[3d124a7] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[341696] | 2 | version="$Id$"; |
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[49998f] | 3 | category="Linear Algebra"; |
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[5480da] | 4 | info=" |
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[8942a5] | 5 | LIBRARY: matrix.lib Elementary Matrix Operations |
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[3d124a7] | 6 | |
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[f34c37c] | 7 | PROCEDURES: |
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[3d124a7] | 8 | compress(A); matrix, zero columns from A deleted |
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| 9 | concat(A1,A2,..); matrix, concatenation of matrices A1,A2,... |
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| 10 | diag(p,n); matrix, nxn diagonal matrix with entries poly p |
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| 11 | dsum(A1,A2,..); matrix, direct sum of matrices A1,A2,... |
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| 12 | flatten(A); ideal, generated by entries of matrix A |
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[6f2edc] | 13 | genericmat(n,m[,id]); generic nxm matrix [entries from id] |
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[3d124a7] | 14 | is_complex(c); 1 if list c is a complex, 0 if not |
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| 15 | outer(A,B); matrix, outer product of matrices A and B |
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[5dc4ea] | 16 | power(A,n); matrix/intmat, n-th power of matrix/intmat A |
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[3d124a7] | 17 | skewmat(n[,id]); generic skew-symmetric nxn matrix [entries from id] |
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[6f2edc] | 18 | submat(A,r,c); submatrix of A with rows/cols specified by intvec r/c |
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[3d124a7] | 19 | symmat(n[,id]); generic symmetric nxn matrix [entries from id] |
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| 20 | tensor(A,B); matrix, tensor product of matrices A nd B |
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| 21 | unitmat(n); unit square matrix of size n |
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[d26ec4] | 22 | gauss_col(A); transform a matrix into col-reduced Gauss normal form |
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| 23 | gauss_row(A); transform a matrix into row-reduced Gauss normal form |
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[3d124a7] | 24 | addcol(A,c1,p,c2); add p*(c1-th col) to c2-th column of matrix A, p poly |
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| 25 | addrow(A,r1,p,r2); add p*(r1-th row) to r2-th row of matrix A, p poly |
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| 26 | multcol(A,c,p); multiply c-th column of A with poly p |
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| 27 | multrow(A,r,p); multiply r-th row of A with poly p |
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| 28 | permcol(A,i,j); permute i-th and j-th columns |
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| 29 | permrow(A,i,j); permute i-th and j-th rows |
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[63be42] | 30 | rowred(A[,any]); reduction of matrix A with elementary row-operations |
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| 31 | colred(A[,any]); reduction of matrix A with elementary col-operations |
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[7f3ad4] | 32 | linear_relations(E); find linear relations between homogeneous vectors |
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[63be42] | 33 | rm_unitrow(A); remove unit rows and associated columns of A |
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| 34 | rm_unitcol(A); remove unit columns and associated rows of A |
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[d3699d] | 35 | headStand(A); A[n-i+1,m-j+1]:=A[i,j] |
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[2ed3a3] | 36 | symmetricBasis(n,k[,s]); basis of k-th symmetric power of n-dim v.space |
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| 37 | exteriorBasis(n,k[,s]); basis of k-th exterior power of n-dim v.space |
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| 38 | symmetricPower(A,k); k-th symmetric power of a module/matrix A |
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[7f3ad4] | 39 | exteriorPower(A,k); k-th exterior power of a module/matrix A |
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[63be42] | 40 | (parameters in square brackets [] are optional) |
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[5480da] | 41 | "; |
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[3d124a7] | 42 | |
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| 43 | LIB "inout.lib"; |
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| 44 | LIB "ring.lib"; |
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| 45 | LIB "random.lib"; |
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[bb7a4d] | 46 | LIB "general.lib"; // for sort |
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[407fdc0] | 47 | LIB "nctools.lib"; // for superCommutative |
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[bb7a4d] | 48 | |
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[3d124a7] | 49 | /////////////////////////////////////////////////////////////////////////////// |
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| 50 | |
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| 51 | proc compress (A) |
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[d2b2a7] | 52 | "USAGE: compress(A); A matrix/ideal/module/intmat/intvec |
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[6f2edc] | 53 | RETURN: same type, zero columns/generators from A deleted |
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[63be42] | 54 | (if A=intvec, zero elements are deleted) |
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[3d124a7] | 55 | EXAMPLE: example compress; shows an example |
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[d2b2a7] | 56 | " |
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[3d124a7] | 57 | { |
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[6f2edc] | 58 | if( typeof(A)=="matrix" ) { return(matrix(simplify(A,2))); } |
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| 59 | if( typeof(A)=="intmat" or typeof(A)=="intvec" ) |
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[3d124a7] | 60 | { |
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| 61 | ring r=0,x,lp; |
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[6f2edc] | 62 | if( typeof(A)=="intvec" ) { intmat C=transpose(A); kill A; intmat A=C; } |
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| 63 | module m = matrix(A); |
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[b9b906] | 64 | if ( size(m) == 0) |
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[d26ec4] | 65 | { intmat B; } |
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| 66 | else |
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| 67 | { intmat B[nrows(A)][size(m)]; } |
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[3d124a7] | 68 | int i,j; |
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[bf5ba90] | 69 | for( i=1; i<=ncols(A); i++ ) |
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[6f2edc] | 70 | { |
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| 71 | if( m[i]!=[0] ) |
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| 72 | { |
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[bf5ba90] | 73 | j++; |
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[3d124a7] | 74 | B[1..nrows(A),j]=A[1..nrows(A),i]; |
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| 75 | } |
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| 76 | } |
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[6f2edc] | 77 | if( defined(C) ) { return(intvec(B)); } |
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[3d124a7] | 78 | return(B); |
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| 79 | } |
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| 80 | return(simplify(A,2)); |
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| 81 | } |
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| 82 | example |
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| 83 | { "EXAMPLE:"; echo = 2; |
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| 84 | ring r=0,(x,y,z),ds; |
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| 85 | matrix A[3][4]=1,0,3,0,x,0,z,0,x2,0,z2,0; |
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| 86 | print(A); |
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| 87 | print(compress(A)); |
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| 88 | module m=module(A); show(m); |
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| 89 | show(compress(m)); |
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| 90 | intmat B[3][4]=1,0,3,0,4,0,5,0,6,0,7,0; |
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| 91 | compress(B); |
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[6f2edc] | 92 | intvec C=0,0,1,2,0,3; |
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| 93 | compress(C); |
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[3d124a7] | 94 | } |
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[63be42] | 95 | /////////////////////////////////////////////////////////////////////////////// |
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[3d124a7] | 96 | proc concat (list #) |
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[d2b2a7] | 97 | "USAGE: concat(A1,A2,..); A1,A2,... matrices |
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[d26ec4] | 98 | RETURN: matrix, concatenation of A1,A2,.... Number of rows of result matrix |
