)9 b(L)p 1451 1354 4 100 v 1467 1354 V 1572 1324 a Fr(0)p 1210 1357 506 4 v 1210 1374 V 1252 1444 a Fq(L)1309 1456 y Fl(B)s(L)p 1451 1473 4 100 v 1467 1473 V 1510 1444 a Fq(L)1567 1456 y Fl(B)s(R)1716 1268 y Fn(\023)1805 1385 y Fr(and)27 b Fq(B)g Fp(!)2162 1268 y Fn(\022)2267 1324 y Fq(B)2330 1336 y Fl(T)p 2224 1357 204 4 v 2224 1374 V 2265 1444 a Fq(B)2328 1456 y Fl(B)2427 1268 y Fn(\023)675 1543 y Fk(where)59 b Fq(L)1040 1555 y Fl(T)9 b(L)1165 1543 y Fr(is)26 b(0)18 b Fp(\002)g Fr(0)28 b(and)f Fq(B)1685 1555 y Fl(T)1765 1543 y Fr(has)g(0)g(ro)n(ws) 532 1784 y Fk(while)54 b Fq(m)p Fr(\()p Fq(L)968 1796 y Fl(T)9 b(L)1065 1784 y Fr(\))24 b Fp(6)p Fr(=)e Fq(m)p Fr(\()p Fq(L)p Fr(\))56 b Fk(do)705 1884 y(Determine)30 b(blo)s(c)m(k)h(size)h Fq(b)705 1983 y Fk(Repartition)848 2131 y Fn(\022)953 2187 y Fq(L)1010 2199 y Fl(T)9 b(L)p 1149 2217 4 100 v 1166 2217 V 1270 2187 a Fr(0)p 909 2220 506 4 v 909 2237 V 950 2307 a Fq(L)1007 2319 y Fl(B)s(L)p 1149 2337 4 100 v 1166 2337 V 1209 2307 a Fq(L)1266 2319 y Fl(B)s(R)1415 2131 y Fn(\023)1499 2248 y Fp(!)1605 2081 y Fn(0)1605 2230 y(@)1719 2136 y Fq(L)1776 2148 y Fm(00)p 1886 2166 V 1902 2166 V 1988 2136 a Fr(0)p 2112 2166 V 168 w(0)p 1677 2169 647 4 v 1677 2186 V 1719 2255 a Fq(L)1776 2267 y Fm(10)p 1886 2285 4 100 v 1902 2285 V 1945 2255 a Fq(L)2002 2267 y Fm(11)p 2112 2285 V 2198 2255 a Fr(0)p 1677 2289 647 4 v 1719 2358 a Fq(L)1776 2370 y Fm(20)p 1886 2388 4 100 v 1902 2388 V 1961 2358 a Fq(l)1986 2370 y Fm(21)p 2112 2388 V 2155 2358 a Fq(L)2212 2370 y Fm(22)2324 2081 y Fn(1)2324 2230 y(A)2424 2248 y Fr(and)2585 2131 y Fn(\022)2690 2187 y Fq(B)2753 2199 y Fl(T)p 2646 2220 204 4 v 2646 2237 V 2688 2307 a Fq(B)2751 2319 y Fl(B)2850 2131 y Fn(\023)2934 2248 y Fp(!)3040 2081 y Fn(0)3040 2230 y(@)3154 2136 y Fq(B)3217 2148 y Fm(0)p 3113 2169 184 4 v 3113 2186 V 3154 2255 a Fq(B)3217 2267 y Fm(1)p 3113 2289 V 3154 2358 a Fq(B)3217 2370 y Fm(2)3296 2081 y Fn(1)3296 2230 y(A)991 2458 y Fk(where)87 b Fq(B)1390 2470 y Fm(1)1454 2458 y Fr(has)28 b Fq(b)f Fr(ro)n(ws)f(and)h Fq(L)2076 2470 y Fm(11)2174 2458 y Fr(is)f Fq(b)18 b Fp(\002)g Fq(b)p 705 2560 2341 4 v 705 2552 V 746 2660 a(B)809 2672 y Fm(1)869 2660 y Fr(:=)23 b Fq(L)1037 2625 y Fj(\000)p Fm(1)1037 2683 y(11)1126 2660 y Fq(B)1189 2672 y Fm(1)746 2760 y Fq(B)809 2772 y Fm(2)869 2760 y Fr(:=)g Fq(B)1043 2772 y Fm(2)1099 2760 y Fp(\000)18 b Fq(L)1239 2772 y Fm(21)1309 2760 y Fq(B)1372 2772 y Fm(1)p 705 2862 V 705 2854 V 705 2959 a Fk(Con)m(tin)m(ue)31 