1 | /**************************************** |
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2 | * Computer Algebra System SINGULAR * |
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3 | ****************************************/ |
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4 | /* $Id: matpol.cc,v 1.24 1999-06-29 13:03:31 pohl Exp $ */ |
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5 | |
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6 | /* |
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7 | * ABSTRACT: |
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8 | */ |
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9 | |
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10 | #include <stdio.h> |
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11 | #include <limits.h> |
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12 | #include <math.h> |
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13 | |
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14 | #include "mod2.h" |
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15 | #include "tok.h" |
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16 | #include "lists.h" |
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17 | #include "polys.h" |
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18 | #include "mmemory.h" |
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19 | #include "febase.h" |
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20 | #include "numbers.h" |
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21 | #include "ideals.h" |
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22 | #include "ipid.h" |
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23 | #include "subexpr.h" |
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24 | #include "intvec.h" |
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25 | #include "ring.h" |
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26 | #include "sparsmat.h" |
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27 | #include "matpol.h" |
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28 | |
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29 | /*0 implementation*/ |
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30 | |
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31 | typedef int perm[100]; |
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32 | static poly mpDivide(poly a, poly b); |
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33 | static void mpReplace(int j, int n, int &sign, int *perm); |
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34 | static float mpPolyWeight(poly p); |
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35 | static int mpNextperm(perm * z, int max); |
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36 | static poly mpLeibnitz(matrix a); |
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37 | static poly minuscopy (poly p); |
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38 | static poly pInsert(poly p1, poly p2); |
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39 | static poly mpSelect (poly fro, poly what); |
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40 | static poly mpExdiv ( poly m, poly d); |
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41 | |
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42 | /*2 |
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43 | * create a r x c zero-matrix |
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44 | */ |
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45 | #ifdef MDEBUG |
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46 | matrix mpDBNew(int r, int c, char *f, int l) |
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47 | #else |
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48 | matrix mpNew(int r, int c) |
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49 | #endif |
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50 | { |
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51 | if (r<=0) r=1; |
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52 | if ( (((int)(INT_MAX/sizeof(poly))) / r) <= c) |
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53 | { |
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54 | Werror("internal error: creating matrix[%d][%d]",r,c); |
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55 | return NULL; |
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56 | } |
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57 | #ifdef MDEBUG |
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58 | matrix rc = (matrix)mmDBAllocBlock(sizeof(ip_smatrix),f,l); |
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59 | #else |
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60 | matrix rc = (matrix)Alloc(sizeof(ip_smatrix)); |
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61 | #endif |
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62 | rc->nrows = r; |
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63 | rc->ncols = c; |
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64 | rc->rank = r; |
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65 | if (c != 0) |
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66 | { |
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67 | int s=r*c*sizeof(poly); |
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68 | #ifdef MDEBUG |
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69 | rc->m = (polyset)mmDBAllocBlock0(s,f,l); |
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70 | #else |
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71 | rc->m = (polyset)Alloc0(s); |
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72 | #endif |
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73 | //if (rc->m==NULL) |
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74 | //{ |
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75 | // Werror("internal error: creating matrix[%d][%d]",r,c); |
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76 | // return NULL; |
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77 | //} |
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78 | } |
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79 | return rc; |
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80 | } |
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81 | |
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82 | /*2 |
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83 | *copies matrix a to b |
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84 | */ |
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85 | matrix mpCopy (matrix a) |
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86 | { |
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87 | poly t; |
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88 | int i, m=MATROWS(a), n=MATCOLS(a); |
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89 | matrix b = mpNew(m, n); |
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90 | |
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91 | for (i=m*n-1; i>=0; i--) |
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92 | { |
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93 | t = a->m[i]; |
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94 | pNormalize(t); |
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95 | b->m[i] = pCopy(t); |
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96 | } |
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97 | b->rank=a->rank; |
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98 | return b; |