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[63be42] | 99 | is max(nrows(A1),nrows(A2),...) |
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[3d124a7] | 100 | EXAMPLE: example concat; shows an example |
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[d2b2a7] | 101 | " |
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[3d124a7] | 102 | { |
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| 103 | int i; |
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[bf5ba90] | 104 | for( i=size(#);i>0; i-- ) { #[i]=module(#[i]); } |
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| 105 | module B=#[1..size(#)]; |
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[3d124a7] | 106 | return(matrix(B)); |
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| 107 | } |
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| 108 | example |
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| 109 | { "EXAMPLE:"; echo = 2; |
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| 110 | ring r=0,(x,y,z),ds; |
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| 111 | matrix A[3][3]=1,2,3,x,y,z,x2,y2,z2; |
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| 112 | matrix B[2][2]=1,0,2,0; matrix C[1][4]=4,5,x,y; |
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| 113 | print(A); |
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| 114 | print(B); |
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| 115 | print(C); |
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| 116 | print(concat(A,B,C)); |
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| 117 | } |
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[63be42] | 118 | /////////////////////////////////////////////////////////////////////////////// |
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[3d124a7] | 119 | |
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| 120 | proc diag (list #) |
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[d2b2a7] | 121 | "USAGE: diag(p,n); p poly, n integer |
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[3d124a7] | 122 | diag(A); A matrix |
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[7dd7549] | 123 | RETURN: diag(p,n): diagonal matrix, p times unit matrix of size n. |
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[d26ec4] | 124 | @* diag(A) : n*m x n*m diagonal matrix with entries all the entries of |
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| 125 | the nxm matrix A, taken from the 1st row, 2nd row etc of A |
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[3d124a7] | 126 | EXAMPLE: example diag; shows an example |
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[d2b2a7] | 127 | " |
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[3d124a7] | 128 | { |
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| 129 | if( size(#)==2 ) { return(matrix(#[1]*freemodule(#[2]))); } |
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| 130 | if( size(#)==1 ) |
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| 131 | { |
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| 132 | int i; ideal id=#[1]; |
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| 133 | int n=ncols(id); matrix A[n][n]; |
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[bf5ba90] | 134 | for( i=1; i<=n; i++ ) { A[i,i]=id[i]; } |
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[3d124a7] | 135 | } |
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| 136 | return(A); |
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| 137 | } |
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| 138 | example |
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| 139 | { "EXAMPLE:"; echo = 2; |
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[d26ec4] | 140 | ring r = 0,(x,y,z),ds; |
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[3d124a7] | 141 | print(diag(xy,4)); |
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[d26ec4] | 142 | matrix A[3][2] = 1,2,3,4,5,6; |
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[3d124a7] | 143 | print(A); |
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| 144 | print(diag(A)); |
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| 145 | } |
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[63be42] | 146 | /////////////////////////////////////////////////////////////////////////////// |
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[3d124a7] | 147 | |
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| 148 | proc dsum (list #) |
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[d2b2a7] | 149 | "USAGE: dsum(A1,A2,..); A1,A2,... matrices |
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[3d124a7] | 150 | RETURN: matrix, direct sum of A1,A2,... |
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| 151 | EXAMPLE: example dsum; shows an example |
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[d2b2a7] | 152 | " |
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[3d124a7] | 153 | { |
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| 154 | int i,N,a; |
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| 155 | list L; |
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[bf5ba90] | 156 | for( i=1; i<=size(#); i++ ) { N=N+nrows(#[i]); } |
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| 157 | for( i=1; i<=size(#); i++ ) |
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[6f2edc] | 158 | { |
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| 159 | matrix B[N][ncols(#[i])]; |
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[3d124a7] | 160 | B[a+1..a+nrows(#[i]),1..ncols(#[i])]=#[i]; |
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| 161 | a=a+nrows(#[i]); |
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| 162 | L[i]=B; kill B; |
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| 163 | } |
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| 164 | return(concat(L)); |
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| 165 | } |
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| 166 | example |
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| 167 | { "EXAMPLE:"; echo = 2; |
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[d26ec4] | 168 | ring r = 0,(x,y,z),ds; |
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| 169 | matrix A[3][3] = 1,2,3,4,5,6,7,8,9; |
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| 170 | matrix B[2][2] = 1,x,y,z; |
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[3d124a7] | 171 | print(A); |
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| 172 | print(B); |
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[d26ec4] | 173 | print(dsum(A,B)); |
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[3d124a7] | 174 | } |
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[63be42] | 175 | /////////////////////////////////////////////////////////////////////////////// |
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[6f2edc] | 176 | |
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| 177 | proc flatten (matrix A) |
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[d2b2a7] | 178 | "USAGE: flatten(A); A matrix |
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[6f2edc] | 179 | RETURN: ideal, generated by all entries from A |
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| 180 | EXAMPLE: example flatten; shows an example |
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[d2b2a7] | 181 | " |
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[6f2edc] | 182 | { |
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| 183 | return(ideal(A)); |
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| 184 | } |
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| 185 | example |
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| 186 | { "EXAMPLE:"; echo = 2; |
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[d26ec4] | 187 | ring r = 0,(x,y,z),ds; |
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| 188 | matrix A[2][3] = 1,2,x,y,z,7; |
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[6f2edc] | 189 | print(A); |
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| 190 | flatten(A); |
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| 191 | } |
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[63be42] | 192 | /////////////////////////////////////////////////////////////////////////////// |
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[6f2edc] | 193 | |
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| 194 | proc genericmat (int n,int m,list #) |
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[d2b2a7] | 195 | "USAGE: genericmat(n,m[,id]); n,m=integers, id=ideal |
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[d26ec4] | 196 | RETURN: nxm matrix, with entries from id. |
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[6f2edc] | 197 | NOTE: if id has less than nxm elements, the matrix is filled with 0's, |
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[d26ec4] | 198 | (default: id=maxideal(1)). |
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[6f2edc] | 199 | genericmat(n,m); creates the generic nxm matrix |
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| 200 | EXAMPLE: example genericmat; shows an example |
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[d2b2a7] | 201 | " |
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[6f2edc] | 202 | { |
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| 203 | if( size(#)==0 ) { ideal id=maxideal(1); } |
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| 204 | if( size(#)==1 ) { ideal id=#[1]; } |
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| 205 | if( size(#)>=2 ) { "// give 3 arguments, 3-rd argument must be an ideal"; } |
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| 206 | matrix B[n][m]=id; |
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| 207 | return(B); |
