1 | // $Id: matrix.lib,v 1.1.1.1 1997-04-25 15:13:26 obachman Exp $ |
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2 | //(GMG+BM) |
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3 | /////////////////////////////////////////////////////////////////////////////// |
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4 | LIBRARY: matrix.lib PROCEDURES FOR MATRIX OPERATIONS |
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5 | |
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6 | compress(A); matrix, zero columns from A deleted |
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7 | concat(A1,A2,..); matrix, concatenation of matrices A1,A2,... |
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8 | diag(p,n); matrix, nxn diagonal matrix with entries poly p |
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9 | dsum(A1,A2,..); matrix, direct sum of matrices A1,A2,... |
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10 | flatten(A); ideal, generated by entries of matrix A |
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11 | is_complex(c); 1 if list c is a complex, 0 if not |
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12 | outer(A,B); matrix, outer product of matrices A and B |
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13 | skewmat(n[,id]); generic skew-symmetric nxn matrix [entries from id] |
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14 | submat(A,r,c); submatrix of A with rows/cols specified by intvec r/c |
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15 | symmat(n[,id]); generic symmetric nxn matrix [entries from id] |
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16 | tensor(A,B); matrix, tensor product of matrices A nd B |
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17 | unitmat(n); unit square matrix of size n |
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18 | gauss_col(A); transform constant matrix A into col-reduced nf |
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19 | gauss_row(A); transform constant matrix A into row-reduced nf |
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20 | addcol(A,c1,p,c2); add p*(c1-th col) to c2-th column of matrix A, p poly |
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21 | addrow(A,r1,p,r2); add p*(r1-th row) to r2-th row of matrix A, p poly |
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22 | multcol(A,c,p); multiply c-th column of A with poly p |
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23 | multrow(A,r,p); multiply r-th row of A with poly p |
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24 | permcol(A,i,j); permute i-th and j-th columns |
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25 | permrow(A,i,j); permute i-th and j-th rows |
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26 | (parameters in square brackets [] are optional) |
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27 | |
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28 | LIB "inout.lib"; |
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29 | LIB "ring.lib"; |
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30 | LIB "random.lib"; |
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31 | /////////////////////////////////////////////////////////////////////////////// |
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32 | |
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33 | proc compress (A) |
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34 | USAGE: compress(A); A matrix/intmat/ideal/module |
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35 | RETURN: matrix/intmat/ideal/module, zero columns/generators from A deleted |
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36 | EXAMPLE: example compress; shows an example |
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37 | { |
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38 | if( typeof(A)=="matrix" ) { return(matrix(simplify(A,2))); } |
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39 | if( typeof(A)=="intmat" ) |
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40 | { |
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41 | ring r=0,x,lp; |
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42 | module m=module(matrix(A)); |
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43 | int c= size(m); |
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44 | intmat B[nrows(A)][c]; |
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45 | int i,j; |
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46 | for( i=1; i<=ncols(A); i++ ) |
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47 | { |
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48 | if( m[i]!=[0] ) |
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49 | { |
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50 | j=j+1; |
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51 | B[1..nrows(A),j]=A[1..nrows(A),i]; |
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52 | } |
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53 | } |
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54 | return(B); |
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55 | } |
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56 | return(simplify(A,2)); |
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57 | } |
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58 | example |
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59 | { "EXAMPLE:"; echo = 2; |
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60 | ring r=0,(x,y,z),ds; |
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61 | matrix A[3][4]=1,0,3,0,x,0,z,0,x2,0,z2,0; |
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62 | print(A); |
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63 | print(compress(A)); |
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64 | module m=module(A); show(m); |
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65 | show(compress(m)); |
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66 | intmat B[3][4]=1,0,3,0,4,0,5,0,6,0,7,0; |
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67 | compress(B); |
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68 | } |
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69 | //////////////////////////////////////////////////////////////////////////////// |
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70 | proc concat (list #) |
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71 | USAGE: concat(A1,A2,..); A1,A2,... matrices |
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72 | RETURN: matrix, concatenation of A1,A2,... . Number of rows of result matrix is |
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73 | max(nrows(A1),nrows(A2),...) |
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74 | EXAMPLE: example concat; shows an example |
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75 | { |
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76 | int i; |
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77 | module B=module(#[1]); |
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78 | for( i=2; i<=size(#); i++ ) { B=B,module(#[i]); } |
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79 | return(matrix(B)); |
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80 | } |
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81 | example |