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99 | } |
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100 | |
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101 | /*2 |
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102 | * make it a p * unit matrix |
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103 | */ |
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104 | matrix mpInitP(int r, int c, poly p) |
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105 | { |
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106 | matrix rc = mpNew(r,c); |
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107 | int i=min(r,c), n = c*(i-1)+i-1, inc = c+1; |
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108 | |
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109 | pNormalize(p); |
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110 | while (n>0) |
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111 | { |
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112 | rc->m[n] = pCopy(p); |
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113 | n -= inc; |
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114 | } |
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115 | rc->m[0]=p; |
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116 | return rc; |
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117 | } |
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118 | |
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119 | /*2 |
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120 | * make it a v * unit matrix |
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121 | */ |
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122 | matrix mpInitI(int r, int c, int v) |
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123 | { |
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124 | return mpInitP(r,c,pISet(v)); |
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125 | } |
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126 | |
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127 | /*2 |
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128 | * c = f*a |
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129 | */ |
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130 | matrix mpMultI(matrix a, int f) |
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131 | { |
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132 | int k, n = a->nrows, m = a->ncols; |
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133 | poly p = pISet(f); |
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134 | matrix c = mpNew(n,m); |
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135 | |
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136 | for (k=m*n-1; k>0; k--) |
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137 | c->m[k] = pMult(pCopy(a->m[k]), pCopy(p)); |
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138 | c->m[0] = pMult(pCopy(a->m[0]), p); |
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139 | return c; |
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140 | } |
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141 | |
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142 | /*2 |
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143 | * multiply a matrix 'a' by a poly 'p', destroy the args |
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144 | */ |
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145 | matrix mpMultP(matrix a, poly p) |
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146 | { |
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147 | int k, n = a->nrows, m = a->ncols; |
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148 | |
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149 | pNormalize(p); |
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150 | for (k=m*n-1; k>0; k--) |
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151 | a->m[k] = pMult(a->m[k], pCopy(p)); |
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152 | a->m[0] = pMult(a->m[0], p); |
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153 | return a; |
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154 | } |
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155 | |
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156 | matrix mpAdd(matrix a, matrix b) |
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157 | { |
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158 | int k, n = a->nrows, m = a->ncols; |
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159 | if ((n != b->nrows) || (m != b->ncols)) |
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160 | { |
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161 | /* |
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162 | * Werror("cannot add %dx%d matrix and %dx%d matrix", |
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163 | * m,n,b->cols(),b->rows()); |
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164 | */ |
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165 | return NULL; |
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166 | } |
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167 | matrix c = mpNew(n,m); |
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168 | for (k=m*n-1; k>=0; k--) |
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169 | c->m[k] = pAdd(pCopy(a->m[k]), pCopy(b->m[k])); |
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170 | return c; |
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171 | } |
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172 | |
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173 | matrix mpSub(matrix a, matrix b) |
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174 | { |
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175 | int k, n = a->nrows, m = a->ncols; |
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176 | if ((n != b->nrows) || (m != b->ncols)) |
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177 | { |
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178 | /* |
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179 | * Werror("cannot sub %dx%d matrix and %dx%d matrix", |
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180 | * m,n,b->cols(),b->rows()); |
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181 | */ |
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182 | return NULL; |
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183 | } |
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184 | matrix c = mpNew(n,m); |
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185 | for (k=m*n-1; k>=0; k--) |
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186 | c->m[k] = pSub(pCopy(a->m[k]), pCopy(b->m[k])); |
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187 | return c; |
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188 | } |
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189 | |
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190 | matrix mpMult(matrix a, matrix b) |
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191 | { |
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192 | int i, j, k; |
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193 | poly s, t, aik, bkj; |
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194 | int m = MATROWS(a); |
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195 | int p = MATCOLS(a); |
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196 | int q = MATCOLS(b); |
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197 | |