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| 208 | } |
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| 209 | example |
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| 210 | { "EXAMPLE:"; echo = 2; |
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[d26ec4] | 211 | ring R = 0,x(1..16),lp; |
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| 212 | print(genericmat(3,3)); // the generic 3x3 matrix |
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| 213 | ring R1 = 0,(a,b,c,d),dp; |
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| 214 | matrix A = genericmat(3,4,maxideal(1)^3); |
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[6f2edc] | 215 | print(A); |
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[d26ec4] | 216 | int n,m = 3,2; |
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[6f2edc] | 217 | ideal i = ideal(randommat(1,n*m,maxideal(1),9)); |
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[d26ec4] | 218 | print(genericmat(n,m,i)); // matrix of generic linear forms |
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[6f2edc] | 219 | } |
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[3d124a7] | 220 | /////////////////////////////////////////////////////////////////////////////// |
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| 221 | |
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| 222 | proc is_complex (list c) |
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[d2b2a7] | 223 | "USAGE: is_complex(c); c = list of size-compatible modules or matrices |
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[63be42] | 224 | RETURN: 1 if c[i]*c[i+1]=0 for all i, 0 if not, hence checking whether the |
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[d26ec4] | 225 | list of matrices forms a complex. |
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| 226 | NOTE: Ideals are treated internally as 1-line matrices. |
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| 227 | If printlevel > 0, the position where c is not a complex is shown. |
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[3d124a7] | 228 | EXAMPLE: example is_complex; shows an example |
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[d2b2a7] | 229 | " |
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[3d124a7] | 230 | { |
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| 231 | int i; |
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| 232 | module @test; |
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[bf5ba90] | 233 | for( i=1; i<=size(c)-1; i++ ) |
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[3d124a7] | 234 | { |
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| 235 | c[i]=matrix(c[i]); c[i+1]=matrix(c[i+1]); |
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| 236 | @test=c[i]*c[i+1]; |
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| 237 | if (size(@test)!=0) |
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| 238 | { |
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[d26ec4] | 239 | dbprint(printlevel-voice+2,"// not a complex at position " +string(i)); |
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[3d124a7] | 240 | return(0); |
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| 241 | } |
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| 242 | } |
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| 243 | return(1); |
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| 244 | } |
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| 245 | example |
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[6f2edc] | 246 | { "EXAMPLE:"; echo = 2; |
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[d26ec4] | 247 | ring r = 32003,(x,y,z),ds; |
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| 248 | ideal i = x4+y5+z6,xyz,yx2+xz2+zy7; |
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| 249 | list L = nres(i,0); |
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[3d124a7] | 250 | is_complex(L); |
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[d26ec4] | 251 | L[4] = matrix(i); |
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[3d124a7] | 252 | is_complex(L); |
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| 253 | } |
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[63be42] | 254 | /////////////////////////////////////////////////////////////////////////////// |
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[3d124a7] | 255 | |
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| 256 | proc outer (matrix A, matrix B) |
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[d2b2a7] | 257 | "USAGE: outer(A,B); A,B matrices |
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[d26ec4] | 258 | RETURN: matrix, outer (tensor) product of A and B |
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[3d124a7] | 259 | EXAMPLE: example outer; shows an example |
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[d2b2a7] | 260 | " |
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[3d124a7] | 261 | { |
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| 262 | int i,j; list L; |
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[6f2edc] | 263 | int triv = nrows(B)*ncols(B); |
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| 264 | if( triv==1 ) |
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| 265 | { |
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| 266 | return(B[1,1]*A); |
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| 267 | } |
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| 268 | else |
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| 269 | { |
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| 270 | int N = nrows(A)*nrows(B); |
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| 271 | matrix C[N][ncols(B)]; |
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[840876] | 272 | for( i=ncols(A);i>0; i-- ) |
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[6f2edc] | 273 | { |
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[bf5ba90] | 274 | for( j=1; j<=nrows(A); j++ ) |
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[6f2edc] | 275 | { |
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| 276 | C[(j-1)*nrows(B)+1..j*nrows(B),1..ncols(B)]=A[j,i]*B; |
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| 277 | } |
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| 278 | L[i]=C; |
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| 279 | } |
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| 280 | return(concat(L)); |
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[3d124a7] | 281 | } |
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| 282 | } |
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| 283 | example |
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| 284 | { "EXAMPLE:"; echo = 2; |
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| 285 | ring r=32003,(x,y,z),ds; |
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| 286 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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| 287 | matrix B[2][2]=x,y,0,z; |
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| 288 | print(A); |
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| 289 | print(B); |
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| 290 | print(outer(A,B)); |
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| 291 | } |
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[63be42] | 292 | /////////////////////////////////////////////////////////////////////////////// |
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[3d124a7] | 293 | |
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[5dc4ea] | 294 | proc power ( A, int n) |
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[65784c6] | 295 | "USAGE: power(A,n); A a square-matrix of type intmat or matrix, n=integer>=0 |
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[c2aa97] | 296 | RETURN: intmat resp. matrix, the n-th power of A |
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| 297 | NOTE: for A=intmat and big n the result may be wrong because of int overflow |
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[5dc4ea] | 298 | EXAMPLE: example power; shows an example |
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[d2b2a7] | 299 | " |
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[5dc4ea] | 300 | { |
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| 301 | //---------------------------- type checking ---------------------------------- |
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| 302 | if( typeof(A)!="matrix" and typeof(A)!="intmat" ) |
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| 303 | { |
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[49f94f] | 304 | ERROR("no matrix or intmat!"); |
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[5dc4ea] | 305 | } |
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| 306 | if( ncols(A) != nrows(A) ) |
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| 307 | { |
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[49f94f] | 308 | ERROR("not a square matrix!"); |
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[5dc4ea] | 309 | } |
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| 310 | //---------------------------- trivial cases ---------------------------------- |
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| 311 | int ii; |
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[82716e] | 312 | if( n <= 0 ) |
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[5dc4ea] | 313 | { |
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[82716e] | 314 | if( typeof(A)=="matrix" ) |
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| 315 | { |
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| 316 | return (unitmat(nrows(A))); |
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[5dc4ea] | 317 | } |
