{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 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 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 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 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 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 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 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 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 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 18 "" 0 "" {TEXT 256 33 "LU decomposition and Det erminants" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 21 "Math 230, Winter 1998" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "First load the linear algebra package. The colon suppresses ou tput." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "LU Factorization" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "Enter a Matrix" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 " Here is a 3 by 3 matrix." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := ma trix([[1,2,1],[-5,2,3],[5,0,-4]]);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Factor the matrix." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The c ommand to find an LU factorization of a matrix is " }}{PARA 0 "" 0 "" {TEXT -1 57 "LUdecomp. It has many possible parameters. See the help " }}{PARA 0 "" 0 "" {TEXT -1 29 "for LUdecomp for the details." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "To find t he LU decomposition for " }{TEXT 274 1 "A" }{TEXT -1 35 " one can use \+ the following command." }}{PARA 0 "" 0 "" {TEXT -1 14 "the variables \+ " }{TEXT 275 1 "x" }{TEXT -1 5 " and " }{TEXT 276 1 "u" }{TEXT -1 41 " will contain the upper-triangular part, " }{TEXT 277 1 "l" }{TEXT -1 14 " contains the " }}{PARA 0 "" 0 "" {TEXT -1 26 "lower triangular pa rt and " }{TEXT 278 1 "p" }{TEXT -1 49 " contains the transpose of the permutation matrix" }}{PARA 0 "" 0 "" {TEXT -1 55 "as defined in the \+ book. The factorization obtained is " }{XPPEDIT 18 0 "A = plu" "/%\"A G%$pluG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "x := LUdecomp(A,L='l',U='u',P='p');\nprint(p,l ,u);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "One can easily check the \+ decomposition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(A - (p &* l &* u));" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Determinants" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "E nter a matrix" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "B := matrix ([[1,2,-3],[2,-1,3],[2,0,-6]]);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Take the determinant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " det(B);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Determinants with va riables." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "One can use Maple to f ind formulas for determinants." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "C := matrix([[a,b],[c,d]]);\ndet(C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "One can verify formulas for cases of special matrices." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "E := matrix([[a,c,b],[0,d,e],[0,0,f ]]);\ndet(E);" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Simple Exerci ses" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Find a " }{XPPEDIT 18 0 "PA \+ = LU" "/%#PAG%#LUG" }{TEXT -1 18 " decomposition of " }{XPPEDIT 18 0 " MATRIX([[-4,6,-5],[2,-3,-2],[6,-3,2]])" "-%'MATRIXG6#7%7%,$\"\"%!\"\" \"\"',$\"\"&F)7%\"\"#,$\"\"$F),$\"\"#F)7%\"\"',$\"\"$F)\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Verify Theorems 2.6, 2 .7 and 2.8 of Hill for arbitrary 3 by 3 matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Check Theorem 2.10 of Hill for one example each o f a permutation matrix, " }}{PARA 0 "" 0 "" {TEXT -1 71 "a row additio n matrix and a row multiplication matrix using the matrix " }{TEXT 279 1 "A" }{TEXT -1 12 " from above." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Check Fact 2.15 with 2 random 10 by 10 matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Check Theorem 2.19 for 5 random 6 by 6 ma trices. (If you do not know how to" }}{PARA 0 "" 0 "" {TEXT -1 52 "fi nd a transpose with Maple, use the help facility.)" }}}}}{MARK "1 1 0 " 13 }{VIEWOPTS 1 1 0 1 1 1803 }