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198 | if (p!=MATROWS(b)) |
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199 | { |
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200 | /* |
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201 | * Werror("cannot multiply %dx%d matrix and %dx%d matrix", |
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202 | * m,p,b->rows(),q); |
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203 | */ |
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204 | return NULL; |
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205 | } |
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206 | matrix c = mpNew(m,q); |
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207 | |
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208 | for (i=1; i<=m; i++) |
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209 | { |
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210 | for (j=1; j<=q; j++) |
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211 | { |
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212 | t = NULL; |
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213 | for (k=1; k<=p; k++) |
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214 | { |
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215 | aik = pCopy(MATELEM(a,i,k)); |
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216 | bkj = pCopy(MATELEM(b,k,j)); |
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217 | s = pMult(aik,bkj); |
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218 | t = pAdd(t,s); |
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219 | } |
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220 | pNormalize(t); |
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221 | MATELEM(c,i,j) = t; |
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222 | } |
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223 | } |
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224 | return c; |
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225 | } |
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226 | |
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227 | matrix mpTransp(matrix a) |
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228 | { |
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229 | int i, j, r = MATROWS(a), c = MATCOLS(a); |
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230 | poly *p; |
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231 | matrix b = mpNew(c,r); |
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232 | |
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233 | p = b->m; |
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234 | for (i=0; i<c; i++) |
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235 | { |
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236 | for (j=0; j<r; j++) |
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237 | { |
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238 | *p++ = pCopy(a->m[j*c+i]); |
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239 | } |
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240 | } |
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241 | return b; |
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242 | } |
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243 | |
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244 | /*2 |
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245 | *returns the trace of matrix a |
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246 | */ |
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247 | poly mpTrace ( matrix a) |
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248 | { |
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249 | int i; |
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250 | int n = (MATCOLS(a)<MATROWS(a)) ? MATCOLS(a) : MATROWS(a); |
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251 | poly t = NULL; |
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252 | |
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253 | for (i=1; i<=n; i++) |
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254 | t = pAdd(t, pCopy(MATELEM(a,i,i))); |
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255 | return t; |
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256 | } |
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257 | |
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258 | /*2 |
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259 | *returns the trace of the product of a and b |
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260 | */ |
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261 | poly TraceOfProd ( matrix a, matrix b, int n) |
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262 | { |
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263 | int i, j; |
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264 | poly p, t = NULL; |
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265 | |
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266 | for (i=1; i<=n; i++) |
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267 | { |
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268 | for (j=1; j<=n; j++) |
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269 | { |
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270 | p = pMult(pCopy(MATELEM(a,i,j)), pCopy(MATELEM(b,j,i))); |
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271 | t = pAdd(t, p); |
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272 | } |
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273 | } |
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274 | return t; |
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275 | } |
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276 | |
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277 | /* |
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278 | * C++ classes for Bareiss algorithm |
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279 | */ |
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280 | class row_col_weight |
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281 | { |
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282 | private: |
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283 | int ym, yn; |
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284 | public: |
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285 | float *wrow, *wcol; |
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286 | row_col_weight() : ym(0) {} |
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287 | row_col_weight(int, int); |
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288 | ~row_col_weight(); |
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289 | }; |
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290 | |
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291 | /*2 |
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292 | * a submatrix M of a matrix X[m,n]: |
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293 | * 0 <= i < s_m <= a_m |
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294 | * 0 <= j < s_n <= a_n |
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295 | * M = ( Xarray[qrow[i],qcol[j]] ) |
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296 | * if a_m = a_n and s_m = s_n |
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297 | * det(X) = sign*div^(s_m-1)*det(M) |
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298 | * resticted pivot for elimination |
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299 | * 0 <= j < piv_s |