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[82716e] | 318 | if( typeof(A)=="intmat" ) |
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| 319 | { |
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[5dc4ea] | 320 | intmat B[nrows(A)][nrows(A)]; |
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| 321 | for( ii=1; ii<=nrows(A); ii++ ) |
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| 322 | { |
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| 323 | B[ii,ii] = 1; |
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| 324 | } |
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[82716e] | 325 | return (B); |
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[5dc4ea] | 326 | } |
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| 327 | } |
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| 328 | if( n == 1 ) { return (A); } |
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| 329 | //---------------------------- sub procedure ---------------------------------- |
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| 330 | proc matpow (A, int n) |
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| 331 | { |
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| 332 | def B = A*A; |
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| 333 | int ii= 2; |
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| 334 | int jj= 4; |
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| 335 | while( jj <= n ) |
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| 336 | { |
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| 337 | B=B*B; |
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| 338 | ii=jj; |
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| 339 | jj=2*jj; |
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| 340 | } |
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| 341 | return(B,n-ii); |
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| 342 | } |
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| 343 | //----------------------------- main program ---------------------------------- |
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| 344 | list L = matpow(A,n); |
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| 345 | def B = L[1]; |
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| 346 | ii = L[2]; |
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| 347 | while( ii>=2 ) |
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| 348 | { |
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| 349 | L = matpow(A,ii); |
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| 350 | B = B*L[1]; |
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| 351 | ii= L[2]; |
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| 352 | } |
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| 353 | if( ii == 0) { return(B); } |
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| 354 | if( ii == 1) { return(A*B); } |
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| 355 | } |
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| 356 | example |
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| 357 | { "EXAMPLE:"; echo = 2; |
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| 358 | intmat A[3][3]=1,2,3,4,5,6,7,8,9; |
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| 359 | print(power(A,3));""; |
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| 360 | ring r=0,(x,y,z),dp; |
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[d26ec4] | 361 | matrix B[3][3]=0,x,y,z,0,0,y,z,0; |
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[5dc4ea] | 362 | print(power(B,3));""; |
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| 363 | } |
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[63be42] | 364 | /////////////////////////////////////////////////////////////////////////////// |
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[5dc4ea] | 365 | |
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[3d124a7] | 366 | proc skewmat (int n, list #) |
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[d2b2a7] | 367 | "USAGE: skewmat(n[,id]); n integer, id ideal |
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[6f2edc] | 368 | RETURN: skew-symmetric nxn matrix, with entries from id |
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[3d124a7] | 369 | (default: id=maxideal(1)) |
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[7dd7549] | 370 | skewmat(n); creates the generic skew-symmetric matrix |
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[b9b906] | 371 | NOTE: if id has less than n*(n-1)/2 elements, the matrix is |
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[d26ec4] | 372 | filled with 0's, |
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[3d124a7] | 373 | EXAMPLE: example skewmat; shows an example |
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[d2b2a7] | 374 | " |
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[3d124a7] | 375 | { |
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| 376 | matrix B[n][n]; |
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| 377 | if( size(#)==0 ) { ideal id=maxideal(1); } |
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| 378 | else { ideal id=#[1]; } |
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| 379 | id = id,B[1..n,1..n]; |
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| 380 | int i,j; |
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[bf5ba90] | 381 | for( i=0; i<=n-2; i++ ) |
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[6f2edc] | 382 | { |
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| 383 | B[i+1,i+2..n]=id[j+1..j+n-i-1]; |
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| 384 | j=j+n-i-1; |
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[3d124a7] | 385 | } |
---|
| 386 | matrix A=transpose(B); |
---|
| 387 | B=B-A; |
---|
| 388 | return(B); |
---|
| 389 | } |
---|
| 390 | example |
---|
| 391 | { "EXAMPLE:"; echo = 2; |
---|
| 392 | ring R=0,x(1..5),lp; |
---|
| 393 | print(skewmat(4)); // the generic skew-symmetric matrix |
---|
[d26ec4] | 394 | ring R1 = 0,(a,b,c),dp; |
---|
| 395 | matrix A=skewmat(4,maxideal(1)^2); |
---|
[3d124a7] | 396 | print(A); |
---|
[d26ec4] | 397 | int n=3; |
---|
[18dd47] | 398 | ideal i = ideal(randommat(1,n*(n-1) div 2,maxideal(1),9)); |
---|
[3d124a7] | 399 | print(skewmat(n,i)); // skew matrix of generic linear forms |
---|
| 400 | kill R1; |
---|
| 401 | } |
---|
[63be42] | 402 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 403 | |
---|
| 404 | proc submat (matrix A, intvec r, intvec c) |
---|
[d2b2a7] | 405 | "USAGE: submat(A,r,c); A=matrix, r,c=intvec |
---|
[d26ec4] | 406 | RETURN: matrix, submatrix of A with rows specified by intvec r |
---|
| 407 | and columns specified by intvec c. |
---|
[3d124a7] | 408 | EXAMPLE: example submat; shows an example |
---|
[d2b2a7] | 409 | " |
---|
[3d124a7] | 410 | { |
---|
| 411 | matrix B[size(r)][size(c)]=A[r,c]; |
---|
| 412 | return(B); |
---|
| 413 | } |
---|
| 414 | example |
---|
| 415 | { "EXAMPLE:"; echo = 2; |
---|
| 416 | ring R=32003,(x,y,z),lp; |
---|
| 417 | matrix A[4][4]=x,y,z,0,1,2,3,4,5,6,7,8,9,x2,y2,z2; |
---|
| 418 | print(A); |
---|
| 419 | intvec v=1,3,4; |
---|
| 420 | matrix B=submat(A,v,1..3); |
---|
| 421 | print(B); |
---|
| 422 | } |
---|
[63be42] | 423 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 424 | |
---|
| 425 | proc symmat (int n, list #) |
---|
[d2b2a7] | 426 | "USAGE: symmat(n[,id]); n integer, id ideal |
---|
[3d124a7] | 427 | RETURN: symmetric nxn matrix, with entries from id (default: id=maxideal(1)) |
---|
| 428 | NOTE: if id has less than n*(n+1)/2 elements, the matrix is filled with 0's, |
---|
| 429 | symmat(n); creates the generic symmetric matrix |
---|
| 430 | EXAMPLE: example symmat; shows an example |
---|
[d2b2a7] | 431 | " |
---|
[3d124a7] | 432 | { |
---|
| 433 | matrix B[n][n]; |
---|
| 434 | if( size(#)==0 ) { ideal id=maxideal(1); } |
---|
| 435 | else { ideal id=#[1]; } |
---|
| 436 | id = id,B[1..n,1..n]; |
---|
| 437 | int i,j; |
---|
[bf5ba90] | 438 | for( i=0; i<=n-1; i++ ) |
---|
[6f2edc] | 439 | { |
---|
| 440 | B[i+1,i+1..n]=id[j+1..j+n-i]; |
---|
| 441 | j=j+n-i; |
---|
[3d124a7] | 442 | } |
---|
| 443 | matrix A=transpose(B); |
---|
[bf5ba90] | 444 | for( i=1; i<=n; i++ ) { A[i,i]=0; } |
---|
[3d124a7] | 445 | B=A+B; |
---|
| 446 | return(B); |
---|
| 447 | } |
---|
| 448 | example |
---|
| 449 | { "EXAMPLE:"; echo = 2; |
---|
| 450 | ring R=0,x(1..10),lp; |
---|
[6f2edc] | 451 | print(symmat(4)); // the generic symmetric matrix |
---|
[d26ec4] | 452 | ring R1 = 0,(a,b,c),dp; |
---|
| 453 | matrix A=symmat(4,maxideal(1)^3); |
---|
[3d124a7] | 454 | print(A); |
---|
| 455 | int n=3; |
---|
[18dd47] | 456 | ideal i = ideal(randommat(1,n*(n+1) div 2,maxideal(1),9)); |
---|
[3d124a7] | 457 | print(symmat(n,i)); // symmetric matrix of generic linear forms |
---|
| 458 | kill R1; |
---|
| 459 | } |
---|
[63be42] | 460 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 461 | |
---|
| 462 | proc tensor (matrix A, matrix B) |
---|
[d2b2a7] | 463 | "USAGE: tensor(A,B); A,B matrices |
---|
[3d124a7] | 464 | RETURN: matrix, tensor product of A and B |
---|
| 465 | EXAMPLE: example tensor; shows an example |
---|
[d2b2a7] | 466 | " |
---|
[3d124a7] | 467 | { |
---|
[7dd7549] | 468 | if (ncols(A)==0) |
---|
| 469 | { |
---|
| 470 | int q=nrows(A)*nrows(B); |
---|
| 471 | matrix D[q][0]; |
---|
| 472 | return(D); |
---|
| 473 | } |
---|
[3c4dcc] | 474 | |
---|
[3d124a7] | 475 | int i,j; |
---|
[8942a5] | 476 | matrix C,D; |
---|
[b9b906] | 477 | for( i=1; i<=nrows(A); i++ ) |
---|
| 478 | { |
---|
[8942a5] | 479 | C = A[i,1]*B; |
---|
| 480 | for( j=2; j<=ncols(A); j++ ) |
---|
| 481 | { |
---|
| 482 | C = concat(C,A[i,j]*B); |
---|
[b9b906] | 483 | } |
---|
[8942a5] | 484 | D = concat(D,transpose(C)); |
---|
[3d124a7] | 485 | } |
---|
[8942a5] | 486 | D = transpose(D); |
---|
| 487 | return(submat(D,2..nrows(D),1..ncols(D))); |
---|
[3d124a7] | 488 | } |
---|
| 489 | example |
---|
| 490 | { "EXAMPLE:"; echo = 2; |
---|
| 491 | ring r=32003,(x,y,z),(c,ds); |
---|
| 492 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
---|
| 493 | matrix B[2][2]=x,y,0,z; |
---|
| 494 | print(A); |
---|
| 495 | print(B); |
---|
| 496 | print(tensor(A,B)); |
---|
| 497 | } |
---|
[63be42] | 498 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 499 | |
---|
| 500 | proc unitmat (int n) |
---|
[d2b2a7] | 501 | "USAGE: unitmat(n); n integer >= 0 |
---|
[3d124a7] | 502 | RETURN: nxn unit matrix |
---|
| 503 | NOTE: needs a basering, diagonal entries are numbers (=1) in the basering |
---|
| 504 | EXAMPLE: example unitmat; shows an example |
---|
[d2b2a7] | 505 | " |
---|
[3d124a7] | 506 | { |
---|
| 507 | return(matrix(freemodule(n))); |
---|
| 508 | } |
---|
| 509 | example |
---|
| 510 | { "EXAMPLE:"; echo = 2; |
---|
| 511 | ring r=32003,(x,y,z),lp; |
---|
| 512 | print(xyz*unitmat(4)); |
---|
| 513 | print(unitmat(5)); |
---|
| 514 | } |
---|
| 515 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 516 | |
---|
[d26ec4] | 517 | proc gauss_col (matrix A, list #) |
---|
[741214] | 518 | "USAGE: gauss_col(A[,e]); A a matrix, e any type |
---|
[d26ec4] | 519 | RETURN: - a matrix B, if called with one argument; B is the complete column- |
---|
[b9b906] | 520 | reduced upper-triangular normal form of A if A is constant, |
---|
[cb1e43b] | 521 | (resp. as far as this is possible if A is a polynomial matrix; |
---|
| 522 | no division by polynomials). |
---|
[b9b906] | 523 | @* - a list L of two matrices, if called with two arguments; |
---|
[7af4ad9] | 524 | L satisfies L[1] = A * L[2] with L[1] the column-reduced form of A |
---|
[d26ec4] | 525 | and L[2] the transformation matrix. |
---|
| 526 | NOTE: * The procedure just applies interred to A with ordering (C,dp). |
---|
| 527 | The transformation matrix is obtained by applying 'lift'. |
---|
[e3e3df] | 528 | This should be faster than the procedure colred. |
---|
[d26ec4] | 529 | @* * It should only be used with exact coefficient field (there is no |
---|
| 530 | pivoting and rounding error treatment). |
---|
| 531 | @* * Parameters are allowed. Hence, if the entries of A are parameters, |
---|
| 532 | B is the column-reduced form of A over the rational function field. |
---|
[7af4ad9] | 533 | SEE ALSO: colred |
---|
[3d124a7] | 534 | EXAMPLE: example gauss_col; shows an example |
---|
[d2b2a7] | 535 | " |
---|
[3d124a7] | 536 | { |
---|
[d26ec4] | 537 | def R=basering; int u; |
---|
[741214] | 538 | string mp = string(minpoly); |
---|
[d26ec4] | 539 | int n = nrows(A); |
---|
| 540 | int m = ncols(A); |
---|
| 541 | module M = A; |
---|
[63be42] | 542 | intvec v = option(get); |
---|
[7af4ad9] | 543 | //------------------------ change ordering if necessary ---------------------- |
---|
[717885] | 544 | if( ordstr(R) != ("C,dp("+string(nvars(R))+")") ) |
---|
[d26ec4] | 545 | { |
---|
[daa83b] | 546 | def @R=changeord("C,dp",R); setring @R; u=1; |
---|
[741214] | 547 | execute("minpoly="+mp+";"); |
---|
[d26ec4] | 548 | matrix A = imap(R,A); |
---|
| 549 | module M = A; |
---|
| 550 | } |
---|
[7af4ad9] | 551 | //------------------------------ start computation --------------------------- |
---|
[d26ec4] | 552 | option(redSB); |
---|
| 553 | M = simplify(interred(M),1); |
---|
[b9b906] | 554 | if(size(#) != 0) |
---|
[cb1e43b] | 555 | { |
---|
[7af4ad9] | 556 | module N = lift(A,M); |
---|
[d26ec4] | 557 | } |
---|
[7af4ad9] | 558 | //--------------- reset ring and options and return -------------------------- |
---|
[d26ec4] | 559 | if ( u==1 ) |
---|
| 560 | { |
---|
| 561 | setring R; |
---|
| 562 | M=imap(@R,M); |
---|
| 563 | if (size(#) != 0) |
---|
| 564 | { |
---|
| 565 | module N = imap(@R,N); |
---|
| 566 | } |
---|
| 567 | kill @R; |
---|
| 568 | } |
---|
[63be42] | 569 | option(set,v); |
---|
[d26ec4] | 570 | // M = sort(M,size(M)..1)[1]; |
---|
| 571 | A = matrix(M,n,m); |
---|
| 572 | if (size(#) != 0) |
---|
[b9b906] | 573 | { |
---|
[d26ec4] | 574 | list L= A,matrix(N,m,m); |
---|
[b9b906] | 575 | return(L); |
---|
[d26ec4] | 576 | } |
---|
| 577 | return(matrix(M,n,m)); |
---|
[6f2edc] | 578 | } |
---|
| 579 | example |
---|
[3d124a7] | 580 | { "EXAMPLE:"; echo = 2; |
---|
[d26ec4] | 581 | ring r=(0,a,b),(A,B,C),dp; |
---|
| 582 | matrix m[8][6]= |
---|
[b9b906] | 583 | 0, 2*C, 0, 0, 0, 0, |
---|
| 584 | 0, -4*C,a*A, 0, 0, 0, |
---|
| 585 | b*B, -A, 0, 0, 0, 0, |
---|
| 586 | -A, B, 0, 0, 0, 0, |
---|
| 587 | -4*C, 0, B, 2, 0, 0, |
---|
| 588 | 2*A, B, 0, 0, 0, 0, |
---|
| 589 | 0, 3*B, 0, 0, 2b, 0, |
---|
[d26ec4] | 590 | 0, AB, 0, 2*A,A, 2a;""; |
---|
| 591 | list L=gauss_col(m,1); |
---|
| 592 | print(L[1]); |
---|
| 593 | print(L[2]); |
---|
| 594 | |
---|
| 595 | ring S=0,x,(c,dp); |
---|
| 596 | matrix A[5][4] = |
---|
| 597 | 3, 1, 1, 1, |
---|
| 598 | 13, 8, 6,-7, |
---|
| 599 | 14,10, 6,-7, |
---|
| 600 | 7, 4, 3,-3, |
---|
| 601 | 2, 1, 0, 3; |
---|
[6f2edc] | 602 | print(gauss_col(A)); |
---|
[3d124a7] | 603 | } |
---|
| 604 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 605 | |
---|
[d26ec4] | 606 | proc gauss_row (matrix A, list #) |
---|
[7af4ad9] | 607 | "USAGE: gauss_row(A [,e]); A matrix, e any type |
---|
[d26ec4] | 608 | RETURN: - a matrix B, if called with one argument; B is the complete row- |
---|
[cb1e43b] | 609 | reduced lower-triangular normal form of A if A is constant, |
---|
| 610 | (resp. as far as this is possible if A is a polynomial matrix; |
---|
| 611 | no division by polynomials). |
---|
[e3e3df] | 612 | @* - a list L of two matrices, if called with two arguments; |
---|
[7dd7549] | 613 | L satisfies transpose(L[2])*A=transpose(L[1]) |
---|
[3c4dcc] | 614 | with L[1] the row-reduced form of A |
---|
[d26ec4] | 615 | and L[2] the transformation matrix. |
---|
| 616 | NOTE: * This procedure just applies gauss_col to the transposed matrix. |
---|
| 617 | The transformation matrix is obtained by applying lift. |
---|
[e3e3df] | 618 | This should be faster than the procedure rowred. |
---|
[d26ec4] | 619 | @* * It should only be used with exact coefficient field (there is no |
---|
| 620 | pivoting and rounding error treatment). |
---|
| 621 | @* * Parameters are allowed. Hence, if the entries of A are parameters, |
---|
| 622 | B is the row-reduced form of A over the rational function field. |
---|
[7af4ad9] | 623 | SEE ALSO: rowred |
---|
[3d124a7] | 624 | EXAMPLE: example gauss_row; shows an example |
---|
[d2b2a7] | 625 | " |
---|
[3d124a7] | 626 | { |
---|
[d26ec4] | 627 | if(size(#) > 0) |
---|
| 628 | { |
---|
| 629 | list L = gauss_col(transpose(A),1); |
---|
| 630 | return(L); |
---|
| 631 | } |
---|
[6f2edc] | 632 | A = gauss_col(transpose(A)); |
---|
| 633 | return(transpose(A)); |
---|
| 634 | } |
---|
| 635 | example |
---|
[3d124a7] | 636 | { "EXAMPLE:"; echo = 2; |
---|
[d26ec4] | 637 | ring r=(0,a,b),(A,B,C),dp; |
---|
| 638 | matrix m[6][8]= |
---|
[b9b906] | 639 | 0, 0, b*B, -A,-4C,2A,0, 0, |
---|
[d26ec4] | 640 | 2C,-4C,-A,B, 0, B, 3B,AB, |
---|
[b9b906] | 641 | 0,a*A, 0, 0, B, 0, 0, 0, |
---|
[d26ec4] | 642 | 0, 0, 0, 0, 2, 0, 0, 2A, |
---|
[b9b906] | 643 | 0, 0, 0, 0, 0, 0, 2b, A, |
---|
[d26ec4] | 644 | 0, 0, 0, 0, 0, 0, 0, 2a;""; |
---|
| 645 | print(gauss_row(m));""; |
---|
[6f2edc] | 646 | ring S=0,x,dp; |
---|
| 647 | matrix A[4][5] = 3, 1,1,-1,2, |
---|
| 648 | 13, 8,6,-7,1, |
---|
| 649 | 14,10,6,-7,1, |
---|
| 650 | 7, 4,3,-3,3; |
---|
[d26ec4] | 651 | list L=gauss_row(A,1); |
---|
| 652 | print(L[1]); |
---|
| 653 | print(L[2]); |
---|
[3d124a7] | 654 | } |
---|
[63be42] | 655 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 656 | |
---|
| 657 | proc addcol (matrix A, int c1, poly p, int c2) |
---|
[d2b2a7] | 658 | "USAGE: addcol(A,c1,p,c2); A matrix, p poly, c1, c2 positive integers |
---|
[3d124a7] | 659 | RETURN: matrix, A being modified by adding p times column c1 to column c2 |
---|
| 660 | EXAMPLE: example addcol; shows an example |
---|
[d2b2a7] | 661 | " |
---|
[3d124a7] | 662 | { |
---|
[5411c8] | 663 | int k=nrows(A); |
---|
| 664 | A[1..k,c2]=A[1..k,c2]+p*A[1..k,c1]; |
---|
[3d124a7] | 665 | return(A); |
---|
| 666 | } |
---|
| 667 | example |
---|
| 668 | { "EXAMPLE:"; echo = 2; |
---|
| 669 | ring r=32003,(x,y,z),lp; |
---|
| 670 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
---|
| 671 | print(A); |
---|
| 672 | print(addcol(A,1,xy,2)); |
---|
| 673 | } |
---|
[63be42] | 674 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 675 | |
---|
| 676 | proc addrow (matrix A, int r1, poly p, int r2) |
---|
[d2b2a7] | 677 | "USAGE: addcol(A,r1,p,r2); A matrix, p poly, r1, r2 positive integers |
---|
[3d124a7] | 678 | RETURN: matrix, A being modified by adding p times row r1 to row r2 |
---|
| 679 | EXAMPLE: example addrow; shows an example |
---|
[d2b2a7] | 680 | " |
---|
[3d124a7] | 681 | { |
---|
[5411c8] | 682 | int k=ncols(A); |
---|
| 683 | A[r2,1..k]=A[r2,1..k]+p*A[r1,1..k]; |
---|
[3d124a7] | 684 | return(A); |
---|
| 685 | } |
---|
| 686 | example |
---|
| 687 | { "EXAMPLE:"; echo = 2; |
---|
| 688 | ring r=32003,(x,y,z),lp; |
---|
| 689 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
---|
| 690 | print(A); |
---|
| 691 | print(addrow(A,1,xy,3)); |
---|
| 692 | } |
---|
[63be42] | 693 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 694 | |
---|
| 695 | proc multcol (matrix A, int c, poly p) |
---|
[d2b2a7] | 696 | "USAGE: addcol(A,c,p); A matrix, p poly, c positive integer |
---|
[7df1eb] | 697 | RETURN: matrix, A being modified by multiplying column c by p |
---|
[3d124a7] | 698 | EXAMPLE: example multcol; shows an example |
---|
[d2b2a7] | 699 | " |
---|
[3d124a7] | 700 | { |
---|
[5411c8] | 701 | int k=nrows(A); |
---|
| 702 | A[1..k,c]=p*A[1..k,c]; |
---|
[3d124a7] | 703 | return(A); |
---|
| 704 | } |
---|
| 705 | example |
---|
| 706 | { "EXAMPLE:"; echo = 2; |
---|
| 707 | ring r=32003,(x,y,z),lp; |
---|
| 708 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
---|
| 709 | print(A); |
---|
| 710 | print(multcol(A,2,xy)); |
---|
| 711 | } |
---|
[63be42] | 712 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 713 | |
---|
| 714 | proc multrow (matrix A, int r, poly p) |
---|
[abb58e] | 715 | "USAGE: multrow(A,r,p); A matrix, p poly, r positive integer |
---|
[7df1eb] | 716 | RETURN: matrix, A being modified by multiplying row r by p |
---|
[3d124a7] | 717 | EXAMPLE: example multrow; shows an example |
---|
[d2b2a7] | 718 | " |
---|
[3d124a7] | 719 | { |
---|
[5411c8] | 720 | int k=ncols(A); |
---|
| 721 | A[r,1..k]=p*A[r,1..k]; |
---|
[3d124a7] | 722 | return(A); |
---|
| 723 | } |
---|
| 724 | example |
---|
| 725 | { "EXAMPLE:"; echo = 2; |
---|
| 726 | ring r=32003,(x,y,z),lp; |
---|
| 727 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
---|
| 728 | print(A); |
---|
| 729 | print(multrow(A,2,xy)); |
---|
| 730 | } |
---|
[63be42] | 731 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 732 | |
---|
| 733 | proc permcol (matrix A, int c1, int c2) |
---|
[d2b2a7] | 734 | "USAGE: permcol(A,c1,c2); A matrix, c1,c2 positive integers |
---|
[7df1eb] | 735 | RETURN: matrix, A being modified by permuting columns c1 and c2 |
---|
[3d124a7] | 736 | EXAMPLE: example permcol; shows an example |
---|
[d2b2a7] | 737 | " |
---|
[3d124a7] | 738 | { |
---|
| 739 | matrix B=A; |
---|
[5411c8] | 740 | int k=nrows(B); |
---|
| 741 | B[1..k,c1]=A[1..k,c2]; |
---|
| 742 | B[1..k,c2]=A[1..k,c1]; |
---|
[3d124a7] | 743 | return(B); |
---|
| 744 | } |
---|
| 745 | example |
---|
| 746 | { "EXAMPLE:"; echo = 2; |
---|
| 747 | ring r=32003,(x,y,z),lp; |
---|
| 748 | matrix A[3][3]=1,x,3,4,y,6,7,z,9; |
---|
| 749 | print(A); |
---|
| 750 | print(permcol(A,2,3)); |
---|
| 751 | } |
---|
[63be42] | 752 | /////////////////////////////////////////////////////////////////////////////// |
---|
[3d124a7] | 753 | |
---|
| 754 | proc permrow (matrix A, int r1, int r2) |
---|
[d2b2a7] | 755 | "USAGE: permrow(A,r1,r2); A matrix, r1,r2 positive integers |
---|
[979c4c] | 756 | RETURN: matrix, A being modified by permuting rows r1 and r2 |
---|
[3d124a7] | 757 | EXAMPLE: example permrow; shows an example |
---|
[d2b2a7] | 758 | " |
---|
[3d124a7] | 759 | { |
---|
| 760 | matrix B=A; |
---|
[5411c8] | 761 | int k=ncols(B); |
---|
| 762 | B[r1,1..k]=A[r2,1..k]; |
---|
| 763 | B[r2,1..k]=A[r1,1..k]; |
---|
[3d124a7] | 764 | return(B); |
---|
| 765 | } |
---|
| 766 | example |
---|
| 767 | { "EXAMPLE:"; echo = 2; |
---|
| 768 | ring r=32003,(x,y,z),lp; |
---|
| 769 | matrix A[3][3]=1,2,3,x,y,z,7,8,9; |
---|
| 770 | print(A); |
---|
| 771 | print(permrow(A,2,1)); |
---|
| 772 | } |
---|
[63be42] | 773 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 774 | |
---|
| 775 | proc rowred (matrix A,list #) |
---|
[d26ec4] | 776 | "USAGE: rowred(A[,e]); A matrix, e any type |
---|
[b9b906] | 777 | RETURN: - a matrix B, being the row reduced form of A, if rowred is called |
---|
[cb1e43b] | 778 | with one argument. |
---|
[b9b906] | 779 | (as far as this is possible over the polynomial ring; no division |
---|
[cb1e43b] | 780 | by polynomials) |
---|
[d26ec4] | 781 | @* - a list L of two matrices, such that L[1] = L[2] * A with L[1] |
---|
| 782 | the row-reduced form of A and L[2] the transformation matrix |
---|
| 783 | (if rowred is called with two arguments). |
---|
[e70f32] | 784 | ASSUME: The entries of A are in the base field. It is not verified whether |
---|
| 785 | this assumption holds. |
---|
[7af4ad9] | 786 | NOTE: * This procedure is designed for teaching purposes mainly. |
---|
[b9b906] | 787 | @* * The straight forward Gaussian algorithm is implemented in the |
---|
[7af4ad9] | 788 | library (no standard basis computation). |
---|
[e3e3df] | 789 | The transformation matrix is obtained by concatenating a unit |
---|
| 790 | matrix to A. proc gauss_row should be faster. |
---|
[d26ec4] | 791 | @* * It should only be used with exact coefficient field (there is no |
---|
| 792 | pivoting) over the polynomial ring (ordering lp or dp). |
---|
| 793 | @* * Parameters are allowed. Hence, if the entries of A are parameters |
---|
| 794 | the computation takes place over the field of rational functions. |
---|
[7af4ad9] | 795 | SEE ALSO: gauss_row |
---|
[63be42] | 796 | EXAMPLE: example rowred; shows an example |
---|
| 797 | " |
---|
| 798 | { |
---|
| 799 | int m,n=nrows(A),ncols(A); |
---|
| 800 | int i,j,k,l,rk; |
---|
| 801 | poly p; |
---|
| 802 | matrix d[m][n]; |
---|
[bf5ba90] | 803 | for (i=1;i<=m;i++) |
---|
| 804 | { for (j=1;j<=n;j++) |
---|
[63be42] | 805 | { p = A[i,j]; |
---|
| 806 | if (ord(p)==0) |
---|
| 807 | { if (deg(p)==0) { d[i,j]=p; } |
---|
| 808 | } |
---|
| 809 | } |
---|
| 810 | } |
---|
| 811 | matrix b = A; |
---|
[d26ec4] | 812 | if (size(#) != 0) { b = concat(b,unitmat(m)); } |
---|
| 813 | for (l=1;l<=n;l=l+1) |
---|
[e70f32] | 814 | { pmat(d); |
---|
[63be42] | 815 | k = findfirst(ideal(d[l]),rk+1); |
---|
| 816 | if (k) |
---|
| 817 | { rk = rk+1; |
---|
[e7a0fa] | 818 | b = permrow(b,rk,k); |
---|
[63be42] | 819 | p = b[rk,l]; p = 1/p; |
---|
[e7a0fa] | 820 | b = multrow(b,rk,p); |
---|
[bf5ba90] | 821 | for (i=1;i<=m;i++) |
---|
[63be42] | 822 | { |
---|
[e7a0fa] | 823 | if (rk-i) { b = addrow(b,rk,-b[i,l],i);} |
---|
[63be42] | 824 | } |
---|
[e7a0fa] | 825 | d = 0; |
---|
[bf5ba90] | 826 | for (i=rk+1;i<=m;i++) |
---|
| 827 | { for (j=l+1;j<=n;j++) |
---|
[e7a0fa] | 828 | { p = b[i,j]; |
---|
| 829 | if (ord(p)==0) |
---|
| 830 | { if (deg(p)==0) { d[i,j]=p; } |
---|
| 831 | } |
---|
| 832 | } |
---|
| 833 | } |
---|
| 834 | |
---|
[63be42] | 835 | } |
---|
| 836 | } |
---|
| 837 | d = submat(b,1..m,1..n); |
---|
[b9b906] | 838 | if (size(#)) |
---|
[d26ec4] | 839 | { |
---|
| 840 | list L=d,submat(b,1..m,n+1..n+m); |
---|
| 841 | return(L); |
---|
| 842 | } |
---|
[63be42] | 843 | return(d); |
---|
| 844 | } |
---|
| 845 | example |
---|
| 846 | { "EXAMPLE:"; echo = 2; |
---|
[d26ec4] | 847 | ring r=(0,a,b),(A,B,C),dp; |
---|
| 848 | matrix m[6][8]= |
---|
[b9b906] | 849 | 0, 0, b*B, -A,-4C,2A,0, 0, |
---|
[d26ec4] | 850 | 2C,-4C,-A,B, 0, B, 3B,AB, |
---|
[b9b906] | 851 | 0,a*A, 0, 0, B, 0, 0, 0, |
---|
[d26ec4] | 852 | 0, 0, 0, 0, 2, 0, 0, 2A, |
---|
[b9b906] | 853 | 0, 0, 0, 0, 0, 0, 2b, A, |
---|
[d26ec4] | 854 | 0, 0, 0, 0, 0, 0, 0, 2a;""; |
---|
| 855 | print(rowred(m));""; |
---|
[63be42] | 856 | list L=rowred(m,1); |
---|
[d26ec4] | 857 | print(L[1]); |
---|
| 858 | print(L[2]); |
---|
[63be42] | 859 | } |
---|
| 860 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 861 | |
---|
| 862 | proc colred (matrix A,list #) |
---|
[d26ec4] | 863 | "USAGE: colred(A[,e]); A matrix, e any type |
---|
[b9b906] | 864 | RETURN: - a matrix B, being the column reduced form of A, if colred is |
---|
[cb1e43b] | 865 | called with one argument. |
---|
[b9b906] | 866 | (as far as this is possible over the polynomial ring; |
---|
[cb1e43b] | 867 | no division by polynomials) |
---|
[d26ec4] | 868 | @* - a list L of two matrices, such that L[1] = A * L[2] with L[1] |
---|
| 869 | the column-reduced form of A and L[2] the transformation matrix |
---|
| 870 | (if colred is called with two arguments). |
---|
[e70f32] | 871 | ASSUME: The entries of A are in the base field. It is not verified whether |
---|
| 872 | this assumption holds. |
---|
[7af4ad9] | 873 | NOTE: * This procedure is designed for teaching purposes mainly. |
---|
[e3e3df] | 874 | @* * It applies rowred to the transposed matrix. |
---|
| 875 | proc gauss_col should be faster. |
---|
[d26ec4] | 876 | @* * It should only be used with exact coefficient field (there is no |
---|
| 877 | pivoting) over the polynomial ring (ordering lp or dp). |
---|
| 878 | @* * Parameters are allowed. Hence, if the entries of A are parameters |
---|
| 879 | the computation takes place over the field of rational functions. |
---|
[7af4ad9] | 880 | SEE ALSO: gauss_col |
---|
[63be42] | 881 | EXAMPLE: example colred; shows an example |
---|
| 882 | " |
---|
| 883 | { |
---|
| 884 | A = transpose(A); |
---|
| 885 | if (size(#)) |
---|
| 886 | { list L = rowred(A,1); return(transpose(L[1]),transpose(L[2]));} |
---|
| 887 | else |
---|
| 888 | { return(transpose(rowred(A)));} |
---|
| 889 | } |
---|
| 890 | example |
---|
| 891 | { "EXAMPLE:"; echo = 2; |
---|
[d26ec4] | 892 | ring r=(0,a,b),(A,B,C),dp; |
---|
| 893 | matrix m[8][6]= |
---|
[b9b906] | 894 | 0, 2*C, 0, 0, 0, 0, |
---|
| 895 | 0, -4*C,a*A, 0, 0, 0, |
---|
| 896 | b*B, -A, 0, 0, 0, 0, |
---|
| 897 | -A, B, 0, 0, 0, 0, |
---|
| 898 | -4*C, 0, B, 2, 0, 0, |
---|
| 899 | 2*A, B, 0, 0, 0, 0, |
---|
| 900 | 0, 3*B, 0, 0, 2b, 0, |
---|
[d26ec4] | 901 | 0, AB, 0, 2*A,A, 2a;""; |
---|
| 902 | print(colred(m));""; |
---|
[63be42] | 903 | list L=colred(m,1); |
---|
[d26ec4] | 904 | print(L[1]); |
---|
| 905 | print(L[2]); |
---|
[63be42] | 906 | } |
---|
| 907 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 908 | |
---|
[0fe830] | 909 | proc linear_relations(module M) |
---|
| 910 | "USAGE: linear_relations(M); |
---|
| 911 | M: a module |
---|
[7f3ad4] | 912 | ASSUME: All non-zero entries of M are homogeneous polynomials of the same |
---|
[d2f488] | 913 | positive degree. The base field must be an exact field (not real |
---|
[6840b2] | 914 | or complex). |
---|
[0fe830] | 915 | It is not checked whether these assumptions hold. |
---|
| 916 | RETURN: a maximal module R such that M*R is formed by zero vectors. |
---|
| 917 | EXAMPLE: example linear_relations; shows an example. |
---|
| 918 | " |
---|
| 919 | { int n = ncols(M); |
---|
[6840b2] | 920 | def BaseR = basering; |
---|
| 921 | def br = changeord("dp",basering); |
---|
| 922 | setring br; |
---|
| 923 | module M = imap(BaseR,M); |
---|
[2ec0d5] | 924 | ideal vars = maxideal(1); |
---|
| 925 | ring tmpR = 0, ('y(1..n)), dp; |
---|
| 926 | def newR = br + tmpR; |
---|
| 927 | setring newR; |
---|
| 928 | module M = imap(br,M); |
---|
| 929 | ideal vars = imap(br,vars); |
---|
| 930 | attrib(vars,"isSB",1); |
---|
| 931 | for (int i = 1; i<=n; i++) { |
---|
| 932 | M[i] = M[i] + 'y(i)*gen(1); |
---|
[0fe830] | 933 | } |
---|
[2ec0d5] | 934 | M = interred(M); |
---|
| 935 | module redM = NF(M,vars); |
---|
| 936 | module REL; |
---|
| 937 | int sizeREL; |
---|
| 938 | int j; |
---|
| 939 | for (i=1; i<=n; i++) { |
---|
| 940 | if (M[i][1]==redM[i][1]) { //-- relation found! |
---|
| 941 | sizeREL++; |
---|
| 942 | REL[sizeREL]=0; |
---|
| 943 | for (j=1; j<=n; j++) { |
---|
| 944 | REL[sizeREL] = REL[sizeREL] + (M[i][1]/'y(j))*gen(j); |
---|
[0fe830] | 945 | } |
---|
| 946 | } |
---|
| 947 | } |
---|
[6840b2] | 948 | setring BaseR; |
---|
[2ec0d5] | 949 | return(minbase(imap(newR,REL))); |
---|
[0fe830] | 950 | } |
---|
| 951 | example |
---|
| 952 | { "EXAMPLE:"; echo = 2; |
---|
| 953 | ring r = (3,w), (a,b,c,d),dp; |
---|
| 954 | minpoly = w2-w-1; |
---|
| 955 | module M = [a2,b2],[wab,w2c2+2b2],[(w-2)*a2+wab,wb2+w2c2]; |
---|
| 956 | module REL = linear_relations(M); |
---|
| 957 | pmat(REL); |
---|
| 958 | pmat(M*REL); |
---|
[7f3ad4] | 959 | } |
---|
[0fe830] | 960 | |
---|
| 961 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 962 | |
---|
[63be42] | 963 | static proc findfirst (ideal i,int t) |
---|
| 964 | { |
---|
| 965 | int n,k; |
---|
| 966 | for (n=t;n<=ncols(i);n=n+1) |
---|
| 967 | { |
---|
| 968 | if (i[n]!=0) { k=n;break;} |
---|
| 969 | } |
---|
| 970 | return(k); |
---|
| 971 | } |
---|
| 972 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 973 | |
---|
| 974 | proc rm_unitcol(matrix A) |
---|
[d26ec4] | 975 | "USAGE: rm_unitcol(A); A matrix (being row-reduced) |
---|
[63be42] | 976 | RETURN: matrix, obtained from A by deleting unit columns (having just one 1 |
---|
| 977 | and else 0 as entries) and associated rows |
---|
| 978 | EXAMPLE: example rm_unitcol; shows an example |
---|
| 979 | " |
---|
| 980 | { |
---|
| 981 | int l,j; |
---|
| 982 | intvec v; |
---|
[bf5ba90] | 983 | for (j=1;j<=ncols(A);j++) |
---|
[63be42] | 984 | { |
---|
| 985 | if (gen(l+1)==module(A)[j]) {l=l+1;} |
---|
| 986 | else { v=v,j;} |
---|
| 987 | } |
---|
| 988 | if (size(v)>1) |
---|
| 989 | { v = v[2..size(v)]; |
---|
| 990 | return(submat(A,l+1..nrows(A),v)); |
---|
| 991 | } |
---|
| 992 | else |
---|
| 993 | { return(0);} |
---|
| 994 | } |
---|
| 995 | example |
---|
| 996 | { "EXAMPLE:"; echo = 2; |
---|
| 997 | ring r=0,(A,B,C),dp; |
---|
[d26ec4] | 998 | matrix m[6][8]= |
---|
| 999 | 0, 0, A, 0, 1,0, 0,0, |
---|
| 1000 | 0, 0, -C2, 0, 0,0, 1,0, |
---|
| 1001 | 0, 0, 0,1/2B, 0,0, 0,1, |
---|
| 1002 | 0, 0, B, -A, 0,2A, 0,0, |
---|
| 1003 | 2C,-4C, -A, B, 0,B, 0,0, |
---|
| 1004 | 0, A, 0, 0, 0,0, 0,0; |
---|
[63be42] | 1005 | print(rm_unitcol(m)); |
---|
| 1006 | } |
---|
| 1007 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 1008 | |
---|
| 1009 | proc rm_unitrow (matrix A) |
---|
[d26ec4] | 1010 | "USAGE: rm_unitrow(A); A matrix (being col-reduced) |
---|
[63be42] | 1011 | RETURN: matrix, obtained from A by deleting unit rows (having just one 1 |
---|
| 1012 | and else 0 as entries) and associated columns |
---|
| 1013 | EXAMPLE: example rm_unitrow; shows an example |
---|
| 1014 | " |
---|
| 1015 | { |
---|
| 1016 | int l,j; |
---|
| 1017 | intvec v; |
---|
| 1018 | module M = transpose(A); |
---|
[bf5ba90] | 1019 | for (j=1; j <= nrows(A); j++) |
---|
[63be42] | 1020 | { |
---|
| 1021 | if (gen(l+1) == M[j]) { l=l+1; } |
---|
| 1022 | else { v=v,j; } |
---|
| 1023 | } |
---|
| 1024 | if (size(v) > 1) |
---|
| 1025 | { v = v[2..size(v)]; |
---|
| 1026 | return(submat(A,v,l+1..ncols(A))); |
---|
| 1027 | } |
---|
| 1028 | else |
---|
| 1029 | { return(0);} |
---|
| 1030 | } |
---|
| 1031 | example |
---|
| 1032 | { "EXAMPLE:"; echo = 2; |
---|
| 1033 | ring r=0,(A,B,C),dp; |
---|
[d26ec4] | 1034 | matrix m[8][6]= |
---|
| 1035 | 0,0, 0, 0, 2C, 0, |
---|
| 1036 | 0,0, 0, 0, -4C,A, |
---|
| 1037 | A,-C2,0, B, -A, 0, |
---|
| 1038 | 0,0, 1/2B,-A,B, 0, |
---|
| 1039 | 1,0, 0, 0, 0, 0, |
---|
| 1040 | 0,0, 0, 2A,B, 0, |
---|
| 1041 | 0,1, 0, 0, 0, 0, |
---|
| 1042 | 0,0, 1, 0, 0, 0; |
---|
[63be42] | 1043 | print(rm_unitrow(m)); |
---|
| 1044 | } |
---|
| 1045 | ////////////////////////////////////////////////////////////////////////////// |
---|
[7dd7549] | 1046 | proc headStand(matrix M) |
---|
[76aca2] | 1047 | "USAGE: headStand(M); M matrix |
---|
| 1048 | RETURN: matrix B such that B[i][j]=M[n-i+1,m-j+1], n=nrows(M), m=ncols(M) |
---|
| 1049 | EXAMPLE: example headStand; shows an example |
---|
| 1050 | " |
---|
[7dd7549] | 1051 | { |
---|
| 1052 | int i,j; |
---|
| 1053 | int n=nrows(M); |
---|
| 1054 | int m=ncols(M); |
---|
| 1055 | matrix B[n][m]; |
---|
| 1056 | for(i=1;i<=n;i++) |
---|
| 1057 | { |
---|
| 1058 | for(j=1;j<=m;j++) |
---|
| 1059 | { |
---|
| 1060 | B[n-i+1,m-j+1]=M[i,j]; |
---|
| 1061 | } |
---|
| 1062 | } |
---|
| 1063 | return(B); |
---|
| 1064 | } |
---|
[76aca2] | 1065 | example |
---|
| 1066 | { "EXAMPLE:"; echo = 2; |
---|
| 1067 | ring r=0,(A,B,C),dp; |
---|
| 1068 | matrix M[2][3]= |
---|
| 1069 | 0,A, B, |
---|
| 1070 | A2, B2, C; |
---|
| 1071 | print(M); |
---|
| 1072 | print(headStand(M)); |
---|
| 1073 | } |
---|
[7dd7549] | 1074 | ////////////////////////////////////////////////////////////////////////////// |
---|
[bb7a4d] | 1075 | |
---|
[2ed3a3] | 1076 | // Symmetric/Exterior powers thanks to Oleksandr Iena for his persistence ;-) |
---|
[bb7a4d] | 1077 | |
---|
[2ed3a3] | 1078 | proc symmetricBasis(int n, int k, list #) |
---|
| 1079 | "USAGE: symmetricBasis(n, k[,s]); n int, k int, s string |
---|
[7f3ad4] | 1080 | RETURN: ring, poynomial ring containing the ideal \"symBasis\", |
---|
[2ed3a3] | 1081 | being a basis of the k-th symmetric power of an n-dim vector space. |
---|
[7f3ad4] | 1082 | NOTE: The output polynomial ring has characteristics 0 and n variables |
---|
[2ed3a3] | 1083 | named \"S(i)\", where the base variable name S is either given by the |
---|
| 1084 | optional string argument(which must not contain brackets) or equal to |
---|
| 1085 | "e" by default. |
---|
[bb7a4d] | 1086 | SEE ALSO: exteriorBasis |
---|
| 1087 | KEYWORDS: symmetric basis |
---|
| 1088 | EXAMPLE: example symmetricBasis; shows an example" |
---|
| 1089 | { |
---|
[2ed3a3] | 1090 | //------------------------ handle optional base variable name--------------- |
---|
| 1091 | string S = "e"; |
---|
| 1092 | if( size(#) > 0 ) |
---|
[bb7a4d] | 1093 | { |
---|
[2ed3a3] | 1094 | if( typeof(#[1]) != "string" ) |
---|
| 1095 | { |
---|
| 1096 | ERROR("Wrong optional argument: must be a string"); |
---|
| 1097 | } |
---|
| 1098 | S = #[1]; |
---|
| 1099 | if( (find(S, "(") + find(S, ")")) > 0 ) |
---|
| 1100 | { |
---|
[7f3ad4] | 1101 | ERROR("Wrong optional argument: must be a string without brackets"); |
---|
[2ed3a3] | 1102 | } |
---|
[bb7a4d] | 1103 | } |
---|
| 1104 | |
---|
[2ed3a3] | 1105 | //------------------------- create ring container for symmetric power basis- |
---|
| 1106 | execute("ring @@@SYM_POWER_RING_NAME=(0),("+S+"(1.."+string(n)+")),dp;"); |
---|
| 1107 | |
---|
| 1108 | //------------------------- choose symmetric basis ------------------------- |
---|
| 1109 | ideal symBasis = maxideal(k); |
---|
[bb7a4d] | 1110 | |
---|
[2ed3a3] | 1111 | //------------------------- export and return ------------------------- |
---|
| 1112 | export symBasis; |
---|
| 1113 | return(basering); |
---|
[bb7a4d] | 1114 | } |
---|
| 1115 | example |
---|
| 1116 | { "EXAMPLE:"; echo = 2; |
---|
| 1117 | |
---|
[2ed3a3] | 1118 | // basis of the 3-rd symmetricPower of a 4-dim vector space: |
---|
[7f3ad4] | 1119 | def R = symmetricBasis(4, 3, "@e"); setring R; |
---|
[2ed3a3] | 1120 | R; // container ring: |
---|
| 1121 | symBasis; // symmetric basis: |
---|
[bb7a4d] | 1122 | } |
---|
| 1123 | |
---|
| 1124 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 1125 | |
---|
[2ed3a3] | 1126 | proc exteriorBasis(int n, int k, list #) |
---|
| 1127 | "USAGE: exteriorBasis(n, k[,s]); n int, k int, s string |
---|
[7f3ad4] | 1128 | RETURN: qring, an exterior algebra containing the ideal \"extBasis\", |
---|
[2ed3a3] | 1129 | being a basis of the k-th exterior power of an n-dim vector space. |
---|
[7f3ad4] | 1130 | NOTE: The output polynomial ring has characteristics 0 and n variables |
---|
[2ed3a3] | 1131 | named \"S(i)\", where the base variable name S is either given by the |
---|
| 1132 | optional string argument(which must not contain brackets) or equal to |
---|
| 1133 | "e" by default. |
---|
[bb7a4d] | 1134 | SEE ALSO: symmetricBasis |
---|
| 1135 | KEYWORDS: exterior basis |
---|
| 1136 | EXAMPLE: example exteriorBasis; shows an example" |
---|
| 1137 | { |
---|
[2ed3a3] | 1138 | //------------------------ handle optional base variable name--------------- |
---|
| 1139 | string S = "e"; |
---|
| 1140 | if( size(#) > 0 ) |
---|
[bb7a4d] | 1141 | { |
---|
[2ed3a3] | 1142 | if( typeof(#[1]) != "string" ) |
---|
| 1143 | { |
---|
| 1144 | ERROR("Wrong optional argument: must be a string"); |
---|
| 1145 | } |
---|
| 1146 | S = #[1]; |
---|
| 1147 | if( (find(S, "(") + find(S, ")")) > 0 ) |
---|
| 1148 | { |
---|
[7f3ad4] | 1149 | ERROR("Wrong optional argument: must be a string without brackets"); |
---|
[2ed3a3] | 1150 | } |
---|
[bb7a4d] | 1151 | } |
---|
[2ed3a3] | 1152 | |
---|
| 1153 | //------------------------- create ring container for symmetric power basis- |
---|
| 1154 | execute("ring @@@EXT_POWER_RING_NAME=(0),("+S+"(1.."+string(n)+")),dp;"); |
---|
| 1155 | |
---|
| 1156 | //------------------------- choose exterior basis ------------------------- |
---|
[407fdc0] | 1157 | def T = superCommutative(); setring T; |
---|
[2ed3a3] | 1158 | ideal extBasis = simplify( NF(maxideal(k), std(0)), 1 + 2 + 8 ); |
---|
[bb7a4d] | 1159 | |
---|
[2ed3a3] | 1160 | //------------------------- export and return ------------------------- |
---|
| 1161 | export extBasis; |
---|
| 1162 | return(basering); |
---|
[bb7a4d] | 1163 | } |
---|
| 1164 | example |
---|
| 1165 | { "EXAMPLE:"; echo = 2; |
---|
[2ed3a3] | 1166 | // basis of the 3-rd symmetricPower of a 4-dim vector space: |
---|
[7f3ad4] | 1167 | def r = exteriorBasis(4, 3, "@e"); setring r; |
---|
[2ed3a3] | 1168 | r; // container ring: |
---|
| 1169 | extBasis; // exterior basis: |
---|
| 1170 | } |
---|
[bb7a4d] | 1171 | |
---|
[a6b576] | 1172 | ////////////////////////////////////////////////////////////////////////////// |
---|
[bb7a4d] | 1173 | |
---|
[2ed3a3] | 1174 | static proc chooseSafeVarName(string prefix, string suffix) |
---|
| 1175 | "USAGE: give appropreate prefix for variable names |
---|
| 1176 | RETURN: safe variable name (repeated prefix + suffix) |
---|
| 1177 | " |
---|
| 1178 | { |
---|
| 1179 | string V = varstr(basering); |
---|
| 1180 | string S = suffix; |
---|
| 1181 | while( find(V, S) > 0 ) |
---|
| 1182 | { |
---|
| 1183 | S = prefix + S; |
---|
| 1184 | } |
---|
| 1185 | return(S); |
---|
[bb7a4d] | 1186 | } |
---|
| 1187 | |
---|
[a6b576] | 1188 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 1189 | |
---|
[2ed3a3] | 1190 | static proc mapPower(int p, module A, int k, def Tn, def Tm) |
---|
| 1191 | "USAGE: by both symmetric- and exterior-Power" |
---|
| 1192 | NOTE: everything over the basering! |
---|
| 1193 | module A (matrix of the map), int k (power) |
---|
| 1194 | rings Tn is source- and Tm is image-ring with bases |
---|
| 1195 | resp. Ink and Imk. |
---|
[7f3ad4] | 1196 | M = max dim of Image, N - dim. of source |
---|
[2ed3a3] | 1197 | SEE ALSO: symmetricPower, exteriorPower" |
---|
[bb7a4d] | 1198 | { |
---|
| 1199 | def save = basering; |
---|
| 1200 | |
---|
[2ed3a3] | 1201 | int n = nvars(save); |
---|
[bb7a4d] | 1202 | int M = nrows(A); |
---|
| 1203 | int N = ncols(A); |
---|
| 1204 | |
---|
| 1205 | int i, j; |
---|
| 1206 | |
---|
[2ed3a3] | 1207 | //------------------------- compute matrix of single images ------------------ |
---|
[7f3ad4] | 1208 | def Rm = save + Tm; setring Rm; |
---|
[bb7a4d] | 1209 | dbprint(p-2, "Temporary Working Ring", Rm); |
---|
| 1210 | |
---|
| 1211 | module A = imap(save, A); |
---|
| 1212 | |
---|
| 1213 | ideal B; poly t; |
---|
| 1214 | |
---|
| 1215 | for( i = N; i > 0; i-- ) |
---|
| 1216 | { |
---|
| 1217 | t = 0; |
---|
| 1218 | for( j = M; j > 0; j-- ) |
---|
| 1219 | { |
---|
[2ed3a3] | 1220 | t = t + A[i][j] * var(n + j); |
---|
[bb7a4d] | 1221 | } |
---|
| 1222 | |
---|
| 1223 | B[i] = t; |
---|
| 1224 | } |
---|
| 1225 | |
---|
[2ed3a3] | 1226 | dbprint(p-1, "Matrix of single images", B); |
---|
[bb7a4d] | 1227 | |
---|
[2ed3a3] | 1228 | //------------------------- compute image --------------------- |
---|
| 1229 | // apply S^k(A): Tn -> Rm to Source basis vectors Ink: |
---|
[7f3ad4] | 1230 | map TMap = Tn, B; |
---|
[bb7a4d] | 1231 | |
---|
[7f3ad4] | 1232 | ideal C = NF(TMap(Ink), std(0)); |
---|
[bb7a4d] | 1233 | dbprint(p-1, "Image Matrix: ", C); |
---|
| 1234 | |
---|
| 1235 | |
---|
[2ed3a3] | 1236 | //------------------------- write it in Image basis --------------------- |
---|
[bb7a4d] | 1237 | ideal Imk = imap(Tm, Imk); |
---|
| 1238 | |
---|
| 1239 | module D; poly lm; vector tt; |
---|
| 1240 | |
---|
| 1241 | for( i = ncols(C); i > 0; i-- ) |
---|
| 1242 | { |
---|
| 1243 | t = C[i]; |
---|
| 1244 | tt = 0; |
---|
| 1245 | |
---|
| 1246 | while( t != 0 ) |
---|
| 1247 | { |
---|
| 1248 | lm = leadmonom(t); |
---|
| 1249 | // lm; |
---|
| 1250 | for( j = ncols(Imk); j > 0; j-- ) |
---|
| 1251 | { |
---|
| 1252 | if( lm / Imk[j] != 0 ) |
---|
| 1253 | { |
---|
| 1254 | tt = tt + (lead(t) / Imk[j]) * gen(j); |
---|
| 1255 | break; |
---|
| 1256 | } |
---|
| 1257 | } |
---|
| 1258 | t = t - lead(t); |
---|
| 1259 | } |
---|
| 1260 | |
---|
| 1261 | D[i] = tt; |
---|
| 1262 | } |
---|
| 1263 | |
---|
[2ed3a3] | 1264 | //------------------------- map it back and return --------------------- |
---|
| 1265 | setring save; |
---|
| 1266 | return( imap(Rm, D) ); |
---|
| 1267 | } |
---|
[bb7a4d] | 1268 | |
---|
| 1269 | |
---|
[2ed3a3] | 1270 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 1271 | |
---|
| 1272 | proc symmetricPower(module A, int k) |
---|
| 1273 | "USAGE: symmetricPower(A, k); A module, k int |
---|
| 1274 | RETURN: module: the k-th symmetric power of A |
---|
[7f3ad4] | 1275 | NOTE: the chosen bases and most of intermediate data will be shown if |
---|
[2ed3a3] | 1276 | printlevel is big enough |
---|
| 1277 | SEE ALSO: exteriorPower |
---|
| 1278 | KEYWORDS: symmetric power |
---|
| 1279 | EXAMPLE: example symmetricPower; shows an example" |
---|
| 1280 | { |
---|
| 1281 | int p = printlevel - voice + 2; |
---|
| 1282 | |
---|
| 1283 | def save = basering; |
---|
| 1284 | |
---|
| 1285 | int M = nrows(A); |
---|
| 1286 | int N = ncols(A); |
---|
[bb7a4d] | 1287 | |
---|
[2ed3a3] | 1288 | string S = chooseSafeVarName("@", "@_e"); |
---|
[bb7a4d] | 1289 | |
---|
[2ed3a3] | 1290 | //------------------------- choose source basis ------------------------- |
---|
| 1291 | def Tn = symmetricBasis(N, k, S); setring Tn; |
---|
| 1292 | ideal Ink = symBasis; |
---|
| 1293 | export Ink; |
---|
| 1294 | dbprint(p-3, "Temporary Source Ring", basering); |
---|
| 1295 | dbprint(p, "S^k(Source Basis)", Ink); |
---|
[bb7a4d] | 1296 | |
---|
[2ed3a3] | 1297 | //------------------------- choose image basis ------------------------- |
---|
| 1298 | def Tm = symmetricBasis(M, k, S); setring Tm; |
---|
| 1299 | ideal Imk = symBasis; |
---|
| 1300 | export Imk; |
---|
| 1301 | dbprint(p-3, "Temporary Image Ring", basering); |
---|
| 1302 | dbprint(p, "S^k(Image Basis)", Imk); |
---|
| 1303 | |
---|
| 1304 | //------------------------- compute and return S^k(A) in chosen bases -- |
---|
[7f3ad4] | 1305 | setring save; |
---|
[bb7a4d] | 1306 | |
---|
[2ed3a3] | 1307 | return(mapPower(p, A, k, Tn, Tm)); |
---|
[bb7a4d] | 1308 | } |
---|
| 1309 | example |
---|
| 1310 | { "EXAMPLE:"; echo = 2; |
---|
| 1311 | |
---|
| 1312 | ring r = (0),(a, b, c, d), dp; r; |
---|
[2ed3a3] | 1313 | module B = a*gen(1) + c* gen(2), b * gen(1) + d * gen(2); print(B); |
---|
[bb7a4d] | 1314 | |
---|
[2ed3a3] | 1315 | // symmetric power over a commutative K-algebra: |
---|
[bb7a4d] | 1316 | print(symmetricPower(B, 2)); |
---|
| 1317 | print(symmetricPower(B, 3)); |
---|
| 1318 | |
---|
[2ed3a3] | 1319 | // symmetric power over an exterior algebra: |
---|
[407fdc0] | 1320 | def g = superCommutative(); setring g; g; |
---|
[bb7a4d] | 1321 | |
---|
[2ed3a3] | 1322 | module B = a*gen(1) + c* gen(2), b * gen(1) + d * gen(2); print(B); |
---|
[bb7a4d] | 1323 | |
---|
[2ed3a3] | 1324 | print(symmetricPower(B, 2)); // much smaller! |
---|
| 1325 | print(symmetricPower(B, 3)); // zero! (over an exterior algebra!) |
---|
[bb7a4d] | 1326 | |
---|
| 1327 | } |
---|
| 1328 | |
---|
| 1329 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 1330 | |
---|
| 1331 | proc exteriorPower(module A, int k) |
---|
| 1332 | "USAGE: exteriorPower(A, k); A module, k int |
---|
| 1333 | RETURN: module: the k-th exterior power of A |
---|
[7f3ad4] | 1334 | NOTE: the chosen bases and most of intermediate data will be shown if |
---|
[2ed3a3] | 1335 | printlevel is big enough. Last rows will be invisible if zero. |
---|
[bb7a4d] | 1336 | SEE ALSO: symmetricPower |
---|
| 1337 | KEYWORDS: exterior power |
---|
| 1338 | EXAMPLE: example exteriorPower; shows an example" |
---|
| 1339 | { |
---|
| 1340 | int p = printlevel - voice + 2; |
---|
| 1341 | def save = basering; |
---|
| 1342 | |
---|
| 1343 | int M = nrows(A); |
---|
| 1344 | int N = ncols(A); |
---|
| 1345 | |
---|
[2ed3a3] | 1346 | string S = chooseSafeVarName("@", "@_e"); |
---|
[bb7a4d] | 1347 | |
---|
[2ed3a3] | 1348 | //------------------------- choose source basis ------------------------- |
---|
| 1349 | def Tn = exteriorBasis(N, k, S); setring Tn; |
---|
| 1350 | ideal Ink = extBasis; |
---|
| 1351 | export Ink; |
---|
[bb7a4d] | 1352 | dbprint(p-3, "Temporary Source Ring", basering); |
---|
| 1353 | dbprint(p, "E^k(Source Basis)", Ink); |
---|
| 1354 | |
---|
[2ed3a3] | 1355 | //------------------------- choose image basis ------------------------- |
---|
| 1356 | def Tm = exteriorBasis(M, k, S); setring Tm; |
---|
| 1357 | ideal Imk = extBasis; |
---|
| 1358 | export Imk; |
---|
[bb7a4d] | 1359 | dbprint(p-3, "Temporary Image Ring", basering); |
---|
| 1360 | dbprint(p, "E^k(Image Basis)", Imk); |
---|
| 1361 | |
---|
[2ed3a3] | 1362 | //------------------------- compute and return E^k(A) in chosen bases -- |
---|
[bb7a4d] | 1363 | setring save; |
---|
[2ed3a3] | 1364 | return(mapPower(p, A, k, Tn, Tm)); |
---|
[bb7a4d] | 1365 | } |
---|
| 1366 | example |
---|
| 1367 | { "EXAMPLE:"; echo = 2; |
---|
[7f3ad4] | 1368 | ring r = (0),(a, b, c, d, e, f), dp; |
---|
[2ed3a3] | 1369 | r; "base ring:"; |
---|
[bb7a4d] | 1370 | |
---|
[7f3ad4] | 1371 | module B = a*gen(1) + c*gen(2) + e*gen(3), |
---|
| 1372 | b*gen(1) + d*gen(2) + f*gen(3), |
---|
[2ed3a3] | 1373 | e*gen(1) + f*gen(3); |
---|
[bb7a4d] | 1374 | |
---|
[2ed3a3] | 1375 | print(B); |
---|
[7f3ad4] | 1376 | print(exteriorPower(B, 2)); |
---|
| 1377 | print(exteriorPower(B, 3)); |
---|
[bb7a4d] | 1378 | |
---|
[407fdc0] | 1379 | def g = superCommutative(); setring g; g; |
---|
[bb7a4d] | 1380 | |
---|
[2ed3a3] | 1381 | module A = a*gen(1), b * gen(1), c*gen(2), d * gen(2); |
---|
| 1382 | print(A); |
---|
[bb7a4d] | 1383 | |
---|
| 1384 | print(exteriorPower(A, 2)); |
---|
| 1385 | |
---|
[7f3ad4] | 1386 | module B = a*gen(1) + c*gen(2) + e*gen(3), |
---|
| 1387 | b*gen(1) + d*gen(2) + f*gen(3), |
---|
[2ed3a3] | 1388 | e*gen(1) + f*gen(3); |
---|
| 1389 | print(B); |
---|
[bb7a4d] | 1390 | |
---|
| 1391 | print(exteriorPower(B, 2)); |
---|
| 1392 | print(exteriorPower(B, 3)); |
---|
| 1393 | |
---|
| 1394 | } |
---|
| 1395 | |
---|
| 1396 | ////////////////////////////////////////////////////////////////////////////// |
---|