b(with)848 3090 y Fn(\022)953 3147 y Fq(L)1010 3159 y Fl(T)9 b(L)p 1149 3177 4 100 v 1166 3177 V 1270 3147 a Fr(0)p 909 3180 506 4 v 909 3197 V 950 3266 a Fq(L)1007 3278 y Fl(B)s(L)p 1149 3296 4 100 v 1166 3296 V 1209 3266 a Fq(L)1266 3278 y Fl(B)s(R)1415 3090 y Fn(\023)1499 3207 y Fp( )1605 3041 y Fn(0)1605 3190 y(@)1719 3095 y Fq(L)1776 3107 y Fm(00)p 1886 3125 V 1971 3095 a Fr(0)p 2095 3125 V 2112 3125 V 185 w(0)p 1677 3129 647 4 v 1719 3198 a Fq(L)1776 3210 y Fm(10)p 1886 3228 4 100 v 1929 3198 a Fq(L)1986 3210 y Fm(11)p 2095 3228 V 2112 3228 V 2198 3198 a Fr(0)p 1677 3232 647 4 v 1677 3248 V 1719 3318 a Fq(L)1776 3330 y Fm(20)p 1886 3348 4 100 v 1929 3318 a Fq(L)1986 3330 y Fm(21)p 2095 3348 V 2112 3348 V 2155 3318 a Fq(L)2212 3330 y Fm(22)2324 3041 y Fn(1)2324 3190 y(A)2424 3207 y Fr(and)2585 3090 y Fn(\022)2690 3147 y Fq(B)2753 3159 y Fl(T)p 2646 3180 204 4 v 2646 3197 V 2688 3266 a Fq(B)2751 3278 y Fl(B)2850 3090 y Fn(\023)2934 3207 y Fp( )3040 3041 y Fn(0)3040 3190 y(@)3154 3095 y Fq(B)3217 3107 y Fm(0)p 3113 3129 184 4 v 3154 3198 a Fq(B)3217 3210 y Fm(1)p 3113 3232 V 3113 3248 V 3154 3318 a Fq(B)3217 3330 y Fm(2)3296 3041 y Fn(1)3296 3190 y(A)532 3418 y Fk(enddo)925 3689 y Fr(Figure)26 b(9:)37 b(Blo)r(c)n(k)n(ed)25 b(algorithm)f(for)j Fq(B)g Fr(:=)c Fq(L)2365 3659 y Fj(\000)p Fm(1)2453 3689 y Fq(B)32 b Fr(\(V)-7 b(arian)n(t)26 b(2\).)1908 5356 y(19)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF
Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents:

E. Michigan >> CH >> 3419 (Fall, 2009)
Copyright 1989 Information Access Company; Copyright Scientific American Inc. 1989 Scientific American June, 1989 SECTION: Vol. 260 ; No. 6 ; Pg. 122; ISSN: 0036-8...
E. Michigan >> CH >> 419 (Fall, 2009)
Copyright 1989 Information Access Company; Copyright Scientific American Inc. 1989 Scientific American June, 1989 SECTION: Vol. 260 ; No. 6 ; Pg. 122; ISSN: 0036-8...
E. Michigan >> CH >> 3419 (Fall, 2009)
Handout for Chapter 4.2 IE 339/429 February 3, 2006 1 Matrix/Diagram Practice Consider the following transition matrix, and draw a graph of it off to the side. A dot indicates a zero. 0 1 0 . 1.0 1 0.7 0.1 2 0.6 . 3 . . 4 . . 2 3 4 . . . 0.2 . . ....
E. Michigan >> CH >> 419 (Fall, 2009)
Handout for Chapter 4.2 IE 339/429 February 3, 2006 1 Matrix/Diagram Practice Consider the following transition matrix, and draw a graph of it off to the side. A dot indicates a zero. 0 1 0 . 1.0 1 0.7 0.1 2 0.6 . 3 . . 4 . . 2 3 4 . . . 0.2 . . ....
