{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 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 288 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Introduction to LU decomposition " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 40 "load extra procedures for linear systems" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 288 8 "linsys.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the o ne that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/linsys.m \";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Finding an LU decomposition from 1st principles" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "A = matrix([[1, 1, -1], [1, 2, -2], [-2, 1, 1]]);" "6#/ %\"AG-%'matrixG6#7%7%\"\"\"F*,$F*!\"\"7%F*\"\"#,$F.F,7%,$F.F,F*F*" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 25 "We find two 3x3 matrice s " }{XPPEDIT 18 0 "L = matrix([[1, 0, 0], [m[2,1], 1, 0], [m[3,1], m[ 3,2], 1]]);" "6#/%\"LG-%'matrixG6#7%7%\"\"\"\"\"!F+7%&%\"mG6$\"\"#F*F* F+7%&F.6$\"\"$F*&F.6$F4F0F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "U = m atrix([[u[1,1], u[1,2], u[1,3]], [0, u[2,2], u[2,3]], [0, 0, u[3,3]]]) ;" "6#/%\"UG-%'matrixG6#7%7%&%\"uG6$\"\"\"F-&F+6$F-\"\"#&F+6$F-\"\"$7% \"\"!&F+6$F0F0&F+6$F0F37%F5F5&F+6$F3F3" }{TEXT -1 8 ", where " }{TEXT 262 1 "L" }{TEXT -1 4 " is " }{TEXT 261 16 "lower triangular" }{TEXT -1 5 " and " }{TEXT 263 1 "U" }{TEXT -1 4 " is " }{TEXT 261 16 "upper \+ triangular" }{TEXT -1 12 ", such that " }{XPPEDIT 18 0 "L*`.`*U=A" "6# /*(%\"LG\"\"\"%\".GF&%\"UGF&%\"AG" }{TEXT -1 10 ", that is," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1, 0, 0], [m[2,1], 1, 0], [m[3,1], m[3,2], 1]]) *`.`* matrix([[u[1,1], u[1,2], u[1,3]], \+ [0, u[2,2], u[2,3]], [0, 0, u[3,3]]]) = matrix([[1, 1, -1], [1, 2, -2] , [-2, 1, 1]])" "6#/*(-%'matrixG6#7%7%\"\"\"\"\"!F+7%&%\"mG6$\"\"#F*F* F+7%&F.6$\"\"$F*&F.6$F4F0F*F*%\".GF*-F&6#7%7%&%\"uG6$F*F*&F=6$F*F0&F=6 $F*F47%F+&F=6$F0F0&F=6$F0F47%F+F+&F=6$F4F4F*-F&6#7%7%F*F*,$F*!\"\"7%F* F0,$F0FP7%,$F0FPF*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The two matrices " }{TEXT 264 1 "L" } {TEXT -1 5 " and " }{TEXT 265 1 "U" }{TEXT -1 20 " will constitute an \+ " }{TEXT 261 16 "LU decomposition" }{TEXT -1 4 " of " }{TEXT 266 1 "A " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Because the first row of " }{TEXT 267 1 "L" }{TEXT -1 32 " is [1 0 0], the first row of " }{TEXT 268 1 "U" }{TEXT -1 38 " mus t be the same as the first row of " }{TEXT 269 1 "A" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " } {TEXT 270 1 "U" }{TEXT -1 14 " has the form " }{XPPEDIT 18 0 "matrix([ [1, 1, -1], [0, u[2,2], u[2,3]], [0, 0, u[3,3]]])" "6#-%'matrixG6#7%7% \"\"\"F(,$F(!\"\"7%\"\"!&%\"uG6$\"\"#F0&F.6$F0\"\"$7%F,F,&F.6$F3F3" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "m[2,1]*u[1,1]=1" "6#/*&&%\"mG6$\" \"#\"\"\"F)&%\"uG6$F)F)F)F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[1,1 ]=1" "6#/&%\"uG6$\"\"\"F'F'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "m[2,1]=1" "6#/&%\"mG6$\"\"#\"\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 6 "Since " }{XPPEDIT 18 0 "m[3,1]*u[1,1] = -2;" "6#/*&&%\"m G6$\"\"$\"\"\"F)&%\"uG6$F)F)F),$\"\"#!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "u[1,1]=1" "6#/&%\"uG6$\"\"\"F'F'" }{TEXT -1 10 ", we ha ve " }{XPPEDIT 18 0 "m[3,1] = -2;" "6#/&%\"mG6$\"\"$\"\"\",$\"\"#!