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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "On Problem 8, page
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{MPLTEXT 1 0 85 "A := Matrix([[1,0,0,0,1],[-1,1,0,0,1],[-1,-1,1,0,1],[
-1,-1,-1,1,1],[-1,-1,-1,-1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"AG-%'RTABLEG6%\"*'4!zX\"-%'MATRIXG6#7'7'\"\"\"\"\"!F/F/F.7'!\"\"F.F/
F/F.7'F1F1F.F/F.7'F1F1F1F.F.7'F1F1F1F1F.%'MatrixG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 100 "(P,L,U) := LUDecomposition(A); # LU Decomposi
tion \n                               # without pivoting" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>6%%\"PG%\"LG%\"UG6%-%'RTABLEG6%\"*/7$o9-%'MATRI
XG6#7'7'\"\"\"\"\"!F3F3F37'F3F2F3F3F37'F3F3F2F3F37'F3F3F3F2F37'F3F3F3F
3F2%'MatrixG-F*6%\"*_+-Z\"-F.6#7'F17'!\"\"F2F3F3F37'F@F@F2F3F37'F@F@F@
F2F37'F@F@F@F@F2F8-F*6%\"*Ki.Z\"-F.6#7'7'F2F3F3F3F27'F3F2F3F3\"\"#7'F3
F3F2F3\"\"%7'F3F3F3F2\"\")7'F3F3F3F3\"#;F8" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 " ConditionNumber(1.0*A,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+SM#RA#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "s := ConditionNumber(1.0*L,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"sG$\"+?SvUH!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "t \+
:= ConditionNumber(1.0*U,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG
$\"+^8\"G8#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "s*t;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+x\"RjF'!\"(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The error in solving
  P.L.Ux=b  is substantially larger than solving Ax=b, without pivotin
g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(P1,L1,U1) := LUDecompositio
n(1.0*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%#P1G%#L1G%#U1G6%-%'RT
ABLEG6%\"*SuCZ\"-%'MATRIXG6#7'7'\"\"\"\"\"!F3F3F37'F3F2F3F3F37'F3F3F2F
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3F3FCFCFC7'$FBF3F@FCFCFC7'FEFEF@FCFC7'FEFEFEF@FC7'FEFEFEFEF@F8-F*6%\"*
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CFP$\"\")F37'FCFCFCFC$\"#;F3F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 22 "With QR decomposition:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "(Q,R) \+
:= QRDecomposition(1.0*A,fullspan);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>6$%\"QG%\"RG6$-%'RTABLEG6%\"*'pe6:-%'MATRIXG6#7'7'$!3sy&**\\&f8sW!#=
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{XPPMATH 20 "6#$\"+++++5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "ConditionNumber(R,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SM#
RA#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 165 "The combined condition number is the same as matrix  A. \+
 Therefore, solving the linear system with QR decomposition is general
ly more accurate than LU decomposition." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 65 "But LU decomposition is about twice \+
as fast as QR decomposition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 13 "Section 5.6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A := Matrix([[4,-1,-1,0],
[-1,4,0,-1],[-1,0,4,-1],[0,-1,-1,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'RTABLEG6%\"*K`J^\"-%'MATRIXG6#7&7&\"\"%!\"\"F/\"\"!7&F/
F.F0F/7&F/F0F.F/7&F0F/F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "E := Matrix([[4,0,0,0],[0,4,0,0],[0,0,4,0],[0,0,0,4]]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6%\"*WRK^\"-%'MA
TRIXG6#7&7&\"\"%\"\"!F/F/7&F/F.F/F/7&F/F/F.F/7&F/F/F/F.%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "C := A-E;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6%\"*?6U^\"-%'MATRIXG6#7&7&\"\"!!\"\"
F/F.7&F/F.F.F/F0F-%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "F := MatrixInverse(E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%
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10 "B := -F.C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"
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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b := Vector([1,2,0,1]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*OJL]\"-%'MATRIXG
6#7&7#\"\"\"7#\"\"#7#\"\"!F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "d := F.b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"dG-%'RTABLEG6%\"*;H#=:-%'MATRIXG6#7&7##\"\"\"\"\"%7##F/\"\"#7#\"\"
!F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 
"x[0] := RandomVector(4,generator=-1.0..1.0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"xG6#\"\"!-%'RTABLEG6%\"*_p*y9-%'MATRIXG6#7&7#$\"3=
pn*4%QIw$*!#>7#$!31K!fEiN+V%!#=7#$!3]4=,+>>\\!)F77#$\"3?!>3X#\\=ZEF7&%
'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for \+
k from 0 to 100 do\n   x[k+1] := B.x[k]+d:\nend do:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "x[101];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'RTABLEG6%\"*K<1Z\"-%'MATRIXG6#7&7#$\"3++++++++]!#=7#$\"3++++++++vF
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{TEXT -1 20 "Gauss-Seidel method:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "P := Matrix([[4,0,0,0],[-1,4
,0,0],[-1,0,4,0],[0,-1,-1,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"PG-%'RTABLEG6%\"*?L'H:-%'MATRIXG6#7&7&\"\"%\"\"!F/F/7&!\"\"F.F/F/7&F
1F/F.F/7&F/F1F1F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "Q := A-P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'RTABLEG6%\"*[
lj^\"-%'MATRIXG6#7&7&\"\"!!\"\"F/F.7&F.F.F.F/F07&F.F.F.F.%'MatrixG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P, MatrixInverse(P);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*?L'H:-%'MATRIXG6#7&7&\"
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B := -MatrixInverse(1.0*P).Q
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"*?g/`\"-%'MAT
RIXG6#7&7&$!\"!\"\"!$\"3++++++++D!#=F1F.7&F.$\"3+++++++]i!#>F5F1F47&F.
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" {MPLTEXT 1 0 28 "d := MatrixInverse(1.0*P).b;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"dG-%'RTABLEG6%\"*C77`\"-%'MATRIXG6#7&7#$\"3++++++++
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