E. Michigan >> CH >> 3419 (Fall, 2009)
> a=1 a = 1 > a a = 1 > b=2 b = 2 > c=a+b c = 3 > a=5 a = 5 > c c = 3 > c=a+2*b c = 9 > % percent is the comment character > % above code shows that equalities > % don\'t get updated automaticall...
E. Michigan >> CH >> 419 (Fall, 2009)
> a=1 a = 1 > a a = 1 > b=2 b = 2 > c=a+b c = 3 > a=5 a = 5 > c c = 3 > c=a+2*b c = 9 > % percent is the comment character > % above code shows that equalities > % don\'t get updated automaticall...
E. Michigan >> CH >> 3419 (Fall, 2009)
...
E. Michigan >> CH >> 3419 (Fall, 2009)
DTMC application in inventory management Rong, Ying I will describe two DTMC applications in inventory management. The rst one is about Standing Order System. In this system, leading time of standing order is eliminated no matter how long product wil...
E. Michigan >> CH >> 419 (Fall, 2009)
DTMC application in inventory management Rong, Ying I will describe two DTMC applications in inventory management. The rst one is about Standing Order System. In this system, leading time of standing order is eliminated no matter how long product wil...
E. Michigan >> CH >> 3419 (Fall, 2009)
Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League Michael A. Jones Michael Jones (jonesm@mail.montclair.edu) received his Ph.D. in game theory from Northwestern University in 1994 under the supervision of Donald...
E. Michigan >> CH >> 419 (Fall, 2009)
Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League Michael A. Jones Michael Jones (jonesm@mail.montclair.edu) received his Ph.D. in game theory from Northwestern University in 1994 under the supervision of Donald...
E. Michigan >> AROSS >> 419 (Fall, 2009)
1 vokraM-ssuaG a ro ssecorp kcebnelhU-nietsnrO na si }0 t : tX{ ssecorp citsahcots A .cipot lartnec eht ot nrut ew ,seiranimilerp htiw desnepsid gnivaH .shtap elpmas ton ,snoitubirtsid tuoba tnemetats a si siht taht etoN . + R + R revo suounitnoc ...
E. Michigan >> CH >> 419 (Fall, 2009)
...
E. Michigan >> CH >> 419 (Fall, 2009)
COMMENTS ON THE ORIGIN AND APPLICATION OF MARKOV DECISION PROCESSES RONALD A. HOWARD Stanford University, Palo Alto, California 94305-4026, rhoward@stanford.edu R ecently, a European mathematician who had spent many years on the theoretical study o...
E. Michigan >> CH >> 419 (Fall, 2009)
...
E. Michigan >> CH >> 419 (Fall, 2009)
...
E. Michigan >> CH >> 419 (Fall, 2009)
Luiz Henrique Gomes, Cristiano Cazita Jussara M. Almeida, Virglio Almeida, Wagner Meira Jr. Department of Computer Science Federal University of Minas Gerais Belo Horizonte - Brazil lhg, cazita, jussara, virgilio, meira @dcc.ufmg.br General Terms Me...
E. Michigan >> CH >> 419 (Fall, 2009)
...
E. Michigan >> CH >> 419 (Fall, 2009)
These papers by Kruse use an M/G/infinity queue to analyze the waiting time in an (s,S) or (S-1,S) inventory system. This is very related to Example 5.16, M/G/infinity, in edition 8 of our textbook (Intro to Prob. Models) ...
E. Michigan >> AROSS >> 419 (Fall, 2009)
The files in these directories are meant to accompany \"Introduction to Probability Models\" by Sheldon Ross. The fundamental idea in creating most of them is that it\'s nice to be able to generate new random samples at the touch of a button, to see ho...
E. Michigan >> AROSS >> 419 (Fall, 2009)
Convolution of Distributions Distrib. A 1 : order 1 : sum 6 : amax 3.5 : mean 2.92 : var 1.71 : std.dev. Pr ReverseIndx RevPr 0 0 6 1 0.17 5 2 0.17 4 3 0.17 3 4 0.17 2 5 0.17 1 6 0.17 0 7 Distrib. B 6 : order Predicted Distrib. C 1 24.5 20.42 4.52...