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }}{PARA 257 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "matrix([[1, 0, 0], [1, 1, 0], [-2, m[3,2], 1]])*`.`*mat rix([[1, 1, -1], [0, u[2,2], u[2,3]], [0, 0, u[3,3]]]) = matrix([[1, 1 , -1], [1, 2, -2], [-2, 1, 1]]);" "6#/*(-%'matrixG6#7%7%\"\"\"\"\"!F+7 %F*F*F+7%,$\"\"#!\"\"&%\"mG6$\"\"$F/F*F*%\".GF*-F&6#7%7%F*F*,$F*F07%F+ &%\"uG6$F/F/&F=6$F/F47%F+F+&F=6$F4F4F*-F&6#7%7%F*F*,$F*F07%F*F/,$F/F07 %,$F/F0F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "1+u[2,2]=2" "6#/,&\"\"\" F%&%\"uG6$\"\"#F)F%F)" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "u[2,2 ]=1" "6#/&%\"uG6$\"\"#F'\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "-1+u[2,3] = -2;" "6#/,&\"\"\"!\"\" &%\"uG6$\"\"#\"\"$F%,$F*F&" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 " u[2,3] = -1;" "6#/&%\"uG6$\"\"#\"\"$,$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This give s " }}{PARA 257 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "matrix([[1, 0, 0], [1, 1, 0], [-2, m[3,2], 1]])*`.`*matrix([[1, 1, -1], [0, 1, -1 ], [0, 0, u[3,3]]]) = matrix([[1, 1, -1], [1, 2, -2], [-2, 1, 1]]);" " 6#/*(-%'matrixG6#7%7%\"\"\"\"\"!F+7%F*F*F+7%,$\"\"#!\"\"&%\"mG6$\"\"$F /F*F*%\".GF*-F&6#7%7%F*F*,$F*F07%F+F*,$F*F07%F+F+&%\"uG6$F4F4F*-F&6#7% 7%F*F*,$F*F07%F*F/,$F/F07%,$F/F0F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "-2+m[3,2] = 1;" "6#/,&\"\"#!\"\"&%\"mG6$\"\"$F%\"\"\"F+" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "m[3,2] = 3" "6#/&%\"mG6$\"\"$\"\"#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "2 -m[3,2]+u[3,3]=1" "6#/,(\"\"#\"\"\"&%\"mG6$\"\"$F%!\"\"&%\"uG6$F*F*F&F &" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "u[3,3]=2" "6#/&%\"uG6$\"\"$F'\" \"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 12 "We now have " }{XPPEDIT 18 0 "L = matrix([[1, 0, 0], [1 , 1, 0], [-2, 3, 1]])" "6#/%\"LG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F*F*F+7 %,$\"\"#!\"\"\"\"$F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "U = matrix([ [1, 1, -1], [0, 1, -1], [0, 0, 2]])" "6#/%\"UG-%'matrixG6#7%7%\"\"\"F* ,$F*!\"\"7%\"\"!F*,$F*F,7%F.F.\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Finally, we check that \+ " }{XPPEDIT 18 0 "L*`.`*U=A" "6#/*(%\"LG\"\"\"%\".GF&%\"UGF&%\"AG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "L := Matrix([[1, 0, 0], [1, 1, 0], [-2, 3, 1]]) ;\nU := Matrix([[1, 1, -1], [0, 1, -1], [0, 0, 2]]);\nL . U;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'RTABLEG6$\")KmT;-%'MATRIXG6#7%7%\" \"\"\"\"!F/7%F.F.F/7%!\"#\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"UG-%'RTABLEG6$\"(#HFF-%'MATRIXG6#7%7%\"\"\"F.!\"\"7%\"\"!F.F/7%F1F1 \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")[I;?-%'MATRIXG 6#7%7%\"\"\"F,!\"\"7%F,\"\"#!\"#7%F0F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 84 "Using Gaussian and Gauss-Jordan elimination to \+ find an LU decomposition of a matrix " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "We find an LU decomposition of the matrix " }{XPPEDIT 18 0 "A = matrix([[1, 1, -1], [1, 2, -2], [-2, 1, 1]]);" "6#/%\"AG-%'matrixG6#7 %7%\"\"\"F*,$F*!\"\"7%F*\"\"#,$F.F,7%,$F.F,F*F*" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := Matrix([[1,1,-1],[1,2,-2],[-2,1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")+`R;-%'MATRIXG6#7%7%\"\"\"F.!