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300 | */ |
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301 | class mp_permmatrix |
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302 | { |
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303 | private: |
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304 | int a_m, a_n, s_m, s_n, sign, piv_s; |
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305 | int *qrow, *qcol; |
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306 | poly *Xarray; |
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307 | void mpInitMat(); |
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308 | poly * mpRowAdr(int); |
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309 | poly * mpColAdr(int); |
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310 | void mpRowWeight(float *); |
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311 | void mpColWeight(float *); |
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312 | void mpRowSwap(int, int); |
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313 | void mpColSwap(int, int); |
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314 | public: |
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315 | mp_permmatrix() : a_m(0) {} |
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316 | mp_permmatrix(matrix); |
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317 | mp_permmatrix(mp_permmatrix *); |
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318 | ~mp_permmatrix(); |
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319 | int mpGetRow(); |
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320 | int mpGetCol(); |
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321 | int mpGetRdim(); |
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322 | int mpGetCdim(); |
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323 | int mpGetSign(); |
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324 | void mpSetSearch(int s); |
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325 | void mpSaveArray(); |
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326 | poly mpGetElem(int, int); |
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327 | void mpSetElem(poly, int, int); |
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328 | void mpDelElem(int, int); |
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329 | void mpElimBareiss(poly); |
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330 | int mpPivotBareiss(row_col_weight *); |
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331 | int mpPivotRow(row_col_weight *, int); |
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332 | void mpToIntvec(intvec *); |
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333 | void mpRowReorder(); |
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334 | void mpColReorder(); |
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335 | }; |
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336 | |
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337 | /*2 |
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338 | * caller of 'Bareiss' algorithm, |
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339 | * return an list of a matrix and an intvec: |
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340 | * the matrix is lower triangular and the result, |
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341 | * the intvec is the performed permutation of columns. |
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342 | */ |
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343 | lists mpBareiss (matrix a, BOOLEAN sw) |
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344 | { |
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345 | poly div; |
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346 | matrix c = mpCopy(a); |
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347 | mp_permmatrix *Bareiss = new mp_permmatrix(c); |
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348 | row_col_weight w(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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349 | intvec *v = new intvec(Bareiss->mpGetCdim()); |
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350 | lists res=(lists)Alloc(sizeof(slists)); |
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351 | |
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352 | if (sw) WarnS(feNotImplemented); |
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353 | /* Bareiss */ |
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354 | div = NULL; |
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355 | while(Bareiss->mpPivotBareiss(&w)) |
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356 | { |
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357 | Bareiss->mpElimBareiss(div); |
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358 | div = Bareiss->mpGetElem(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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359 | } |
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360 | Bareiss->mpToIntvec(v); |
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361 | Bareiss->mpRowReorder(); |
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362 | Bareiss->mpColReorder(); |
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363 | Bareiss->mpSaveArray(); |
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364 | delete Bareiss; |
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365 | |
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366 | res->Init(2); |
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367 | res->m[0].rtyp=MATRIX_CMD; |
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368 | res->m[0].data=(void *)c; |
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369 | res->m[1].rtyp=INTVEC_CMD; |
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370 | res->m[1].data=(void *)v; |
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371 | return res; |
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372 | } |
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373 | |
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374 | /*2 |
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375 | * one step of 'Bareiss' algorithm |
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376 | * for application in minor |
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377 | * assume to have a full matrix |
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378 | * if *H!=0, then divide by *H (the pivot from the step before) |
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379 | * returns the choosen pivot *H=m[*r,*c] |
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380 | * the result has the pivot at the right lower corner |
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381 | */ |
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382 | matrix mpOneStepBareiss (matrix a, poly *H, int *r, int *c) |
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383 | { |
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384 | poly div=*H; |
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385 | matrix re = mpCopy(a); |
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386 | mp_permmatrix *Bareiss = new mp_permmatrix(re); |
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387 | row_col_weight w(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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388 | int row = *r; |
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389 | |