E. Michigan >> AROSS >> 419 (Fall, 2009)
Convolution of Distributions Distrib. A 1 : order 1 : sum 6 : amax 3.5 : mean 2.92 : var 1.71 : std.dev. Pr ReverseIndx RevPr 0 0 6 1 0.17 5 2 0.17 4 3 0.17 3 4 0.17 2 5 0.17 1 6 0.17 0 7 Distrib. B 6 : order Predicted Distrib. C 1 24.5 20.42 4.52...
E. Michigan >> AROSS >> 419 (Fall, 2009)
Excel sheet ideas: DTMC: vector*pmat; Error: transpose pmat powers; Error: too small pmat squared; Error: actually powers steady-state; Error: transpose? simulate; Error: rand->column mapping collect-stats; Error: reverse jumps Poisson Proc: Binomi...
E. Michigan >> AROSS >> 419 (Fall, 2009)
X, Y are IID Normals Normals by inversion: u1 0.73 u2 0.92 X 0.71 Y -0.37 X 1 ...
E. Michigan >> AROSS >> 419 (Fall, 2009)
...
E. Michigan >> AROSS >> 419 (Fall, 2009)
How do you know your spreadsheet is right? Principles, Techniques and Practice of Spreadsheet Style Philip L. Bewig July 28, 2005 You know its true: Spreadsheets have errors like dogs have fleas.1 It is generally accepted2 that nine out of every te...
E. Michigan >> AROSS >> 419 (Fall, 2009)
Spreadsheet Modelling Best Practice by Nick Read and Jonathan Batson BUSINESS DYNAMICS April 1999 This document has been published by the Institute of Chartered Accountants for England and Wales who jointly own the copyright, it should not be repro...
E. Michigan >> CH >> 419 (Fall, 2009)
Multivariate Normal A-matrix name0 name0 name1 0.1 0.5 name1 0 0.5 Covariance Matrix = A*A^T name0 name1 name0 name1 0.01 0.05 0.05 0.5 Correlation Matrix name0 0.1 name0 0.71 name1 0.71 name1 1 0.71 0.71 1 0.1 Data Samples Z0 -0.61 -0.38 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Scaled Random Walks approaching Brownian Motion, or not. deltat 0.1 0.1 warning: keep deltat >= 0 try deltax=deltat or deltax=sqrt(deltat) i Xi tmp_tval t 0 1 0 1 1 0.1 2 1 0.2 3 1 0.3 4 -1 0.4 5 1 0.5 6 -1 0.6 7 1 0.7 8 -1 0.8 9 -1 0.9 10 1 1 11 -1 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Bin Frequency umulative % C 0.09 1 .10% 0.09 1 .20% 0.09 2 .40% 0.1 1 .50% 0.1 9 1.41% 0.1 12 2.61% 0.11 9 3.52% 0.11 11 4.62% 0.11 24 7.04% 0.11 27 9.75% 0.12 44 14.17% 0.12 59 20.10% 0.12 60 26.13% 0.13 72 33.37% 0.13 69 40.30% 0.13 65 46.83% 0.14 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Simulate a CTMC, Way #1 automatic recalculation turned off-use F9 to force recalculation. control-a will store and update the state control-b will just update the state without storing. control-i will initialize the result log Mean Duration Current S...
E. Michigan >> CH >> 419 (Fall, 2009)
Gather data into a DTMC put your data on the \"Data\" sheet, and update the \"From\" row and \"To\" column on this page. control-a will store and update the state control-i will initialize the matrix and start at data row 1 Time Current State Number 1 Err:...
E. Michigan >> CH >> 419 (Fall, 2009)
Simulate a DTMC automatic recalculation turned off-use F9 to force recalculation. control-a will store and update the state control-b will just update the state without storing. control-i will initialize the result log Current State Current Row name1...
E. Michigan >> CH >> 419 (Fall, 2009)
Solve a DTMC Must enter number of states here: Temporary vector 1 1.17 1.83 sum= 4 Final vector 0.25 sum= Row sums: 1 name0 1 name1 1 name2 0 name3 0 name4 0 name5 0 name6 0 name7 0 name8 0 name9 0 name10 0 name11 0 name12 0 name13 0 name14 0 name15 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Simulate a DTMC automatic recalculation turned off-use F9 to force recalculation. control-a will store and update the state control-b will just update the state without storing. To name0 name1 name2 From 0 1 name0 0 0 1 name1 1 0.1 0 name2 2 0.8 0 na...