\"\"7 %F.\"\"#!\"#7%F2F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We use the procedure " }{TEXT 0 19 "GaussianEliminati on" }{TEXT -1 10 " from the " }{TEXT 0 13 "LinearAlgebra" }{TEXT -1 43 " package to apply row operations to reduce " }{TEXT 271 1 "A" } {TEXT -1 7 " to an " }{TEXT 261 23 "upper triangular matrix" }{TEXT -1 1 " " }{TEXT 272 1 "U" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 16 "GaussElimination" }{TEXT -1 43 " from another worksheet could also be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "U := GaussianElimination(A); \n#U := GaussElimination(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG -%'RTABLEG6$\")kPb;-%'MATRIXG6#7%7%\"\"\"F.!\"\"7%\"\"!F.F/7%F1F1\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The problem is to find a \+ " }{TEXT 261 23 "lower triangular matrix" }{TEXT -1 1 " " }{TEXT 273 1 "L" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "L*`.`*U = A" "6#/*(% \"LG\"\"\"%\".GF&%\"UGF&%\"AG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 9 "transp ose" }{TEXT -1 20 " of a square matrix " }{TEXT 274 1 "A" }{TEXT -1 15 " is the matrix " }{XPPEDIT 18 0 "A^t;" "6#)%\"AG%\"tG" }{TEXT -1 51 " obtained by interchanging the rows and columns of " }{TEXT 275 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 276 1 "B " }{TEXT -1 35 " is another square matrix, we have " }{XPPEDIT 18 0 "( A*`.`*B)^t=B^t*`.`*A^t" "6#/)*(%\"AG\"\"\"%\".GF'%\"BGF'%\"tG*()F)F*F' F(F')F&F*F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "Taking th e transpose of each side of this matrix equation " }{XPPEDIT 18 0 "L*` .`*U=A" "6#/*(%\"LG\"\"\"%\".GF&%\"UGF&%\"AG" }{TEXT -1 20 " gives the equation " }{XPPEDIT 18 0 "U^t*`.`*L^t=A^t" "6#/*()%\"UG%\"tG\"\"\"% \".GF()%\"LGF'F()%\"AGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "This equation can then be solved for " }{XPPEDIT 18 0 "L^t;" "6#)% \"LG%\"tG" }{TEXT -1 29 " by Gauss-Jordan elimination." }}{PARA 0 "" 0 "" {TEXT -1 36 "This can be done with the procedure " }{TEXT 0 21 "R educedRowEchelonForm" }{TEXT -1 10 " from the " }{TEXT 0 13 "LinearAlg ebra" }{TEXT -1 33 " package, or using the procedure " }{TEXT 0 11 "Ga ussJordan" }{TEXT -1 24 " from another worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Ut := Transp ose(U);\nAt := Transpose(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#UtG -%'RTABLEG6$\")cT^;-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F.F.F/7%!\"\"F2\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AtG-%'RTABLEG6$\")?1^;-%'MATRIX G6#7%7%\"\"\"F.!\"#7%F.\"\"#F.7%!\"\"F/F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "UA := ;\nR educedRowEchelonForm(UA);\n#GaussJordan(UA);\nDeleteColumn(%,1..3);\nL := Transpose(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#UAG-%'RTABLEG6 $\")7xt@-%'MATRIXG6#7%7(\"\"\"\"\"!F/F.F.!\"#7(F.F.F/F.\"\"#F.7(!\"\"F 4F2F4F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")3'=#=-%'MA TRIXG6#7%7(\"\"\"\"\"!F-F,F,!\"#7(F-F,F-F-F,\"\"$7(F-F-F,F-F-F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")!eM\">-%'MATRIXG6#7%7% \"\"\"F,!\"#7%\"\"!F,\"\"$7%F/F/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"LG-%'RTABLEG6$\")OQY;-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F.F.F/7%!