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390 | /* step of Bareiss */ |
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391 | if(((row!=0) && Bareiss->mpPivotRow(&w,row-1)) || Bareiss->mpPivotBareiss(&w)) |
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392 | { |
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393 | Bareiss->mpElimBareiss(div); |
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394 | div = Bareiss->mpGetElem(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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395 | pDelete(H); |
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396 | *H = pCopy(div); |
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397 | *c = Bareiss->mpGetCol()+1; |
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398 | *r = Bareiss->mpGetRow()+1; |
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399 | Bareiss->mpRowReorder(); |
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400 | Bareiss->mpColReorder(); |
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401 | } |
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402 | else |
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403 | { |
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404 | pDelete(H); |
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405 | *H = NULL; |
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406 | *c = *r = 0; |
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407 | } |
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408 | Bareiss->mpSaveArray(); |
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409 | idTest((ideal)re); |
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410 | delete Bareiss; |
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411 | return re; |
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412 | } |
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413 | |
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414 | /*2 |
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415 | *returns the determinant of the matrix m; |
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416 | *uses Bareiss algorithm |
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417 | */ |
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418 | poly mpDetBareiss (matrix a) |
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419 | { |
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420 | int s; |
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421 | poly div, res; |
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422 | if (MATROWS(a) != MATCOLS(a)) |
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423 | { |
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424 | Werror("det of %d x %d matrix",MATROWS(a),MATCOLS(a)); |
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425 | return NULL; |
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426 | } |
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427 | matrix c = mpCopy(a); |
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428 | mp_permmatrix *Bareiss = new mp_permmatrix(c); |
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429 | row_col_weight w(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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430 | |
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431 | /* Bareiss */ |
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432 | div = NULL; |
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433 | while(Bareiss->mpPivotBareiss(&w)) |
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434 | { |
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435 | Bareiss->mpElimBareiss(div); |
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436 | div = Bareiss->mpGetElem(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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437 | } |
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438 | Bareiss->mpRowReorder(); |
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439 | Bareiss->mpColReorder(); |
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440 | Bareiss->mpSaveArray(); |
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441 | s = Bareiss->mpGetSign(); |
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442 | delete Bareiss; |
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443 | |
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444 | /* result */ |
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445 | res = MATELEM(c,1,1); |
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446 | MATELEM(c,1,1) = NULL; |
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447 | idDelete((ideal *)&c); |
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448 | if (s < 0) |
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449 | res = pNeg(res); |
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450 | return res; |
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451 | } |
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452 | |
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453 | /*2 |
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454 | *returns the determinant of the matrix m; |
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455 | *uses Newtons formulea for symmetric functions |
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456 | */ |
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457 | poly mpDet (matrix m) |
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458 | { |
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459 | int i,j,k,n; |
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460 | poly p,q; |
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461 | matrix a, s; |
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462 | matrix ma[100]; |
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463 | number c=NULL, d=NULL, ONE=NULL; |
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464 | |
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465 | n = MATROWS(m); |
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466 | if (n != MATCOLS(m)) |
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467 | { |
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468 | Werror("det of %d x %d matrix",n,MATCOLS(m)); |
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469 | return NULL; |
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470 | } |
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471 | k=rChar(); |
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472 | if (((k > 0) && (k <= n)) |
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473 | #ifdef SRING |
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474 | || (pSRING) |
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475 | #endif |
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476 | ) |
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477 | return mpLeibnitz(m); |
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478 | ONE = nInit(1); |
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479 | ma[1]=mpCopy(m); |
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480 | k = (n+1) / 2; |
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481 | s = mpNew(1, n); |
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482 | MATELEM(s,1,1) = mpTrace(m); |
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483 | for (i=2; i<=k; i++) |
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484 | { |
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485 | //ma[i] = mpNew(n,n); |
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486 | ma[i]=mpMult(ma[i-1], ma[1]); |