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82 | { "EXAMPLE:"; echo = 2; |
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83 | ring r=0,(x,y,z),ds; |
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84 | matrix A[3][3]=1,2,3,x,y,z,x2,y2,z2; |
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85 | matrix B[2][2]=1,0,2,0; matrix C[1][4]=4,5,x,y; |
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86 | print(A); |
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87 | print(B); |
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88 | print(C); |
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89 | print(concat(A,B,C)); |
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90 | } |
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91 | //////////////////////////////////////////////////////////////////////////////// |
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92 | |
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93 | proc diag (list #) |
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94 | USAGE: diag(p,n); p poly, n integer |
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95 | diag(A); A matrix |
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96 | RETURN: diag(p,n): diagonal matrix, p times unitmatrix of size n |
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97 | diag(A) : n*mxn*m diagonal matrix with entries all the entries of the |
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98 | nxm matrix A, taken from the 1st row, 2nd row etc of A |
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99 | EXAMPLE: example diag; shows an example |
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100 | { |
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101 | if( size(#)==2 ) { return(matrix(#[1]*freemodule(#[2]))); } |
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102 | if( size(#)==1 ) |
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103 | { |
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104 | int i; ideal id=#[1]; |
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105 | int n=ncols(id); matrix A[n][n]; |
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106 | for( i=1; i<=n; i++ ) { A[i,i]=id[i]; } |
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107 | } |
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108 | return(A); |
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109 | } |
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110 | example |
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111 | { "EXAMPLE:"; echo = 2; |
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112 | ring r=0,(x,y,z),ds; |
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113 | print(diag(xy,4)); |
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114 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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115 | print(A); |
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116 | print(diag(A)); |
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117 | } |
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118 | //////////////////////////////////////////////////////////////////////////////// |
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119 | |
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120 | proc flatten (matrix A) |
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121 | USAGE: flatten(A); A matrix |
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122 | RETURN: ideal, generated by all entries from A |
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123 | EXAMPLE: example flatten; shows an example |
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124 | { |
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125 | return(ideal(A)); |
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126 | } |
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127 | example |
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128 | { "EXAMPLE:"; echo = 2; |
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129 | ring r=0,(x,y,z),ds; |
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130 | matrix A[3][3]=1,2,3,x,y,z,7,8,9; |
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131 | print(A); |
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132 | flatten(A); |
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133 | } |
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134 | //////////////////////////////////////////////////////////////////////////////// |
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135 | |
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136 | proc dsum (list #) |
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137 | USAGE: dsum(A1,A2,..); A1,A2,... matrices |
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138 | RETURN: matrix, direct sum of A1,A2,... |
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139 | EXAMPLE: example dsum; shows an example |
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140 | { |
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141 | int i,N,a; |
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142 | list L; |
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143 | for( i=1; i<=size(#); i++ ) { N=N+nrows(#[i]); } |
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144 | for( i=1; i<=size(#); i++ ) |
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145 | { |
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146 | matrix B[N][ncols(#[i])]; |
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147 | B[a+1..a+nrows(#[i]),1..ncols(#[i])]=#[i]; |
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148 | a=a+nrows(#[i]); |
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149 | L[i]=B; kill B; |
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150 | } |
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151 | return(concat(L)); |
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152 | } |
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153 | example |
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154 | { "EXAMPLE:"; echo = 2; |
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155 | ring r=0,(x,y,z),ds; |
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156 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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157 | matrix B[2][2]=1,x,y,z; |
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158 | matrix C[1][4]=4,5,x,y; |
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159 | print(A); |
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160 | print(B); |
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161 | print(C); |
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162 | print(dsum(A,B,C)); |
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163 | } |
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164 | /////////////////////////////////////////////////////////////////////////////// |
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165 | |
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166 | proc is_complex (list c) |
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167 | USAGE: is_complex(c); c = list of size-compatible modules or matrices |
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168 | RETURN: 1 if c[i]*c[i+1]=0 for all i, 0 if not. |