\"#\" \"$F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Now check that " }{XPPEDIT 18 0 "L*`.`*U=A" "6#/*(%\"LG\"\"\"%\".GF&% \"UGF&%\"AG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "L.U;\nA;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")WyX;-%'MATRIXG6#7%7%\"\"\"F,!\"\"7%F,\" \"#!\"#7%F0F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")+`R; -%'MATRIXG6#7%7%\"\"\"F,!\"\"7%F,\"\"#!\"#7%F0F,F," }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "We find an LU \+ decomposition of the matrix " }{XPPEDIT 18 0 "A = matrix([[1, 1, -1, 2 ], [1, 2, -2, 1], [-2, 1, 1, -1], [1, 2, 1, -2]]);" "6#/%\"AG-%'matrix G6#7&7&\"\"\"F*,$F*!\"\"\"\"#7&F*F-,$F-F,F*7&,$F-F,F*F*,$F*F,7&F*F-F*, $F-F," }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A := Matrix([[1,1,-1,2],[1,2,-2,1], [-2,1,1,-1],[1,2,1,-2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%' RTABLEG6%\")%Qsm\"-%'MATRIXG6#7&7&\"\"\"F.!\"\"\"\"#7&F.F0!\"#F.7&F2F. F.F/7&F.F0F.F2%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 7 "We use " }{TEXT 0 19 "GaussianElimination" }{TEXT -1 35 " to apply row operations to reduce " }{TEXT 277 1 "A" }{TEXT -1 7 " to an " }{TEXT 261 23 "upper triangular matrix" }{TEXT -1 1 " \+ " }{TEXT 278 1 "U" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "U := GaussianElimination(A);\n#U := GaussElimination(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'RTABLEG6%\")ke$o\"-%'MATRIXG6#7&7& \"\"\"F.!\"\"\"\"#7&\"\"!F.F/F/7&F2F2F0\"\"'7&F2F2F2!#7%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Taking the transpose of each si de of the matrix equation " }{XPPEDIT 18 0 "L*`.`*U=A" "6#/*(%\"LG\"\" \"%\".GF&%\"UGF&%\"AG" }{TEXT -1 20 " gives the equation " }{XPPEDIT 18 0 "U^t*`.`*L^t=A^t" "6#/*()%\"UG%\"tG\"\"\"%\".GF()%\"LGF'F()%\"AGF '" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "This equation can b e solved for " }{XPPEDIT 18 0 "L^t;" "6#)%\"LG%\"tG" }{TEXT -1 29 " by Gauss-Jordan elimination." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Ut := Transpose(U);\nAt := Transpos e(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#UtG-%'RTABLEG6%\")KM8<-%'M ATRIXG6#7&7&\"\"\"\"\"!F/F/7&F.F.F/F/7&!\"\"F2\"\"#F/7&F3F2\"\"'!#7%'M atrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AtG-%'RTABLEG6%\")70\"p\" -%'MATRIXG6#7&7&\"\"\"F.!\"#F.7&F.\"\"#F.F17&!\"\"F/F.F.7&F1F.F3F/%'Ma trixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "UA := ;\nReducedRowEchelonForm(UA);\n#GaussJo rdan(UA);\nDeleteColumn(%,1..4);\nL := Transpose(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#UAG-%'RTABLEG6%\"))[hr\"-%'MATRIXG6#7&7*\"\"\"\" \"!F/F/F.F.!\"#F.7*F.F.F/F/F.\"\"#F.F27*!\"\"F4F2F/F4F0F.F.7*F2F4\"\"' !#7F2F.F4F0%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\" )o;[<-%'MATRIXG6#7&7*\"\"\"\"\"!F-F-F,F,!\"#F,7*F-F,F-F-F-F,\"\"$F,7*F -F-F,F-F-F-F,#F0\"\"#7*F-F-F-F,F-F-F-F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")CL_<-%'MATRIXG6#7&7&\"\"\"F,!\"#F,7&\"\" !F,\"\"$F,7&F/F/F,#F0\"\"#7&F/F/F/F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'RTABLEG6%\")OD^<-%'MATRIXG6#7&7&\"\"\"\"\"!F/F /7&F.F.F/F/7&!\"#\"\"$F.F/7&F.F.#F3\"\"#F.%'MatrixG" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Now check that " } {XPPEDIT 18 0 "L*`.