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487 | MATELEM(s,1,i) = mpTrace(ma[i]); |
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488 | pTest(MATELEM(s,1,i)); |
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489 | } |
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490 | for (i=k+1; i<=n; i++) |
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491 | { |
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492 | MATELEM(s,1,i) = TraceOfProd(ma[i / 2], ma[(i+1) / 2], n); |
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493 | pTest(MATELEM(s,1,i)); |
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494 | } |
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495 | for (i=1; i<=k; i++) |
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496 | idDelete((ideal *)&(ma[i])); |
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497 | /* the array s contains the traces of the powers of the matrix m, |
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498 | * these are the power sums of the eigenvalues of m */ |
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499 | a = mpNew(1,n); |
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500 | MATELEM(a,1,1) = minuscopy(MATELEM(s,1,1)); |
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501 | for (i=2; i<=n; i++) |
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502 | { |
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503 | p = pCopy(MATELEM(s,1,i)); |
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504 | for (j=i-1; j>=1; j--) |
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505 | { |
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506 | q = pMult(pCopy(MATELEM(s,1,j)), pCopy(MATELEM(a,1,i-j))); |
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507 | pTest(q); |
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508 | p = pAdd(p,q); |
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509 | } |
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510 | // c= -1/i |
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511 | d = nInit(-(int)i); |
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512 | c = nDiv(ONE, d); |
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513 | nDelete(&d); |
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514 | |
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515 | pMultN(p, c); |
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516 | pTest(p); |
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517 | MATELEM(a,1,i) = p; |
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518 | nDelete(&c); |
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519 | } |
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520 | /* the array a contains the elementary symmetric functions of the |
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521 | * eigenvalues of m */ |
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522 | for (i=1; i<=n-1; i++) |
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523 | { |
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524 | //pDelete(&(MATELEM(a,1,i))); |
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525 | pDelete(&(MATELEM(s,1,i))); |
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526 | } |
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527 | pDelete(&(MATELEM(s,1,n))); |
---|
528 | /* up to a sign, the determinant is the n-th elementary symmetric function */ |
---|
529 | if ((n/2)*2 < n) |
---|
530 | { |
---|
531 | d = nInit(-1); |
---|
532 | pMultN(MATELEM(a,1,n), d); |
---|
533 | nDelete(&d); |
---|
534 | } |
---|
535 | nDelete(&ONE); |
---|
536 | idDelete((ideal *)&s); |
---|
537 | poly result=MATELEM(a,1,n); |
---|
538 | MATELEM(a,1,n)=NULL; |
---|
539 | idDelete((ideal *)&a); |
---|
540 | return result; |
---|
541 | } |
---|
542 | |
---|
543 | /*2 |
---|
544 | * compute all ar-minors of the matrix a |
---|
545 | */ |
---|
546 | matrix mpWedge(matrix a, int ar) |
---|
547 | { |
---|
548 | int i,j,k,l; |
---|
549 | int *rowchoise,*colchoise; |
---|
550 | BOOLEAN rowch,colch; |
---|
551 | matrix result; |
---|
552 | matrix tmp; |
---|
553 | poly p; |
---|
554 | |
---|
555 | i = binom(a->nrows,ar); |
---|
556 | j = binom(a->ncols,ar); |
---|
557 | |
---|
558 | rowchoise=(int *)Alloc(ar*sizeof(int)); |
---|
559 | colchoise=(int *)Alloc(ar*sizeof(int)); |
---|
560 | result =mpNew(i,j); |
---|
561 | tmp=mpNew(ar,ar); |
---|
562 | l = 1; /* k,l:the index in result*/ |
---|
563 | idInitChoise(ar,1,a->nrows,&rowch,rowchoise); |
---|
564 | while (!rowch) |
---|
565 | { |
---|
566 | k=1; |
---|
567 | idInitChoise(ar,1,a->ncols,&colch,colchoise); |
---|
568 | while (!colch) |
---|
569 | { |
---|
570 | for (i=1; i<=ar; i++) |
---|
571 | { |
---|
572 | for (j=1; j<=ar; j++) |
---|
573 | { |
---|
574 | MATELEM(tmp,i,j) = MATELEM(a,rowchoise[i-1],colchoise[j-1]); |
---|
575 | } |
---|
576 | } |
---|
577 | p = mpDetBareiss(tmp); |
---|
578 | if ((k+l) & 1) p=pNeg(p); |
---|
579 | MATELEM(result,l,k) = p; |
---|
580 | k++; |
---|
581 | idGetNextChoise(ar,a->ncols,&colch,colchoise); |
---|
582 | } |
---|
583 | idGetNextChoise(ar,a->nrows,&rowch,rowchoise); |
---|
584 | l++; |
---|
585 | } |
---|
586 | /*delete the matrix tmp*/ |
---|
587 | for (i=1; i<=ar; i++) |
---|
588 | { |
---|
589 | for (j=1; j<=ar; j++) MATELEM(tmp,i,j) = NULL; |
---|
590 | } |
---|
591 | idDelete((ideal *) &tmp); |
---|
592 | return (result); |
---|
593 | } |
---|
594 | |
---|
595 | /*2 |
---|
596 | * compute the jacobi matrix of an ideal |
---|
597 | */ |
---|
598 | BOOLEAN mpJacobi(leftv res,leftv a) |
---|
599 | { |
---|
600 | int i,j; |
---|
601 | matrix result; |
---|
602 | ideal id=(ideal)a->Data(); |
---|
603 | |
---|
604 | result =mpNew(IDELEMS(id),pVariables); |
---|
605 | for (i=1; i<=IDELEMS(id); i++) |
---|
606 | { |
---|
607 | for (j=1; j<=pVariables; j++) |
---|
608 | { |
---|
609 | MATELEM(result,i,j) = pDiff(id->m[i-1],j); |
---|
610 | } |
---|
611 | } |
---|
612 | res->data=(char *)result; |
---|
613 | return FALSE; |
---|
614 | } |
---|
615 | |
---|
616 | /*2 |
---|
617 | * returns the Koszul-matrix of degree d of a vectorspace with dimension n |
---|
618 | * uses the first n entrees of id, if id <> NULL |
---|
619 | */ |
---|
620 | BOOLEAN mpKoszul(leftv res,leftv b/*in*/, leftv c/*ip*/,leftv id) |
---|
621 | { |
---|
622 | int n=(int)b->Data(); |
---|
623 | int d=(int)c->Data(); |
---|
624 | int k,l,sign,row,col; |
---|
625 | matrix result; |
---|
626 | ideal temp; |
---|
627 | BOOLEAN bo; |
---|
628 | poly p; |
---|
629 | |
---|
630 | if ((d>n) || (d<1) || (n<1)) |
---|
631 | { |
---|
632 | res->data=(char *)mpNew(1,1); |
---|
633 | return FALSE; |
---|
634 | } |
---|
635 | int *choise = (int*)Alloc(d*sizeof(int)); |
---|
636 | if (id==NULL) |
---|
637 | temp=idMaxIdeal(1); |
---|
638 | else |
---|
639 | temp=(ideal)id->Data(); |
---|
640 | |
---|
641 | k = binom(n,d); |
---|
642 | l = k*d; |
---|
643 | l /= n-d+1; |
---|
644 | result =mpNew(l,k); |
---|
645 | col = 1; |
---|
646 | idInitChoise(d,1,n,&bo,choise); |
---|
647 | while (!bo) |
---|
648 | { |
---|
649 | sign = 1; |
---|
650 | for (l=1;l<=d;l++) |
---|
651 | { |
---|
652 | if (choise[l-1]<=IDELEMS(temp)) |
---|
653 | { |
---|
654 | p = pCopy(temp->m[choise[l-1]-1]); |
---|
655 | if (sign == -1) p = pNeg(p); |
---|
656 | sign *= -1; |
---|
657 | row = idGetNumberOfChoise(l-1,d,1,n,choise); |
---|
658 | MATELEM(result,row,col) = p; |
---|
659 | } |
---|
660 | } |
---|
661 | col++; |
---|
662 | idGetNextChoise(d,n,&bo,choise); |
---|
663 | } |
---|
664 | if (id==NULL) idDelete(&temp); |
---|
665 | |
---|
666 | res->data=(char *)result; |
