{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 "" 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 "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 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "A := Matrix([[8,2,-2],[3,10,4],[5,1,7]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*3pWY\"-%'MATRIXG6 #7%7%\"\")\"\"#!\"#7%\"\"$\"#5\"\"%7%\"\"&\"\"\"\"\"(%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b := Vector([3,-2,4]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*7lVY\"-%'MATRIXG6 #7%7#\"\"$7#!\"#7#\"\"%&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B := Matrix(%id = 145810168);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"*o,\"e9-%'MATRIXG6#7%7%\"\"!#!\" \"\"\"%#\"\"\"F17%#!\"$\"#5F.#!\"#\"\"&7%#!\"&\"\"(#F0F>F.%'MatrixG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "d := Vector([3/8,-1/5,4/7] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG-%'RTABLEG6%\"*/>kY\"-%'MA TRIXG6#7%7##\"\"$\"\")7##!\"\"\"\"&7##\"\"%\"\"(&%'VectorG6#%'columnG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "x[0] := RandomVector(3, generator=-1.0..1.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\" !-%'RTABLEG6%\"*#fOk9-%'MATRIXG6#7%7#$!3!*y)HTnj-Y(!#=7#$\"3\\%Q7:uQe6 )F37#$\"3syN'ystWH'F3&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "for k from 1 to 10 do\n x[k] := B.x[k-1]+d:\nend \+ do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x[9],x[10];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*s5WZ\"-%'MATRIXG6#7%7#$ \"3^xn+hTe>b!#=7#$!3q6)GQV%4KYF.7#$\"35_FObGOICF.&%'VectorG6#%'columnG -F$6%\"*OT^Z\"-F(6#7%7#$\"3R!*yHAVh:bF.7#$!3)R8Z/R?!GYF.7#$\"3qJ@avLXL CF.F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "beta := 6/7;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#\"\"'\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "(beta/(1-beta))*Norm(x[10]-x[9],infinity) ; # error bound on x[10]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+HgUWC! #7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Let's verify that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z := LinearSolve(1.0*A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG- %'RTABLEG6%\"*C&\\x9-%'MATRIXG6#7%7#$\"3sk!eA.Hh^&!#=7#$!3,;X1eA.HYF07 #$\"3]Tx'4(Q[NCF0&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Norm(x[10]-z,infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3p!z4cU&\\I?!#@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "For G-S method:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "E := Matrix([[6,0 ,0],[1,5,0],[2,1,7]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTA BLEG6%\"*_h#y9-%'MATRIXG6#7%7%\"\"'\"\"!F/7%\"\"\"\"\"&F/7%\"\"#F1\"\" (%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S := Matrix([ [0,0,2],[0,0,-1],[0,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG- %'RTABLEG6%\"*%)Q$y9-%'MATRIXG6#7%7%\"\"!F.\"\"#7%F.F.!\"\"7%F.F.F.%'M atrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B := - MatrixInve rse(E).S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"*c?(z 9-%'MATRIXG6#7%7%\"\"!F.#!\"\"\"\"$7%F.F.#\"\"%\"#:7%F.F.#\"\"#\"#N%'M atrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "d := -MatrixInver se(E).b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG-%'RTABLEG6%\"*3e1[ \"-%'MATRIXG6#7%7##!\"\"\"\"#7##\"\"\"F0F-&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "To find the least squares solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "x, y := Vector([1.5,2,2.5,3,3.5]), Vector([5.1,3.5, 1.2,2.8,4.2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"xG%\"yG6$-%'RT ABLEG6%\"*;w(y9-%'MATRIXG6#7'7#$\"#:!\"\"7#\"\"#7#$\"#DF37#\"\"$7#$\"# NF3&%'VectorG6#%'columnG-F)6%\"*'fB![\"-F-6#7'7#$\"#^F3F;7#$\"#7F37#$ \"#GF37#$\"#UF3F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A := M atrix(5,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for k from 1 to 5 do\n A[k,1] := 1:\n A[k,2] := x[k]:\n A[k,3] := x[k]^2:\ne nd do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*[&)eY\"-%'MATRIXG6#7'7%\"\"\"$\"#:! \"\"$\"$D#!\"#7%F,\"\"#\"\"%7%F,$\"#DF/$\"$D'F27%F,\"\"$\"\"*7%F,$\"#N F/$\"%D7F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b := \+ y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*'fB![\"-%'M ATRIXG6#7'7#$\"#^!\"\"7#$\"#NF07#$\"#7F07#$\"#GF07#$\"#UF0&%'VectorG6# %'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a := LeastSqua res(A,b); # least squares solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"aG-%'RTABLEG6%\"*3RaZ\"-%'MATRIXG6#7%7#$\"3'>cG9dGu3#!#;7#$!3m2dG 9dGk9F07#$\"3v;9dG9dGG!#<&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "p := a[1] + a[2]*t + a[3]*t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,($\"3'>cG9dGu3#!#;\"\"\"*&$\"3m2dG9dGk9F(F) %\"tGF)!\"\"*&$\"3v;9dG9dGG!# " 0 "" {MPLTEXT 1 0 15 "plot(p,t=1..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 491 491 491 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"\"\"\"!$\"3kf********* *f!*!#<7$$\"31++]i9Rl5F-$\"3]/xM&41X[)F-7$$\"34+]PC#)GA6F-$\"3QF\"4;d_ M+)F-7$$\"3%****\\Peui=\"F-$\"353c+2iL%[(F-7$$\"33++D'3&o]7F-$\"3e]1#) )Qr^)pF-7$$\"3/+](oX*y98F-$\"3JhL!z\"=o6lF-7$$\"3#***\\P9CAu8F-$\"3kPI ;!GkM4'F-7$$\"3.+]P*zhdV\"F-$\"37'f?Ps\"[\"o&F-7$$\"3++]P>fS*\\\"F-$\" 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