`*U=A" "6#/*(%\"LG\"\"\"%\".GF&%\"UGF&%\"AG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "L.U;\nA;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RT ABLEG6%\")3))e<-%'MATRIXG6#7&7&\"\"\"F,!\"\"\"\"#7&F,F.!\"#F,7&F0F,F,F -7&F,F.F,F0%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\" )%Qsm\"-%'MATRIXG6#7&7&\"\"\"F,!\"\"\"\"#7&F,F.!\"#F,7&F0F,F,F-7&F,F.F ,F0%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "Using an LU decomposition t o solve a linear system of equations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Consider the linear system " }{XPPEDIT 18 0 "A*`.`*x = b" "6#/*(%\"AG\"\"\"%\".GF&%\"xGF& %\"bG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "A = matrix([[1, 1, -1], \+ [1, 2, -2], [-2, 1, 1]])" "6#/%\"AG-%'matrixG6#7%7%\"\"\"F*,$F*!\"\"7% F*\"\"#,$F.F,7%,$F.F,F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = mat rix([[4], [-1], [6]])" "6#/%\"bG-%'matrixG6#7%7#\"\"%7#,$\"\"\"!\"\"7# \"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "It is shown above that the matrix " }{TEXT 279 1 "A " }{TEXT -1 27 " has the LU decomposition " }{XPPEDIT 18 0 "A = L*`.` *U;" "6#/%\"AG*(%\"LG\"\"\"%\".GF'%\"UGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "L = matrix([[1, 0, 0], [1, 1, 0], [-2, 3, 1]])" "6#/%\"LG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F*F*F+7% ,$\"\"#!\"\"\"\"$F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "U = matrix([[ 1, 1, -1], [0, 1, -1], [0, 0, 2]])" "6#/%\"UG-%'matrixG6#7%7%\"\"\"F*, $F*!\"\"7%\"\"!F*,$F*F,7%F.F.\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 " L*`.`*U*`.`*x=b" "6#/*,%\"LG\"\"\"%\".GF&%\"UGF&F'F&%\"xGF&%\"bG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = U*`.`*x" "6#/%\"yG*(%\"UG\"\"\"%\".GF'%\"xGF'" }{TEXT -1 9 " so that \+ " }{XPPEDIT 18 0 "L*`.`*y=b" "6#/*(%\"LG\"\"\"%\".GF&%\"yGF&%\"bG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus we can solve " } {XPPEDIT 18 0 "L*`.`*y = b" "6#/*(%\"LG\"\"\"%\".GF&%\"yGF&%\"bG" } {TEXT -1 5 " for " }{TEXT 280 1 "y" }{TEXT -1 17 ", and then solve " } {XPPEDIT 18 0 "U*`.`*x = y;" "6#/*(%\"UG\"\"\"%\".GF&%\"xGF&%\"yG" } {TEXT -1 5 " for " }{TEXT 281 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 20 "Athough we now have " }{TEXT 261 11 "two systems" } {TEXT -1 54 " to solve, the solution of each system is very simple." } }{PARA 0 "" 0 "" {TEXT -1 15 "(i) The system " }{XPPEDIT 18 0 "L*`.`*y = b;" "6#/*(%\"LG\"\"\"%\".GF&%\"yGF&%\"bG" }{TEXT -1 18 " can be sol ved by " }{TEXT 261 20 "forward substitution" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "If the components of " }{TEXT 282 1 "y" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "y[1], y[2]" "6$&%\"yG6#\"\"\"&F$6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[3]" "6#&%\"yG6#\"\"$" }{TEXT -1 13 ", the system " }{XPPEDIT 18 0 "L*`.`*y = b" "6#/*(%\"LG\"\"\"%\".G F&%\"yGF&%\"bG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "matrix([[1, 0, 0], \+ [1, 1, 0], [-2, 3, 1]])*`.`*matrix([[y[1]], [y[2]], [y[3]]])=matrix([[ 4], [-1], [6]])" "6#/*(-%'matrixG6#7%7%\"\"\"\"\"!F+7%F*F*F+7%,$\"\"#! \"\"\"\"$F*F*%\".GF*-F&6#7%7#&%\"yG6#F*7#&F86#F/7#&F86#F1F*-F&6#7%7#\" \"%7#,$F*F07#\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding linear equations are:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([y[1]=4 ,`` ], [y[1]+y[2]=-1 , ``], [-2*y[1]+3*y[2]+y[3] \+ = 6 ,`` ])" "6#-%*PIECEWISEG6%7$/&%\"yG6#\"\"\"\"\"%%!G7$/,&&F)6#F+F+& F)6#\"\"#F+,$F+!