E. Michigan >> CH >> 419 (Fall, 2009)
Iterating the P-matrix Press control-a to run a macro that copies the result into the intermediate space, and increases the power-counters Press control-i to run an initialization macro. Intermediate matrix power: 1 Result matrix power: 2 Row Sums 1 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Simulate an arbitrary discrete random variable building up to simulating a DTMC Mean 2.6 Value Probability Cumulative ShiftedCumu U(0,1) Value 0.21 Result 0 0 0.3 0.3 0 2 0.2 0.5 0.3 3 0.1 0.6 0.5 4 0.1 0.7 0.6 5 0.3 1 0.7 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Squaring the P-matrix Original Transition Matrix 0.3 0.2 0.2 0.1 0.8 0 0.4 0.2 Square this matrix: 0.3 0.2 0.2 0.1 0.8 0 0.4 0.2 Result: 0.33 0.52 0.32 0.4 0.18 0.11 0.2 0.14 Row Sums 1 1 1 1 Power of Original Matrix 1 0 0.4 0 0.2 0.5 0.3 0.2 0.2 ...
E. Michigan >> CH >> 419 (Fall, 2009)
Vector times Transition Matrix Press control-a to run a macro that copies the result to the log on the next sheet, and then back into the initial vector Press control-i to initialize the log sheet (clear the data and start with the current initial ve...
E. Michigan >> CH >> 419 (Fall, 2009)
Vector times Transition Matrix Press control-a to run a macro that copies the result to the log on the next sheet, and then back into the initial vector Press control-i to initialize the log sheet (clear the data and start with the current initial ve...
E. Michigan >> CH >> 419 (Fall, 2009)
Binomial Process Simulator #steps P-value 1000 0.05 total heads 42 Press F9 to get new random numbers. Notice how gaps in the event times correspond to flat spots on the N(t) graph. Event Times 1.2 1 0 if Tails, 1 if Heads 0.8 0.6 0.4 0.2 0 0 100 20...
E. Michigan >> CH >> 419 (Fall, 2009)
Conditional Distribution of Arrival Times, given exactly 1 arrival Arrival Rate Time span 1 0.5 S1 mean 0.39 S1 std.dev Interactive Histogram: 0.44 Bucket # Bucket from to # S1 Occur 1 0 0.05 301 2 0.05 0.1 32 3 0.1 0.15 32 4 0.15 0.2 35 5 0.2 0.25 3...
E. Michigan >> CH >> 419 (Fall, 2009)
AnM/G/infinitysimulation ArrivalRate 10 perminute MeanSvcDuration 20 minutes SvcDurationStdDev 0 minutes Arrival# Time SvcDur Effect 0 0 20 1 0 20 1 1 0.12 20 1 1 20.12 1 2 0.16 20 1 2 20.16 1 3 0.36 20 1 3 20.36 1 4 0.41 20 1 4 20.41 1 5 0.52 20 1 5...
E. Michigan >> CH >> 419 (Fall, 2009)
The Wrong Way to generate a Non-homogeneous Poisson Process Linear arrival rate function lambda(t) = a*t+b a= 1 b= 1 tmax E[#arrivals] Average number arrivals in (0,tmax)= integral from 0 to tmax of lambda(t) dt equals 0.5*a*tmax^2 + b*tmax # arrival...
E. Michigan >> CH >> 419 (Fall, 2009)
Inspection Paradox 12 Age Age 0.010 0.525 0.367 1.362 Excess 0.025 0.576 0.352 1.382 Inspected Overall Length lifetime 0.512 1.101 0.270 1.484 10 #trials 100 min mean Std.dev max 8 1.002 0.252 6 4 2 Correl: Age Excess Ins. Length Age 1 Err:50...
E. Michigan >> CH >> 419 (Fall, 2009)
Insurance Ruin 200 180 160 140 120 Total Fortune 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Time (months) 1.2 1.4 1.6 1.8 2 The Insurance Ruin Problem Initial Capital Monthly Income Claims per month, avg Avg claim size Avg monthly profit Distribution O...
E. Michigan >> CH >> 419 (Fall, 2009)
Mystery Renewal Process #1 Arrival# 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Inter-Arrivals Arrivals 0.95 0.01 0.89 0.03 0.33 0.66 0.84 0.72 0.05 0.03 0.26 0.77 0.77 0.3 0.37 0.47 0.19 0.57 0.02 0.55 0.36 0.72 0.31 0.6...
E. Michigan >> CH >> 419 (Fall, 2009)
Checking Independence of InterArrival Times 1.2 1 f(x) = 0.82x + 0.92 0.8 R = 0.68 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Superposition of Renewal Processes Here are two Uniform renewal processes: a value 0.8 0.6 b value 1.2 1.4 mean 1 1 std.dev...
E. Michigan >> CH >> 419 (Fall, 2009)
Solve a CTMC Must enter number of states here: Temporary vector 1 0.5 0.13 sum= 1.63 Final vector 0.62 sum= Row sums: 0 name0 0 name1 0 name2 0 name3 0 name4 0 name5 0 name6 0 name7 0 name8 0 name9 0 name10 0 name11 0 name12 0 name13 0 name14 0 name1...
E. Michigan >> CH >> 419 (Fall, 2009)
Transient CTMC by Euler\'s Method Enter number of states here: nstates= sums= Current Time Current Vector 1 0 1 0 0 Time Step Derivative Vector 0 0.1 -1 1 0 #N/A #N/A Next Time Next Vector 1 0.1 0.9 0.1 0 #N/A #N/A Row sums: 0 name0 0 name1 0 name2 0 ...
E. Michigan >> AROSS >> 419 (Fall, 2009)
Convolution of Discrete Distributions Distribution A 1 : order 1 : sum 7 : length 3.5 : mean 2.92 : variance 1.71 : std.dev. Index Pr RevIndex RevPr 0 0 6 0.17 1 0.17 5 0.17 2 0.17 4 0.17 3 0.17 3 0.17 4 0.17 2 0.17 5 0.17 1 0.17 6 0.17 0 0 7 8 9 10 ...
E. Michigan >> AROSS >> 419 (Fall, 2009)
Convolution of Two Distributions the memoryintensive way Alength: Blength: 7 13 Aindex Bindex Pr 0 0 1 0 2 0.03 3 0.06 4 0.08 5 0.11 6 0.14 7 0.17 8 0.14 9 0.11 10 0.08 11 0.06 12 0.03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.17 0 0 0 0.01 0.01 0.02 0.0...
E. Michigan >> CH >> 419 (Fall, 2009)
Lindley\'s Recursion for G/G/1 0.99 : arrival rate 1 : service rate Arrival# InterArrival ServiceDur Wq lag1 Wq 0 0 1 0 0 1 1.01 0.5 0 0 2 2.58 0.43 0 0 3 1.15 1.48 0 0 4 1.72 0.68 0 0.33 5 0.34 0.26 0.33 0 6 1.53 1.03 0 0 7 2.1 1.04 0 0 8 1.13 0.86 ...
Roosevelt >> MATH >> 300 (Fall, 2009)
Vectors, Vector and Matrix Equations Linear Combinations and Linear Systems Span of Vectors Matrix Equations Solution Sets of Homogenous Systems Solution Sets of Non-homogenous Systems 1 In the Last Class Systems of linear equations, lines, planes, ...
Maryville MO >> LPM >> 368 (Fall, 2009)
Storyboard Template-Lauren Magnuson TITLE PAGE (PAGE 1 2) 700x780 0 No: 1-Title Screen, No. 3 -Input Text and No. 8 Audio Interaction Enter first name in input text field in title screen and press enter Effect Welcoming screen will appear featuri...
Maryville MO >> MCM >> 1111 (Fall, 2009)
Advanced Search Help Sheet Guided Style Search Creating an Advanced Search 1. In the first Find field, enter a keyword. 2. Choose the search field from the drop-down list (for example, search in only the Subject Terms field of the citation). 3. Repe...
Stanford >> EE >> 191 (Fall, 2009)
EE191 - Lecture 4 Embedded Systems Compact Flash File Systems Spring 2006 Copyright Statement This work is Copyright 2006, Max Klein and Stanford University. Permission is granted to reprint and redistribute this document in its entirety as long as...
Stanford >> EE >> 191 (Fall, 2009)
Connexions module: m11677 1 Preparing for OrCAD Layout Version 1.5: 2004/01/29 14:20:32.254 US/Central Patrick Frantz This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract Step-by-st...
Stanford >> EE >> 192 (Fall, 2009)
EE192C Lecture 9 Embedded Systems Engineering Real Time Systems Interrupts, Timers April 29, 2008 Max Klein STARLab - VLF Group Ofce: Packard 351, x33789 Lab: Packard 075, x60333 maxklein@stanford.edu EE192C - Klein 1 Coming Up Thursday: I...
Stanford >> EE >> 192 (Fall, 2009)
EE192C Lecture 11 Embedded Systems Engineering IRQ / ISR Follow-up FreeRTOS, Continued May 6, 2008 Max Klein STARLab - VLF Group Ofce: Packard 351, x33789 Lab: Packard 075, x60333 maxklein@stanford.edu EE192C - Klein 1 Follow-up EE192C - K...
Stanford >> EE >> 192 (Fall, 2009)
200708 Spring Quarter EE192C: Embedded Systems Engineering TTh 12:5014:05, Hewlett 102 Tentative Syllabus Week 1: Embedded systems overview, microcontroller review, GPIO. Week 2: GPIO and character LCDs. Development kits issued. Week 3: Ser...
Stanford >> EE >> 292 (Fall, 2009)
EE292C - Lecture 1 EE292C: Embedded Systems Engineering Course Overview & Basics Review April 3, 2007 Max Klein STARLab - VLF Group Office: Packard 351, x33789 Lab: Packard 075, x60333 maxklein@stanford.edu EE292C - Klein 1 What is an Embedded Sys...
Stanford >> EE >> 292 (Fall, 2009)
EE292C - Lecture 9 Embedded Systems Engineering Mass Storage May 1, 2007 Max Klein STARLab - VLF Group Office: Packard 351, x33789 Lab: Packard 075, x60333 maxklein@stanford.edu EE292C - Klein 1 Coming Up. Today: CF, a bit of SD/MMC Thursday: Ti...
Stanford >> EE >> 292 (Fall, 2009)
EE292C - Lecture 10 Embedded Systems Engineering Interrupts, Timers, and Real Time Systems May 3, 2007 Max Klein STARLab - VLF Group Office: Packard 351, x33789 Lab: Packard 075, x60333 maxklein@stanford.edu EE292C - Klein 1 Coming Up. Tuesday: ...
Stanford >> EE >> 292 (Fall, 2009)
EE292C - Lecture 13 Embedded Systems Engineering FreeRTOS Q&A Introduction to Hardware May 15, 2007 Max Klein STARLab - VLF Group Office: Packard 351, x33789 Lab: Packard 075, x60333 maxklein@stanford.edu EE292C - Klein 1 FreeRTOS Questions? OneWi...
Stanford >> EE >> 292 (Fall, 2009)
EE292C - Lecture 17 Embedded Systems Engineering Printed Circuit Board (PCB) Intro Cadence Allegro Layout Plus Tutorial With Significant Contributions from Jeffrey Chang May 29, 2007 Max Klein STARLab - VLF Group Office: Packard 351, x33789 Lab: Pa...
Texas >> CS >> 378 (Fall, 2008)
%!PS-Adobe-2.0 %Creator: dvipsk 5.86 p1.5d Copyright 1996-2001 ASCII Corp.(www-ptex@ascii.co.jp) %based on dvipsk 5.86 Copyright 1999 Radical Eye Software (www.radicaleye.com) %Title: Recipe.dvi %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 ...
Ill. Chicago >> I >> 480 (Fall, 2009)
CS 480: DATABASE SYSTEMS Homework 4 Due on: 29th April., in the class No late submissions accepted Solve the problems from the text book Database System Concepts by Silbershatz, Korth and Sudarshan (Fifth Edition). 1. Exercises 7.1 (page 306) and 7.6...
Ill. Chicago >> CS >> 724 (Fall, 2009)
connectMLS - Connecting Your Real Estate Community http:/mlsni4.connectmls.com/common/PrintableView.jsp?ud=1185223. < 1 of 9 > Detached Single Status: NEW MLS #: 06615355 Address: 724 CARDIGAN CT, NAPERVILLE, 60565 Bedrooms: 4 Price:$539,900 Baths...
What are you waiting for?
View This Document For FREE!