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169 | NOTE: Ideals are treated internally as 1-line matrices |
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170 | EXAMPLE: example is_complex; shows an example |
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171 | { |
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172 | int i; |
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173 | module @test; |
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174 | for( i=1; i<=size(c)-1; i=i+1 ) |
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175 | { |
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176 | c[i]=matrix(c[i]); c[i+1]=matrix(c[i+1]); |
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177 | @test=c[i]*c[i+1]; |
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178 | if (size(@test)!=0) |
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179 | { |
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180 | if( voice==2 ) { "// argument is not a complex at position",i; } |
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181 | return(0); |
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182 | } |
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183 | } |
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184 | if( voice==2 ) { "// argument is a complex"; } |
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185 | return(1); |
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186 | } |
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187 | example |
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188 | { "EXAMPLE:"; echo = 2; |
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189 | ring r=32003,(x,y,z),ds; |
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190 | ideal i=x4+y5+z6,xyz,yx2+xz2+zy7; |
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191 | list L=res(i,0); |
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192 | is_complex(L); |
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193 | L[4]=matrix(i); |
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194 | is_complex(L); |
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195 | } |
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196 | //////////////////////////////////////////////////////////////////////////////// |
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197 | |
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198 | proc outer (matrix A, matrix B) |
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199 | USAGE: outer(A,B); A,B matrices |
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200 | RETURN: matrix, outer product of A and B |
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201 | EXAMPLE: example outer; shows an example |
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202 | { |
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203 | int i,j; list L; |
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204 | int N=nrows(A)*nrows(B); |
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205 | matrix C[N][ncols(B)]; |
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206 | for( i=1; i<=ncols(A); i++ ) |
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207 | { |
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208 | for( j=1; j<=nrows(A); j++ ) |
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209 | { |
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210 | C[(j-1)*nrows(B)+1..j*nrows(B),1..ncols(B)]=A[j,i]*B; |
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211 | } |
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212 | L[i]=C; |
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213 | } |
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214 | return(concat(L)); |
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215 | } |
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216 | example |
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217 | { "EXAMPLE:"; echo = 2; |
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218 | ring r=32003,(x,y,z),ds; |
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219 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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220 | matrix B[2][2]=x,y,0,z; |
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221 | print(A); |
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222 | print(B); |
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223 | print(outer(A,B)); |
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224 | } |
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225 | //////////////////////////////////////////////////////////////////////////////// |
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226 | |
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227 | proc skewmat (int n, list #) |
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228 | USAGE: skewmat(n[,id]); n integer, id ideal |
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229 | RETURN: skew-symmetric nxn matrix, with entries from id |
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230 | (default: id=maxideal(1)) |
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231 | NOTE: if id has less than n*(n-1)/2 elements, the matrix is filled with 0's, |
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232 | skewmat(n); creates the generic skew-symmetric matrix |
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233 | EXAMPLE: example skewmat; shows an example |
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234 | { |
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235 | matrix B[n][n]; |
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236 | if( size(#)==0 ) { ideal id=maxideal(1); } |
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237 | else { ideal id=#[1]; } |
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238 | id = id,B[1..n,1..n]; |
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239 | int i,j; |
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240 | for( i=0; i<=n-2; i++ ) |
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241 | { |
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242 | B[i+1,i+2..n]=id[j+1..j+n-i-1]; |
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243 | j=j+n-i-1; |
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244 | } |
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245 | matrix A=transpose(B); |
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246 | B=B-A; |
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247 | return(B); |
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248 | } |
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249 | example |
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250 | { "EXAMPLE:"; echo = 2; |
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251 | ring R=0,x(1..5),lp; |
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252 | print(skewmat(4)); // the generic skew-symmetric matrix |
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253 | changevar("R1",A_Z("a",5),R); |
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254 | matrix A=skewmat(6,maxideal(1)^2); |
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255 | print(A); |
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256 | int n=4; |
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257 | ideal i = ideal(randommat(1,n*(n-1)/2,maxideal(1),9)); |
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258 | print(skewmat(n,i)); // skew matrix of generic linear forms |
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259 | kill R1; |
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260 | } |
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261 | //////////////////////////////////////////////////////////////////////////////// |
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262 | |
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263 | proc submat (matrix A, intvec r, intvec c) |
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264 | USAGE: submat(A,r,c); A=matrix, r,c=intvec |
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265 | RETURN: matrix, submatrix of A with rows specified by intvec r and columns |
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266 | specified by intvec c |
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267 | EXAMPLE: example submat; shows an example |
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268 | { |
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269 | matrix B[size(r)][size(c)]=A[r,c]; |
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270 | return(B); |
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271 | } |
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272 | example |
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273 | { "EXAMPLE:"; echo = 2; |
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274 | ring R=32003,(x,y,z),lp; |
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275 | matrix A[4][4]=x,y,z,0,1,2,3,4,5,6,7,8,9,x2,y2,z2; |
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276 | print(A); |
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277 | intvec v=1,3,4; |
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278 | matrix B=submat(A,v,1..3); |
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279 | print(B); |
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280 | } |
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281 | //////////////////////////////////////////////////////////////////////////////// |
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282 | |
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283 | proc symmat (int n, list #) |
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284 | USAGE: symmat(n[,id]); n integer, id ideal |
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285 | RETURN: symmetric nxn matrix, with entries from id (default: id=maxideal(1)) |
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286 | NOTE: if id has less than n*(n+1)/2 elements, the matrix is filled with 0's, |
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287 | symmat(n); creates the generic symmetric matrix |
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288 | EXAMPLE: example symmat; shows an example |
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289 | { |
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290 | matrix B[n][n]; |
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291 | if( size(#)==0 ) { ideal id=maxideal(1); } |
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292 | else { ideal id=#[1]; } |
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293 | id = id,B[1..n,1..n]; |
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294 | int i,j; |
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295 | for( i=0; i<=n-1; i++ ) |
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296 | { |
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297 | B[i+1,i+1..n]=id[j+1..j+n-i]; |
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298 | j=j+n-i; |
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299 | } |
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300 | matrix A=transpose(B); |
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301 | for( i=1; i<=n; i++ ) { A[i,i]=0; } |
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302 | B=A+B; |
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303 | return(B); |
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304 | } |
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305 | example |
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306 | { "EXAMPLE:"; echo = 2; |
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307 | ring R=0,x(1..10),lp; |
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308 | print(symmat(4)); // the generic symmetric matrix |
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309 | changevar("R1",A_Z("a",5),R); |
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310 | matrix A=symmat(5,maxideal(1)^2); |
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311 | print(A); |
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312 | int n=3; |
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313 | ideal i = ideal(randommat(1,n*(n+1)/2,maxideal(1),9)); |
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314 | print(symmat(n,i)); // symmetric matrix of generic linear forms |
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315 | kill R1; |
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316 | } |
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317 | //////////////////////////////////////////////////////////////////////////////// |
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318 | |
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319 | proc tensor (matrix A, matrix B) |
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320 | USAGE: tensor(A,B); A,B matrices |
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321 | RETURN: matrix, tensor product of A and B |
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322 | EXAMPLE: example tensor; shows an example |
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323 | { |
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324 | int i,j; |
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325 | matrix C=B; |
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326 | for( i=2; i<=nrows(A); i++ ) { C=dsum(C,B); } |
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327 | matrix D[nrows(C)][ncols(A)*nrows(B)]; |
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328 | for( j=1; j<=nrows(B); j++ ) |
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329 | { |
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330 | for( i=1; i<=nrows(A); i++ ) |
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331 | { |
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332 | D[(i-1)*nrows(B)+j,(j-1)*ncols(A)+1..j*ncols(A)]=A[i,1..ncols(A)]; |
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333 | } |
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334 | } |
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335 | return(concat(C,D)); |
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336 | } |
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337 | example |
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338 | { "EXAMPLE:"; echo = 2; |
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339 | ring r=32003,(x,y,z),(c,ds); |
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340 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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341 | matrix B[2][2]=x,y,0,z; |
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342 | print(A); |