\"\"F-7$/,(*&F5F+&F)6#F+F+F7*&\"\"$F+&F)6#F5F+F+&F)6#F ?F+\"\"'F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Substituting " }{XPPEDIT 18 0 "y[1]=4" "6#/&%\"y G6#\"\"\"\"\"%" }{TEXT -1 57 " from the first equation into the secon d equation gives " }{XPPEDIT 18 0 "4+y[2]=-1" "6#/,&\"\"%\"\"\"&%\"yG6 #\"\"#F&,$F&!\"\"" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "y[2]=-5" "6#/&% \"yG6#\"\"#,$\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then substituting " }{XPPEDIT 18 0 "y[1]=4" "6#/&%\"yG6#\"\"\"\" \"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[2]=-5" "6#/&%\"yG6#\"\"#,$ \"\"&!\"\"" }{TEXT -1 29 " in the third equation gives " }{XPPEDIT 18 0 "-8 - 15 +y[3]=6" "6#/,(\"\")!\"\"\"#:F&&%\"yG6#\"\"$\"\"\"\"\"'" } {TEXT -1 5 ", so " }{XPPEDIT 18 0 "y[3]=29" "6#/&%\"yG6#\"\"$\"#H" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "(ii) The system " }{XPPEDIT 18 0 "U*`.`*x = y" "6#/*(%\"U G\"\"\"%\".GF&%\"xGF&%\"yG" }{TEXT -1 22 " can now be solved by " } {TEXT 261 21 "backward substitution" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 22 " If the components of " }{TEXT 283 1 "x" }{TEXT -1 5 " \+ are " }{XPPEDIT 18 0 "x[1],x[2];" "6$&%\"xG6#\"\"\"&F$6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 13 ", \+ the system " }{XPPEDIT 18 0 "U*`.`*x = y" "6#/*(%\"UG\"\"\"%\".GF&%\"x GF&%\"yG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "matrix([[1, 1, -1], [0, 1 , -1], [0, 0, 2]])*`.`*matrix([[x[1]], [x[2]], [x[3]]]) = matrix([[4], [-5], [29]]);" "6#/*(-%'matrixG6#7%7%\"\"\"F*,$F*!\"\"7%\"\"!F*,$F*F, 7%F.F.\"\"#F*%\".GF*-F&6#7%7#&%\"xG6#F*7#&F86#F17#&F86#\"\"$F*-F&6#7%7 #\"\"%7#,$\"\"&F,7#\"#H" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding linear equations are:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x[1]+x[2]-x[3]=4, ``],[x[2]- x[3] = -5, ``],[2*x[3]=29, ``])" "6#-%*PIECEWISEG6%7$/,(&%\"xG6#\"\"\" F,&F*6#\"\"#F,&F*6#\"\"$!\"\"\"\"%%!G7$/,&&F*6#F/F,&F*6#F2F3,$\"\"&F3F 57$/*&F/F,&F*6#F2F,\"#HF5" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The last equation gives " }{XPPEDIT 18 0 "x[3]=29/2" "6#/ &%\"xG6#\"\"$*&\"#H\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 14 "Substituting " }{XPPEDIT 18 0 "x[3] = 29/2;" "6#/&%\"x G6#\"\"$*&\"#H\"\"\"\"\"#!\"\"" }{TEXT -1 56 " from the last equation into the second equation gives " }{XPPEDIT 18 0 "x[2]-29/2 = -5;" "6# /,&&%\"xG6#\"\"#\"\"\"*&\"#HF)F(!\"\"F,,$\"\"&F," }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "x[2] = 19/2;" "6#/&%\"xG6#\"\"#*&\"#>\"\"\"F'!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then substituting " } {XPPEDIT 18 0 "x[3] = 29/2;" "6#/&%\"xG6#\"\"$*&\"#H\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[2] = 19/2;" "6#/&%\"xG6#\"\"#*& \"#>\"\"\"F'!\"\"" }{TEXT -1 29 " in the first equation gives " } {XPPEDIT 18 0 "x[1]+19/2-29/2 = 4;" "6#/,(&%\"xG6#\"\"\"F(*&\"#>F(\"\" #!\"\"F(*&\"#HF(F+F,F,\"\"%" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "x[1] \+ = 9;" "6#/&%\"xG6#\"\"\"\"\"*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "The solution is " }{XPPEDIT 18 0 "PIECEWISE([x[1] = 9 , ``],[x[2]=19/2, ``],[x[3] = 29/2, ``])" "6#-%*PIECEWISEG6%7$/&%\"xG6 #\"\"\"\"\"*%!G7$/&F)6#\"\"#*&\"#>F+F2!