{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Francis Algrorithm for Schur decomposition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A := RandomMatrix(5,5,generator=-5. 0..5.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*73VY \"-%'MATRIXG6#7'7'$\"3[Mxd[:N(y\"!#<$\"3UI!HREC\\f%F0$\"3+-!))))o/G+$F 0$\"3alF*>N&))[YF0$\"35&4aAY#fB8F07'$\"3[0bdxC$*RVF0$\"3uVbf&Ht?#HF0$! 3!)zerF^Vi9!#=$\"3?wHMa$o]d%F0$\"3F$>!Rh&eP8%F07'$\"3urxoeIH\"\\$F0$\" 3!4n!*=Dbt:%F0$\"3Oc%HC[p;d%F0$\"3g%Q)\\?>:)o%F@$!3BR\\1P=8IPF07'$!3-/ \"eU@$)Gk%F0$!33.DPP<(Q#yF@$\"33chgc2P> z0%F07'$\"3#z'ec\"*pSd:F0$!3kYys8m8\"e$F0$!3; " 0 "" {MPLTEXT 1 0 7 "B := A:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "U := IdentityMa trix(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'RTABLEG6%\"*Kl^Y \"-%'MATRIXG6#7'7'\"\"\"\"\"!F/F/F/7'F/F.F/F/F/7'F/F/F.F/F/7'F/F/F/F.F /7'F/F/F/F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "f or k from 1 to 200 do\n (Q,R) := QRDecomposition(B,fullspan):\n U \+ := U.Q:\n B := R.Q:\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "read \"c:/304/clearmatrix.txt\";" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,ClearMatrixGf*6$'%\"AG%'MatrixG'%$tolG%(numericG6'%\"BG%\"mG% \"nG%\"jG%\"kG6\"F3C&>6$8%8&-_%.LinearAlgebraG%*DimensionG6#9$>8$-F)F6 ?(8'\"\"\"FDF7%%trueG?(8(FDFDF8FE@%2-%$absG6#&F>6$FCFG9%>&F@FN\"\"!>FQ FMOF@F3F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "S := ClearMa trix(B,1E-9); # the Schur form" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"SG-%'RTABLEG6%\"*+7nZ\"-%'MATRIXG6#7'7'$\"3%*e/7(G&zJn!#<$!3ghx\\^rw g\\F0$\"3^w#G_#o'47#F0$!3Y0UoR/?@r!#=$!3*y>x.h1!H\\F07'$\"31`([y\"pc&* [!#>$\"3-')p0a+VkoF0$!3&R[cph3(4vF0$!3q#zO'[Ail>F0$!3I$Rm4L#yT>F07'\" \"!FG$!3))>A2iq]b@F0$\"3'3!*HH=LcM(F0$!3%fJQQ5E`J#F07'FGFG$!3UPFn]^X#o %F0$!3gq$e)ed6ahF7$\"3_ikn&=&Q*)GF07'FGFGFGFG$!3qFR9SK$*>hF7%'MatrixG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C := U.S.Transpose(U); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6%\"*)=1c:-%'MATRI XG6#7'7'$\"3QH)G&[:N(y\"!#<$\"3IiT&QEC\\f%F0$\"3r6`'))o/G+$F0$\"3I]7,_ `))[YF0$\"3Dx0CiCfB8F07'$\"3iQYbxC$*RVF0$\"3#GjjbHt?#HF0$!3auP\"y7NCY \"!#=$\"3m\"*3Na$o]d%F0$\"3\\'\\%Qh&eP8%F07'$\"3Q)3$yeIH\"\\$F0$\"3[Oq .__NdTF0$\"3#R2tC[p;d%F0$\"3Q:)o%F@$!3\"46Qq$=8IPF07'$!3&>!oD9K)G k%F0$!38pdLP<(Q#yF@$\"3.I[gz0%F07'$\" 3u^0q\"*pSd:F0$!35A2_8m8\"e$F0$!3q*GjJ=pQU$F0$!3y%)p0]ffCSF0$\"3us+(R' oBZJF0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Norm(C-A ,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+QF@wL!#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ClearMatrix(Transpose(U).U,1E-12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*?GEb\"-%'MATRIXG6#7'7'$ \"35.++++++5!#<\"\"!F/F/F/7'F/$\"3w2++++++5F.F/F/F/7'F/F/$\"3+1++++++5 F.F/F/7'F/F/F/$\"3A/++++++5F.F/7'F/F/F/F/$\"3W+++++++5F.%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "U, S;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*C7uZ\"-%'MATRIXG6#7'7'$\"36$!3CIo*4u#= t`F.7'$\"3!RW&G/$G(oyF.$!31'e\\S3.WJ#F.$!3OopvAf?T;F.$\"3_(e#o)e6/F&F. $!3e7*>;T^>]\"F.7'$\"33#ozaXKo=*F@$\"3_6!GFAzt%>F.$\"3%pP+jAMTY*F.$\"3 !G\\Z'Qa\"34#F.$!3Kn[ez'RE>\"F.7'$!334uU;%\\F@&F.$!3i>Y\\?x#3Z\"F.$!3h Q&)=m4H1aF@$\"3@myM\\*yy#yF.$\"3^ZAvBx%f,$F.%'MatrixG-F$6%\"*+7nZ\"-F( 6#7'7'$\"3%*e/7(G&zJn!#<$!3ghx\\^rwg\\Fho$\"3^w#G_#o'47#Fho$!3Y0UoR/?@ rF.$!3*y>x.h1!H\\Fho7'$\"31`([y\"pc&*[F@$\"3-')p0a+VkoFho$!3&R[cph3(4v Fho$!3q#zO'[Ail>Fho$!3I$Rm4L#yT>Fho7'\"\"!F]q$!3))>A2iq]b@Fho$\"3'3!*H H=LcM(Fho$!3%fJQQ5E`J#Fho7'F]qF]q$!3UPFn]^X#o%Fho$!3gq$e)ed6ahF.$\"3_i kn&=&Q*)GFho7'F]qF]qF]qF]q$!3qFR9SK$*>hF.F^o" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Example: Find all roo ts of " }{XPPEDIT 18 0 "p(x) = x^4-3*x^3+4*x^2-5*x+7;" "6#/-%\"pG6 #%\"xG,,*$)F'\"\"%\"\"\"F,*&\"\"$F,*$)F'F.F,F,!\"\"*&F+F,*$)F'\"\"#F,F ,F,*&\"\"&F,F'F,F1\"\"(F," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "A := Matrix([[0,0,0, -7],[1,0,0,5],[0,1,0,-4],[0,0,1,3]]); # the companion matrix" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*c:$e:-%'MATRIXG6# 7&7&\"\"!F.F.!\"(7&\"\"\"F.F.\"\"&7&F.F1F.!\"%7&F.F.F1\"\"$%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "B := evalf(A):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for k from 1 to 200 do\n ( Q,R) := QRDecomposition(B,fullspan):\n B := R.Q:\nend do:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*3+!p:-%'MATRIXG6#7&7&$\"3vPN#\\ABTM\"!#< $!3vnVo%3[ui$F.$\"3w#=oxnKW3*!#=$!3_chr@#3o1)F.7&$\"3<\"z+3=CR@#F3$\"3 \\*)4?Hm>3@F.$!3WSFLu5s@[F3$\"3PZ[(yZ4NS$F.7&$\"\"!FA$!35#)*=,z_+k$!#W $\"3KDm>')p/aB!#>$!3?>/+MLt\")=F.7&F@F@$\"3%)Q4VPH$)G5F.$!3Gc]')RKgeZF 3%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e1 := Eigenva lues(B[1..2,1..2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G-%'RTABLE G6%\"*CIOZ\"-%'MATRIXG6#7$7#^$$\"3%QEiq#*fhs\"!#<$\"3D\"QM:b,k5)!#=7#^ $F/$!3D\"QM:b,k5)F4&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e2 := Eigenvalues(B[3..4,3..4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G-%'RTABLEG6%\"*_V`b\"-%'MATRIXG6#7$7#^$$!3W'pA1F* fhA!#=$\"332ph;$4)o8!#<7#^$F/$!332ph;$4)o8F4&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "e := [1..-1,1]; # sh ould be the root vector" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG-%'RT ABLEG6%\"*!o0r:-%'MATRIXG6#7&7#^$$\"3%QEiq#*fhs\"!#<$\"3D\"QM:b,k5)!#= 7#^$F/$!3D\"QM:b,k5)F47#^$$!3W'pA1F*fhAF4$\"332ph;$4)o8F17#^$F;$!332ph ;$4)o8F1&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 " Verify the roots" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p := x^ 4 - 3*x^3 + 4*x^2 - 5*x +7:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for k from 1 to 4 do\n subs(x=e[k],p)\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"\"!\"*$\"\"%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"\"!\"*$!\"%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"%! \"*$\"\"'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"%!\"*$!\"'F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "24 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 146430812 146516532 147671200 155606188 155262820 147741224 155831556 156900008 147363024 155534352 157105680 }{RTABLE M7R0 I6RTABLE_SAVE/146430812X,%)anythingG6"6"[gl!"%!!!#:"&"&$"3[Mxd[:N(y"!#<$"3[0bdx C$*RVF)$"3urxoeIH"\$F)$!3-/"eU@$)Gk%F)$"3#z'ec"*pSd:F)$"3UI!HREC\f%F)$"3uVbf&Ht ?#HF)$"3!4n!*=Dbt:%F)$!33.DPP<(Q#y!#=$!3kYys8m8"e$F)$"3+-!))))o/G+$F)$!3!)zerF^ Vi9F:$"3Oc%HC[p;d%F)$"33chgN&))[YF)$"3?wHMa$o]d% F)$"3g%Q)\?>:)o%F:$!3.;&H8"y,:AF)$!3v/f+]ffCSF)$"35&4aAY#fB8F)$"3F$>!Rh&eP8%F)$ !3BR\1P=8IPF)$"3-#>c2P>z0%F)$"3e*yJR'oBZJF)F& } {RTABLE M7R0 I6RTABLE_SAVE/146516532X,%)anythingG6#%)identityG6"[gl!""!!!#!"&"&F' } {RTABLE M7R0 I6RTABLE_SAVE/147671200X,%)anythingG6"6"[gl!"%!!!#:"&"&$"3%*e/7(G&zJn!#<$"31`([ y"pc&*[!#>""!F-F-$!3ghx\^rwg\F)$"3-')p0a+VkoF)F-F-F-$"3^w#G_#o'47#F)$!3&R[cph3( 4vF)$!3))>A2iq]b@F)$!3UPFn]^X#o%F)F-$!3Y0UoR/?@r!#=$!3q#zO'[Ail>F)$"3'3!*HH=LcM (F)$!3gq$e)ed6ahFx.h1!H\F)$!3I$Rm4L#yT>F)$!3%fJQQ5E`J#F)$"3_ikn&=&Q*)G F)$!3qFR9SK$*>hF