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343 | print(B); |
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344 | print(tensor(A,B)); |
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345 | } |
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346 | //////////////////////////////////////////////////////////////////////////////// |
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347 | |
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348 | proc unitmat (int n) |
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349 | USAGE: unitmat(n); n integer >= 0 |
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350 | RETURN: nxn unit matrix |
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351 | NOTE: needs a basering, diagonal entries are numbers (=1) in the basering |
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352 | EXAMPLE: example unitmat; shows an example |
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353 | { |
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354 | return(matrix(freemodule(n))); |
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355 | } |
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356 | example |
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357 | { "EXAMPLE:"; echo = 2; |
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358 | ring r=32003,(x,y,z),lp; |
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359 | print(xyz*unitmat(4)); |
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360 | print(unitmat(5)); |
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361 | } |
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362 | /////////////////////////////////////////////////////////////////////////////// |
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363 | |
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364 | proc gauss_col (matrix m) |
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365 | USAGE: gauss_col(A); A=matrix with constant coefficients |
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366 | RETURN: matrix = col-reduced normal form of A |
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367 | EXAMPLE: example gauss_col; shows an example |
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368 | { |
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369 | def R=basering; |
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370 | changeord("@R","ds,c",R); |
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371 | option(redSB); option(nointStrategy); |
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372 | matrix m = imap(R,m); |
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373 | m = matrix(std(m),nrows(m),ncols(m)); |
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374 | setring R; |
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375 | m=imap(@R,m); |
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376 | option(noredSB); |
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377 | kill @R; |
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378 | return(m); |
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379 | } |
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380 | example |
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381 | { "EXAMPLE:"; echo = 2; |
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382 | ring S; |
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383 | matrix M[3][4] = 1,3,2,4,2,6,4,8,1,3,4,4; |
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384 | print(M); |
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385 | print(gauss_col(M)); |
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386 | } |
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387 | /////////////////////////////////////////////////////////////////////////////// |
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388 | |
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389 | proc gauss_row (matrix m) |
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390 | USAGE: gauss_row(A); A=matrix with constant coefficients |
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391 | RETURN: matrix = row-reduced normal form of A |
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392 | EXAMPLE: example gauss_row; shows an example |
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393 | { |
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394 | m = gauss_col(transpose(m)); |
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395 | return(transpose(m)); |
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396 | } |
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397 | example |
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398 | { "EXAMPLE:"; echo = 2; |
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399 | ring S; |
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400 | matrix M[3][4] = 1,3,2,4,2,6,4,8,1,3,4,4; |
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401 | print(M); |
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402 | print(gauss_row(M)); |
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403 | } |
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404 | //////////////////////////////////////////////////////////////////////////////// |
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405 | |
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406 | proc addcol (matrix A, int c1, poly p, int c2) |
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407 | USAGE: addcol(A,c1,p,c2); A matrix, p poly, c1, c2 positive integers |
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408 | RETURN: matrix, A being modified by adding p times column c1 to column c2 |
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409 | EXAMPLE: example addcol; shows an example |
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410 | { |
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411 | A[1..nrows(A),c2]=A[1..nrows(A),c2]+p*A[1..nrows(A),c1]; |
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412 | return(A); |
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413 | } |
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414 | example |
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415 | { "EXAMPLE:"; echo = 2; |
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416 | ring r=32003,(x,y,z),lp; |
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417 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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418 | print(A); |
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419 | print(addcol(A,1,xy,2)); |
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420 | } |
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421 | //////////////////////////////////////////////////////////////////////////////// |
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422 | |
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423 | proc addrow (matrix A, int r1, poly p, int r2) |
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424 | USAGE: addcol(A,r1,p,r2); A matrix, p poly, r1, r2 positive integers |
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425 | RETURN: matrix, A being modified by adding p times row r1 to row r2 |
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426 | EXAMPLE: example addrow; shows an example |