\"\"F-7$/&F)6#\"\"$*&\"#HF+F2F5 F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 64 "We can perform this calculation with Maple using the pr ocedures " }{TEXT 0 17 "ForwardSubstitute" }{TEXT -1 5 " and " }{TEXT 0 18 "BackwardSubstitute" }{TEXT -1 8 " in the " }{TEXT 0 13 "LinearAl gebra" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 26 "First se t up the matrices " }{TEXT 284 1 "L" }{TEXT -1 5 " and " }{TEXT 285 1 "U" }{TEXT -1 17 ", and the vector " }{TEXT 286 1 "b" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "(L,U,b) := (Matrix([[1,0,0],[1,1,0],[-2,3,1]]),Matrix([[1,1,-1] ,[0,1,-1],[0,0,2]]),\n Vector([4,-1,6]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"LG%\"UG%\"bG6%-%'RTABLEG6$\")_sq;-%'MATRIXG6 #7%7%\"\"\"\"\"!F37%F2F2F37%!\"#\"\"$F2-F*6$\")cVq;-F.6#7%7%F2F2!\"\"7 %F3F2F?7%F3F3\"\"#-F*6$\")c0^;-F.6#7%7#\"\"%7#F?7#\"\"'" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "with( LinearAlgebra):\ny := ForwardSubstitute(L,b);\nx := BackwardSubstitute (U,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'RTABLEG6$\"(c'HK-%' MATRIXG6#7%7#\"\"%7#!\"&7#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" xG-%'RTABLEG6$\"(?N\"G-%'MATRIXG6#7%7#\"\"*7##\"#>\"\"#7##\"#HF2" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The solut ion can be checked using matrix multiplication with the original coeff icient matrix " }{TEXT 287 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A := Matrix( [[1,1,-1],[1,2,-2],[-2,1,1]]);\nA.x;\nx := 'x':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")CMo;-%'MATRIXG6#7%7%\"\"\"F.!\"\"7 %F.\"\"#!\"#7%F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\" )k[.@-%'MATRIXG6#7%7#\"\"%7#!\"\"7#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The result is the original vector \+ b, as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 77 "(a) Use Gaussian and Gauss-Jordan elimination t o find an LU decomposition of " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "A = matrix([[2, 1, 5], [3, -2, 2], [5, -8, -4]]);" "6#/ %\"AG-%'matrixG6#7%7%\"\"#\"\"\"\"\"&7%\"\"$,$F*!\"\"F*7%F,,$\"\")F0,$ \"\"%F0" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 106 "(b) Use your result from (a), together with forwa rd and backward substitution, to solve the linear system " }{XPPEDIT 18 0 "A*`.`*x = b" "6#/*(%\"AG\"\"\"%\".GF&%\"xGF&%\"bG" }{TEXT -1 8 " , where " }{XPPEDIT 18 0 "b = matrix([[7], [2], [-3]])" "6#/%\"bG-%'ma trixG6#7%7#\"\"(7#\"\"#7#,$\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 77 "(a) Use Gaussian and Gauss-Jordan \+ elimination to find an LU decomposition of " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "A = matrix([[2, 4, -3, 1], [3, -3, 2, 1], [1, -2, -1, 2 ], [-1, 2, 4, 3]])" "6#/%\"AG-%'matrixG6#7&7&\"\"#\"\"%,$\"\"$!\"\"\" \"\"7&F-,$F-F.F*F/7&F/,$F*F.,$F/F.F*7&,$F/F.F*F+F-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 106 "(b) Use your result from (a), together w ith forward and backward substitution, to solve the linear system " } {XPPEDIT 18 0 "A*`.`*x = b" "6#/*(%\"AG\"\"\"%\".GF&%\"xGF&%\"bG" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "b = matrix([[5], [-4], [8], [-2] ]);" "6#/%\"bG-%'matrixG6#7&7#\"\"&7#,$\"\"%!\"\"7#\"\")7#,$\"\"#F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "________________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }