{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 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 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{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 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 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 "" {MPLTEXT 1 0 33 "A := Matrix([[1,3],[2,0],[3,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*7QzX\"-%'MATRIXG6#7%7$ \"\"\"\"\"$7$\"\"#\"\"!7$F/\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "b := Vector([3,2,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*![zd9-%'MATRIXG6#7%7#\"\"$7#\"\"#7#\"\"&&%' VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with( LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transpos e(A).A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*3hkY\"-%'MAT RIXG6#7$7$\"#9\"#:7$F-\"#D%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transpose(A).b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\"*g%)pY\"-%'MATRIXG6#7$7#\"#A7#\"#H&%'VectorG6#%'columnG" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " Example 5.7.2, page 200" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "epsilon := 0.0001;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"\"\"!\"%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "A := Matrix([[1,1,1],[epsilon,0,0],[0,epsilon,0],[0 ,0,epsilon]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\" *g6tY\"-%'MATRIXG6#7&7%\"\"\"F.F.7%$F.!\"%\"\"!F27%F2F0F27%F2F2F0%'Mat rixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ConditionNumber(A,2 );" }}{PARA 8 "" 1 "" {TEXT -1 221 "Error, (in LinearAlgebra:-Conditio nNumber) invalid input: LinearAlgebra:-ConditionNumber expects its 1st argument, M, to be of type Matrix(square) but received Matrix(4, 3, [ [...],[...],[...],[...]], datatype = anything)\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B := Transpose(A).A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"*gZ*o9-%'MATRIXG6#7%7%$\"3%*******4 +++5!#<$\"\"\"\"\"!F17%F1F.F17%F1F1F.%'MatrixG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "ConditionNumber(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P+++S!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A := Matrix([[1,3],[2,0],[3,4]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*Gr,Z\"-%'MATRIXG6 #7%7$\"\"\"\"\"$7$\"\"#\"\"!7$F/\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "b := Vector([3,2,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*'plp9-%'MATRIXG6#7%7#\"\"$7#\"\"#7 #\"\"&&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "(Q,R) := QRDecomposition(1.0*A, fullspan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"QG%\"RG6$-%'RTABLEG6%\"*_/AZ\"-%'MATRIXG6#7%7%$!3 'GCC\">ChsE!#=$\"3%ps0/\\MUX'F3$!31F$**z_tstDPy,)F3$\"3plc`SH]HEF3$\"3Y%\\**f 9jlO&F3%'MatrixG-F)6%\"*7CBZ\"-F-6#7%7$$!3LTRx'Qd;u$!#<$!3hljoG'=*3SFP 7$$\"\"!FU$\"3[%)fLB:2))HFP7$FTFTFF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d := Transpose(Q).b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG-%'RTABLEG6%\"*+[AZ\"-%'MATRIXG6#7%7#$!31OL2Atuze!#<7#$\"33z# )='[Zn\"=F07#$!3'=n**Rw3xd$!#=&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "R1 := R[1..2,1..2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R1G-%'RTABLEG6%\"*?_KZ\"-%'MATRIXG6#7$7$$!3LTRx'Qd;u $!#<$!3hljoG'=*3SF07$$\"\"!F5$\"3[%)fLB:2))HF0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d1 := d[1..2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d1G-%'RTABLEG6%\"*SYAZ\"-%'MATRIXG6#7$7#$!31OL2Atuze !#<7#$\"33z#)='[Zn\"=F0&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "LinearSolve(R1,d1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*![Ds9-%'MATRIXG6#7$7#$\"31(************> *!#=7#$\"31-+++++!3'F.&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "For example 5.7.1, solved by QR decomposition: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A := Matrix([[1,0,0],[0,1,0] ,[0,0,1],[-1,1,0],[0,-1,1],[-1,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*OF[Z\"-%'MATRIXG6#7(7%\"\"\"\"\"!F/7%F/F.F/ 7%F/F/F.7%!\"\"F.F/7%F/F3F.7%F3F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "b := Vector([1,2,3,1,2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*SeAZ\"-%'MATRIXG6#7(7#\"\"\"7#\"\" #7#\"\"$F-F/F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "(Q,R) := QRDecomposition(1.0*A,fullspan); # Step 1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6$%\"QG%\"RG6$-%'RTABLEG6%\"*SZcZ\" -%'MATRIXG6#7(7($!3Kdi*=p-Nx&!#=$!3\"[J>BXT7/#F3$!3JPF$f!R`NNF3$\"3[\" >IH?#R8^F3$!3wGM>a-x]B!#>$\"3k\\3^x^Jy[F37($!\"!\"\"!$!3qWz&pNCP7'F3$! 3'ytKf!R`NNF3$!35\\3^x^Jy[F3$\"3P!>IH?#R8^F3$\"37@M>a-x]BF<7(F@F@$!3Ss a'=\"y1rqF3$!3y?M>a-x]BF<$!3?]3^x^Jy[F3$!3[\">IH?#R8^F37($\"3Kdi*=p-Nx &F3$!3iH'QY!H[#3%F3$\"3ME9Q?ipc=!#N$\"3;8q9gC!Rm'F3$\"3o@o@d-^]:F3$!3M khj#G(e&y\"F37(F@$\"3qWz&pNCP7'F3$!3UQF$f!R`NNF3$\"31khj#G(e&y\"F3Fhn$ !3'>#o@d-^]:F37(FW$\"3\"[J>BXT7/#F3F6$!3o@o@d-^]:F3F\\o$\"3E9q9gC!Rm'F 3%'MatrixG-F)6%\"*+1dZ\"-F-6#7(7%$!3?x)ov!30K " 0 "" {MPLTEXT 1 0 30 "d := Transpose(Q).b; # Step 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG-%'RTABLEG6%\"*_]iZ\" -%'MATRIXG6#7(7#$\"3Kdi*=p-Nx&!#=7#$!3iH'QY!H[#3%F07#$!3x%G>roSEC%!#<7 #$\"3&o)H&)Rv4OLF07#$\"3cqJyU(*[\\%)F07#$!3$[$QO " 0 "" {MPLTEXT 1 0 42 "R1, d1 := R[1..3 ,1..3], d[1..3]; # step 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#R1G %#d1G6$-%'RTABLEG6%\"*oblZ\"-%'MATRIXG6#7%7%$!3?x)ov!30KroSE C%F3&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "x := BackwardSubstitute(R1,d1); # step 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'RTABLEG6%\"*gNgZ\"-%'MATRIXG6#7%7#$\"3y******* *****\\7!#<7#$\"3c************\\ " 0 "" {MPLTEXT 1 0 15 "?LinearAlgebr a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "x, y := Vector([1,2,3,4,5,6]),Vector([.62,.34 ,.28,.36,.46,.43]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"xG%\"yG6$ -%'RTABLEG6%\"*+iAZ\"-%'MATRIXG6#7(7#\"\"\"7#\"\"#7#\"\"$7#\"\"%7#\"\" 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