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427 | { |
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428 | A[r2,1..ncols(A)]=A[r2,1..ncols(A)]+p*A[r1,1..ncols(A)]; |
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429 | return(A); |
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430 | } |
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431 | example |
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432 | { "EXAMPLE:"; echo = 2; |
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433 | ring r=32003,(x,y,z),lp; |
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434 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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435 | print(A); |
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436 | print(addrow(A,1,xy,3)); |
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437 | } |
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438 | //////////////////////////////////////////////////////////////////////////////// |
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439 | |
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440 | proc multcol (matrix A, int c, poly p) |
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441 | USAGE: addcol(A,c,p); A matrix, p poly, c positive integer |
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442 | RETURN: matrix, A being modified by multiplying column c with p |
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443 | EXAMPLE: example multcol; shows an example |
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444 | { |
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445 | A[1..nrows(A),c]=p*A[1..nrows(A),c]; |
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446 | return(A); |
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447 | } |
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448 | example |
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449 | { "EXAMPLE:"; echo = 2; |
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450 | ring r=32003,(x,y,z),lp; |
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451 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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452 | print(A); |
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453 | print(multcol(A,2,xy)); |
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454 | } |
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455 | //////////////////////////////////////////////////////////////////////////////// |
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456 | |
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457 | proc multrow (matrix A, int r, poly p) |
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458 | USAGE: addcol(A,r,p); A matrix, p poly, r positive integer |
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459 | RETURN: matrix, A being modified by multiplying row r with p |
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460 | EXAMPLE: example multrow; shows an example |
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461 | { |
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462 | A[r,1..ncols(A)]=p*A[r,1..ncols(A)]; |
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463 | return(A); |
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464 | } |
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465 | example |
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466 | { "EXAMPLE:"; echo = 2; |
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467 | ring r=32003,(x,y,z),lp; |
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468 | matrix A[3][3]=1,2,3,4,5,6,7,8,9; |
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469 | print(A); |
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470 | print(multrow(A,2,xy)); |
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471 | } |
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472 | //////////////////////////////////////////////////////////////////////////////// |
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473 | |
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474 | proc permcol (matrix A, int c1, int c2) |
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475 | USAGE: permcol(A,c1,c2); A matrix, c1,c2 positive integers |
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476 | RETURN: matrix, A being modified by permuting column c1 and c2 |
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477 | EXAMPLE: example permcol; shows an example |
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478 | { |
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479 | matrix B=A; |
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480 | B[1..nrows(B),c1]=A[1..nrows(A),c2]; |
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481 | B[1..nrows(B),c2]=A[1..nrows(A),c1]; |
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482 | return(B); |
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483 | } |
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484 | example |
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485 | { "EXAMPLE:"; echo = 2; |
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486 | ring r=32003,(x,y,z),lp; |
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487 | matrix A[3][3]=1,x,3,4,y,6,7,z,9; |
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488 | print(A); |
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489 | print(permcol(A,2,3)); |
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490 | } |
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491 | //////////////////////////////////////////////////////////////////////////////// |
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492 | |
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493 | proc permrow (matrix A, int r1, int r2) |
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494 | USAGE: permrow(A,r1,r2); A matrix, r1,r2 positive integers |
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495 | RETURN: matrix, A being modified by permuting row r1 and r2 |
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496 | EXAMPLE: example permrow; shows an example |
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497 | { |
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498 | matrix B=A; |
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499 | B[r1,1..ncols(B)]=A[r2,1..ncols(A)]; |
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500 | B[r2,1..ncols(B)]=A[r1,1..ncols(A)]; |
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501 | return(B); |
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502 | } |
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503 | example |
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504 | { "EXAMPLE:"; echo = 2; |
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505 | ring r=32003,(x,y,z),lp; |
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506 | matrix A[3][3]=1,2,3,x,y,z,7,8,9; |
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507 | print(A); |
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508 | print(permrow(A,2,1)); |
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509 | } |
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510 | //////////////////////////////////////////////////////////////////////////////// |
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