---|
667 | return FALSE; |
---|
668 | } |
---|
669 | |
---|
670 | ///*2 |
---|
671 | //*homogenize all elements of matrix (not the matrix itself) |
---|
672 | //*/ |
---|
673 | //matrix mpHomogen(matrix a, int v) |
---|
674 | //{ |
---|
675 | // int i,j; |
---|
676 | // poly p; |
---|
677 | // |
---|
678 | // for (i=1;i<=MATROWS(a);i++) |
---|
679 | // { |
---|
680 | // for (j=1;j<=MATCOLS(a);j++) |
---|
681 | // { |
---|
682 | // p=pHomogen(MATELEM(a,i,j),v); |
---|
683 | // pDelete(&(MATELEM(a,i,j))); |
---|
684 | // MATELEM(a,i,j)=p; |
---|
685 | // } |
---|
686 | // } |
---|
687 | // return a; |
---|
688 | //} |
---|
689 | |
---|
690 | /*2 |
---|
691 | * corresponds to Maple's coeffs: |
---|
692 | * var has to be the number of a variable |
---|
693 | */ |
---|
694 | matrix mpCoeffs (ideal I, int var) |
---|
695 | { |
---|
696 | poly h,f; |
---|
697 | int l, i, c, m=0; |
---|
698 | matrix co; |
---|
699 | /* look for maximal power m of x_var in I */ |
---|
700 | for (i=IDELEMS(I)-1; i>=0; i--) |
---|
701 | { |
---|
702 | f=I->m[i]; |
---|
703 | while (f!=NULL) |
---|
704 | { |
---|
705 | l=pGetExp(f,var); |
---|
706 | if (l>m) m=l; |
---|
707 | pIter(f); |
---|
708 | } |
---|
709 | } |
---|
710 | co=mpNew((m+1)*I->rank,IDELEMS(I)); |
---|
711 | /* divide each monomial by a power of x_var, |
---|
712 | * remember the power in l and the component in c*/ |
---|
713 | for (i=IDELEMS(I)-1; i>=0; i--) |
---|
714 | { |
---|
715 | f=I->m[i]; |
---|
716 | while (f!=NULL) |
---|
717 | { |
---|
718 | l=pGetExp(f,var); |
---|
719 | pSetExp(f,var,0); |
---|
720 | c=max(pGetComp(f),1); |
---|
721 | pSetComp(f,0); |
---|
722 | pSetm(f); |
---|
723 | /* now add the resulting monomial to co*/ |
---|
724 | h=pNext(f); |
---|
725 | pNext(f)=NULL; |
---|
726 | //MATELEM(co,c*(m+1)-l,i+1) |
---|
727 | // =pAdd(MATELEM(co,c*(m+1)-l,i+1),f); |
---|
728 | MATELEM(co,(c-1)*(m+1)+l+1,i+1) |
---|
729 | =pAdd(MATELEM(co,(c-1)*(m+1)+l+1,i+1),f); |
---|
730 | /* iterate f*/ |
---|
731 | f=h; |
---|
732 | } |
---|
733 | } |
---|
734 | return co; |
---|
735 | } |
---|
736 | |
---|
737 | /*2 |
---|
738 | * given the result c of mpCoeffs(ideal/module i, var) |
---|
739 | * i of rank r |
---|
740 | * build the matrix of the corresponding monomials in m |
---|
741 | */ |
---|
742 | void mpMonomials(matrix c, int r, int var, matrix m) |
---|
743 | { |
---|
744 | /* clear contents of m*/ |
---|
745 | int k,l; |
---|
746 | for (k=MATROWS(m);k>0;k--) |
---|
747 | { |
---|
748 | for(l=MATCOLS(m);l>0;l--) |
---|
749 | { |
---|
750 | pDelete(&MATELEM(m,k,l)); |
---|
751 | } |
---|
752 | } |
---|
753 | Free((ADDRESS)m->m,MATROWS(m)*MATCOLS(m)*sizeof(poly)); |
---|
754 | /* allocate monoms in the right size r x MATROWS(c)*/ |
---|
755 | m->m=(polyset)Alloc0(r*MATROWS(c)*sizeof(poly)); |
---|
756 | MATROWS(m)=r; |
---|
757 | MATCOLS(m)=MATROWS(c); |
---|
758 | m->rank=r; |
---|
759 | /* the maximal power p of x_var: MATCOLS(m)=r*(p+1) */ |
---|
760 | int p=MATCOLS(m)/r-1; |
---|
761 | /* fill in the powers of x_var=h*/ |
---|
762 | poly h=pOne(); |
---|
763 | for(k=r;k>0; k--) |
---|
764 | { |
---|
765 | MATELEM(m,k,k*(p+1))=pOne(); |
---|
766 | } |
---|
767 | for(l=p;l>0; l--) |
---|
768 | { |
---|
769 | pSetExp(h,var,l); |
---|
770 | pSetm(h); |
---|
771 | for(k=r;k>0; k--) |
---|
772 | { |
---|
773 | MATELEM(m,k,k*(p+1)-l)=pCopy(h); |
---|
774 | } |
---|
775 | } |
---|
776 | pDelete(&h); |
---|
777 | } |
---|
778 | |
---|
779 | matrix mpCoeffProc (poly f, poly vars) |
---|
780 | { |
---|
781 | poly sel, h; |
---|
782 | int l, i; |
---|
783 | int pos_of_1 = -1; |
---|
784 | matrix co; |
---|
785 | |
---|
786 | if (f==NULL) |
---|
787 | { |
---|
788 | co = mpNew(2, 1); |
---|
789 | MATELEM(co,1,1) = pOne(); |
---|
790 | MATELEM(co,2,1) = NULL; |
---|
791 | return co; |
---|
792 | } |
---|
793 | sel = mpSelect(f, vars); |
---|
794 | l = pLength(sel); |
---|
795 | co = mpNew(2, l); |
---|
796 | if (pOrdSgn==-1) |
---|
797 | { |
---|
798 | for (i=l; i>=1; i--) |
---|
799 | { |
---|
800 | h = sel; |
---|
801 | pIter(sel); |
---|
802 | pNext(h)=NULL; |
---|
803 | MATELEM(co,1,i) = h; |
---|
804 | MATELEM(co,2,i) = NULL; |
---|
805 | if (pIsConstant(h)) pos_of_1 = i; |
---|
806 | } |
---|
807 | } |
---|
808 | else |
---|
809 | { |
---|
810 | for (i=1; i<=l; i++) |
---|
811 | { |
---|
812 | h = sel; |
---|
813 | pIter(sel); |
---|
814 | pNext(h)=NULL; |
---|
815 | MATELEM(co,1,i) = h; |
---|
816 | MATELEM(co,2,i) = NULL; |
---|
817 | if (pIsConstant(h)) pos_of_1 = i; |
---|
818 | } |
---|
819 | } |
---|
820 | while (f!=NULL) |
---|
821 | { |
---|
822 | i = 1; |
---|
823 | loop |
---|
824 | { |
---|
825 | if (i!=pos_of_1) |
---|
826 | { |
---|
827 | h = mpExdiv(f, MATELEM(co,1,i)); |
---|
828 | if (h!=NULL) |
---|
829 | { |
---|
830 | MATELEM(co,2,i) = pAdd(MATELEM(co,2,i), h); |
---|
831 | break; |
---|
832 | } |
---|
833 | } |
---|
834 | if (i == l) |
---|
835 | { |
---|
836 | // check monom 1 last: |
---|
837 | if (pos_of_1 != -1) |
---|
838 | { |
---|
839 | h = mpExdiv(f, MATELEM(co,1,pos_of_1)); |
---|
840 | if (h!=NULL) |
---|
841 | { |
---|
842 | MATELEM(co,2,pos_of_1) = pAdd(MATELEM(co,2,pos_of_1), h); |
---|
843 | } |
---|
844 | } |
---|
845 | break; |
---|
846 | } |
---|
847 | i ++; |
---|
848 | } |
---|
849 | pIter(f); |
---|
850 | } |
---|
851 | return co; |
---|
852 | } |
---|
853 | |
---|
854 | void mpCoef2(poly v, poly mon, matrix *c, matrix *m) |
---|
855 | { |
---|
856 | polyset s; |
---|
857 | poly p; |
---|
858 | int sl,i,j; |
---|
859 | int l=0; |
---|
860 | poly sel=mpSelect(v,mon); |
---|
861 | |
---|
862 | pVec2Polys(sel,&s,&sl); |
---|
863 | for (i=0; i<sl; i++) |
---|
864 | l=max(l,pLength(s[i])); |
---|
865 | *c=mpNew(sl,l); |
---|
866 | *m=mpNew(sl,l); |
---|
867 | poly h; |
---|
868 | int isConst; |
---|
869 | for (j=1; j<=sl;j++) |
---|
870 | { |
---|
871 | p=s[j-1]; |
---|
872 | if (pIsConstant(p)) /*p != NULL */ |
---|
873 | { |
---|
874 | isConst=-1; |
---|
875 | i=l; |
---|
876 | } |
---|
877 | else |
---|
878 | { |
---|
879 | isConst=1; |
---|
880 | i=1; |
---|
881 | } |
---|
882 | while(p!=NULL) |
---|
883 | { |
---|
884 | h = pHead(p); |
---|
885 | MATELEM(*m,j,i) = h; |
---|
886 | i+=isConst; |
---|
887 | p = p->next; |
---|
888 | } |
---|
889 | } |
---|
890 | while (v!=NULL) |
---|
891 | { |
---|
892 | i = 1; |
---|
893 | j = pGetComp(v); |
---|
894 | loop |
---|
895 | { |
---|
896 | poly mp=MATELEM(*m,j,i); |
---|
897 | if (mp!=NULL) |
---|
898 | { |
---|
899 | h = mpExdiv(v, mp /*MATELEM(*m,j,i)*/); |
---|
900 | if (h!=NULL) |
---|
901 | { |
---|
902 | pSetComp(h,0); |
---|
903 | MATELEM(*c,j,i) = pAdd(MATELEM(*c,j,i), h); |
---|
904 | break; |
---|
905 | } |
---|
906 | } |
---|
907 | if (i < l) |
---|
908 | i++; |
---|
909 | else |
---|
910 | break; |
---|
911 | } |
---|
912 | v = v->next; |
---|
913 | } |
---|
914 | } |
---|
915 | |
---|
916 | |
---|
917 | BOOLEAN mpEqual(matrix a, matrix b) |
---|
918 | { |
---|
919 | if ((MATCOLS(a)!=MATCOLS(b)) || (MATROWS(a)!=MATROWS(b))) |
---|
920 | return FALSE; |
---|
921 | int i=MATCOLS(a)*MATROWS(b)-1; |
---|
922 | while (i>=0) |
---|
923 | { |
---|
924 | if (a->m[i]==NULL) |
---|
925 | { |
---|
926 | if (b->m[i]!=NULL) return FALSE; |
---|
927 | } |
---|
928 | else |
---|
929 | if (pComp(a->m[i],b->m[i])!=0) return FALSE; |
---|
930 | i--; |
---|
931 | } |
---|
932 | i=MATCOLS(a)*MATROWS(b)-1; |
---|
933 | while (i>=0) |
---|
934 | { |
---|
935 | if (pSub(pCopy(a->m[i]),pCopy(b->m[i]))!=NULL) return FALSE; |
---|
936 | i--; |
---|
937 | } |
---|
938 | return TRUE; |
---|
939 | } |
---|
940 | |
---|
941 | /* --------------- internal stuff ------------------- */ |
---|
942 | |
---|
943 | row_col_weight::row_col_weight(int i, int j) |
---|
944 | { |
---|
945 | ym = i; |
---|
946 | yn = j; |
---|
947 | wrow = (float *)Alloc(i*sizeof(float)); |
---|
948 | wcol = (float *)Alloc(j*sizeof(float)); |
---|
949 | } |
---|
950 | |
---|
951 | row_col_weight::~row_col_weight() |
---|
952 | { |
---|
953 | if (ym!=0) |
---|
954 | { |
---|
955 | Free((ADDRESS)wcol, yn*sizeof(float)); |
---|
956 | Free((ADDRESS)wrow, ym*sizeof(float)); |
---|
957 | } |
---|
958 | } |
---|
959 | |
---|
960 | mp_permmatrix::mp_permmatrix(matrix A) : sign(1) |
---|
961 | { |
---|
962 | a_m = A->nrows; |
---|
963 | a_n = A->ncols; |
---|
964 | this->mpInitMat(); |
---|
965 | Xarray = A->m; |
---|
966 | } |
---|
967 | |
---|
968 | mp_permmatrix::mp_permmatrix(mp_permmatrix *M) |
---|
969 | { |
---|
970 | poly p, *athis, *aM; |
---|
971 | int i, j; |
---|
972 | |
---|
973 | a_m = M->s_m; |
---|
974 | a_n = M->s_n; |
---|
975 | sign = M->sign; |
---|
976 | this->mpInitMat(); |
---|
977 | Xarray = (poly *)Alloc0(a_m*a_n*sizeof(poly)); |
---|
978 | for (i=a_m-1; i>=0; i--) |
---|
979 | { |
---|
980 | athis = this->mpRowAdr(i); |
---|
981 | aM = M->mpRowAdr(i); |
---|
982 | for (j=a_n-1; j>=0; j--) |
---|
983 | { |
---|
984 | p = aM[M->qcol[j]]; |
---|
985 | if (p) |
---|
986 | { |
---|
987 | athis[j] = pCopy(p); |
---|
988 | } |
---|
989 | } |
---|
990 | } |
---|
991 | } |
---|
992 | |
---|
993 | mp_permmatrix::~mp_permmatrix() |
---|
994 | { |
---|
995 | int k; |
---|
996 | |
---|
997 | if (a_m != 0) |
---|
998 | { |
---|
999 | Free((ADDRESS)qrow,a_m*sizeof(int)); |
---|
1000 | Free((ADDRESS)qcol,a_n*sizeof(int)); |
---|
1001 | if (Xarray != NULL) |
---|
1002 | { |
---|
1003 | for (k=a_m*a_n-1; k>=0; k--) |
---|
1004 | pDelete(&Xarray[k]); |
---|
1005 | Free((ADDRESS)Xarray,a_m*a_n*sizeof(poly)); |
---|
1006 | } |
---|
1007 | } |
---|
1008 | } |
---|
1009 | |
---|
1010 | int mp_permmatrix::mpGetRdim() { return s_m; } |
---|
1011 | |
---|
1012 | int mp_permmatrix::mpGetCdim() { return s_n; } |
---|
1013 | |
---|
1014 | int mp_permmatrix::mpGetSign() { return sign; } |
---|
1015 | |
---|
1016 | void mp_permmatrix::mpSetSearch(int s) { piv_s = s; } |
---|
1017 | |
---|
1018 | void mp_permmatrix::mpSaveArray() { Xarray = NULL; } |
---|
1019 | |
---|
1020 | poly mp_permmatrix::mpGetElem(int r, int c) |
---|
1021 | { |
---|
1022 | return Xarray[a_n*qrow[r]+qcol[c]]; |
---|
1023 | } |
---|
1024 | |
---|
1025 | void mp_permmatrix::mpSetElem(poly p, int r, int c) |
---|
1026 | { |
---|
1027 | Xarray[a_n*qrow[r]+qcol[c]] = p; |
---|
1028 | } |
---|
1029 | |
---|
1030 | void mp_permmatrix::mpDelElem(int r, int c) |
---|
1031 | { |
---|
1032 | pDelete(&Xarray[a_n*qrow[r]+qcol[c]]); |
---|
1033 | } |
---|
1034 | |
---|
1035 | /* |
---|
1036 | * the Bareiss-type elimination with division by div (div != NULL) |
---|
1037 | */ |
---|
1038 | void mp_permmatrix::mpElimBareiss(poly div) |
---|
1039 | { |
---|
1040 | poly piv, elim, q1, q2, *ap, *a; |
---|
1041 | int i, j, jj; |
---|
1042 | |
---|
1043 | ap = this->mpRowAdr(s_m); |
---|
1044 | piv = ap[qcol[s_n]]; |
---|
1045 | for(i=s_m-1; i>=0; i--) |
---|
1046 | { |
---|
1047 | a = this->mpRowAdr(i); |
---|
1048 | elim = a[qcol[s_n]]; |
---|
1049 | if (elim != NULL) |
---|
1050 | { |
---|
1051 | elim = pNeg(elim); |
---|
1052 | for (j=s_n-1; j>=0; j--) |
---|
1053 | { |
---|
1054 | q2 = NULL; |
---|
1055 | jj = qcol[j]; |
---|
1056 | if (ap[jj] != NULL) |
---|
1057 | { |
---|
1058 | q2 = smMultDiv(ap[jj], elim, div); |
---|
1059 | if (a[jj] != NULL) |
---|
1060 | { |
---|
1061 | q1 = smMultDiv(a[jj], piv, div); |
---|
1062 | pDelete(&a[jj]); |
---|
1063 | q2 = pAdd(q2, q1); |
---|
1064 | } |
---|
1065 | } |
---|
1066 | else if (a[jj] != NULL) |
---|
1067 | { |
---|
1068 | q2 = smMultDiv(a[jj], piv, div); |
---|
1069 | } |
---|
1070 | if ((q2!=NULL) && div) |
---|
1071 | smSpecialPolyDiv(q2, div); |
---|
1072 | a[jj] = q2; |
---|
1073 | } |
---|
1074 | pDelete(&a[qcol[s_n]]); |
---|
1075 | } |
---|
1076 | else |
---|
1077 | { |
---|
1078 | for (j=s_n-1; j>=0; j--) |
---|
1079 | { |
---|
1080 | jj = qcol[j]; |
---|
1081 | if (a[jj] != NULL) |
---|
1082 | { |
---|
1083 | q2 = smMultDiv(a[jj], piv, div); |
---|
1084 | pDelete(&a[jj]); |
---|
1085 | if (div) |
---|
1086 | smSpecialPolyDiv(q2, div); |
---|
1087 | a[jj] = q2; |
---|
1088 | } |
---|
1089 | } |
---|
1090 | } |
---|
1091 | } |
---|
1092 | } |
---|
1093 | |
---|
1094 | /*2 |
---|
1095 | * pivot strategy for Bareiss algorithm |
---|
1096 | */ |
---|
1097 | int mp_permmatrix::mpPivotBareiss(row_col_weight *C) |
---|
1098 | { |
---|
1099 | poly p, *a; |
---|
1100 | int i, j, iopt, jopt; |
---|
1101 | float sum, f1, f2, fo, r, ro, lp; |
---|
1102 | float *dr = C->wrow, *dc = C->wcol; |
---|
1103 | |
---|
1104 | fo = 1.0e20; |
---|
1105 | ro = 0.0; |
---|
1106 | iopt = jopt = -1; |
---|
1107 | |
---|
1108 | s_n--; |
---|
1109 | s_m--; |
---|
1110 | if (s_m == 0) |
---|
1111 | return 0; |
---|
1112 | if (s_n == 0) |
---|
1113 | { |
---|
1114 | for(i=s_m; i>=0; i--) |
---|
1115 | { |
---|
1116 | p = this->mpRowAdr(i)[qcol[0]]; |
---|
1117 | if (p) |
---|
1118 | { |
---|
1119 | f1 = mpPolyWeight(p); |
---|
1120 | if (f1 < fo) |
---|
1121 | { |
---|
1122 | fo = f1; |
---|
1123 | if (iopt >= 0) |
---|
1124 | pDelete(&(this->mpRowAdr(iopt)[qcol[0]])); |
---|
1125 | iopt = i; |
---|
1126 | } |
---|
1127 | else |
---|
1128 | pDelete(&(this->mpRowAdr(i)[qcol[0]])); |
---|
1129 | } |
---|
1130 | } |
---|
1131 | if (iopt >= 0) |
---|
1132 | mpReplace(iopt, s_m, sign, qrow); |
---|
1133 | return 0; |
---|
1134 | } |
---|
1135 | this->mpRowWeight(dr); |
---|
1136 | this->mpColWeight(dc); |
---|
1137 | sum = 0.0; |
---|
1138 | for(i=s_m; i>=0; i--) |
---|
1139 | sum += dr[i]; |
---|
1140 | for(i=s_m; i>=0; i--) |
---|
1141 | { |
---|
1142 | r = dr[i]; |
---|
1143 | a = this->mpRowAdr(i); |
---|
1144 | for(j=s_n; j>=0; j--) |
---|
1145 | { |
---|
1146 | p = a[qcol[j]]; |
---|
1147 | if (p) |
---|
1148 | { |
---|
1149 | lp = mpPolyWeight(p); |
---|
1150 | ro = r - lp; |
---|
1151 | f1 = ro * (dc[j]-lp); |
---|
1152 | if (f1 != 0.0) |
---|
1153 | { |
---|
1154 | f2 = lp * (sum - ro - dc[j]); |
---|
1155 | f2 += f1; |
---|
1156 | } |
---|
1157 | else |
---|
1158 | f2 = lp-r-dc[j]; |
---|
1159 | if (f2 < fo) |
---|
1160 | { |
---|
1161 | fo = f2; |
---|
1162 | iopt = i; |
---|
1163 | jopt = j; |
---|
1164 | } |
---|
1165 | } |
---|
1166 | } |
---|
1167 | } |
---|
1168 | if (iopt < 0) |
---|
1169 | return 0; |
---|
1170 | mpReplace(iopt, s_m, sign, qrow); |
---|
1171 | mpReplace(jopt, s_n, sign, qcol); |
---|
1172 | return 1; |
---|
1173 | } |
---|
1174 | |
---|
1175 | /*2 |
---|
1176 | * pivot strategy for Bareiss algorithm with defined row |
---|
1177 | */ |
---|
1178 | int mp_permmatrix::mpPivotRow(row_col_weight *C, int row) |
---|
1179 | { |
---|
1180 | poly p, *a; |
---|
1181 | int j, iopt, jopt; |
---|
1182 | float sum, f1, f2, fo, r, ro, lp; |
---|
1183 | float *dr = C->wrow, *dc = C->wcol; |
---|
1184 | |
---|
1185 | fo = 1.0e20; |
---|
1186 | ro = 0.0; |
---|
1187 | iopt = jopt = -1; |
---|
1188 | |
---|
1189 | s_n--; |
---|
1190 | s_m--; |
---|
1191 | if (s_m == 0) |
---|
1192 | return 0; |
---|
1193 | if (s_n == 0) |
---|
1194 | { |
---|
1195 | p = this->mpRowAdr(row)[qcol[0]]; |
---|
1196 | if (p) |
---|
1197 | { |
---|
1198 | f1 = mpPolyWeight(p); |
---|
1199 | if (f1 < fo) |
---|
1200 | { |
---|
1201 | fo = f1; |
---|
1202 | if (iopt >= 0) |
---|
1203 | pDelete(&(this->mpRowAdr(iopt)[qcol[0]])); |
---|
1204 | iopt = row; |
---|
1205 | } |
---|
1206 | else |
---|
1207 | pDelete(&(this->mpRowAdr(row)[qcol[0]])); |
---|
1208 | } |
---|
1209 | if (iopt >= 0) |
---|
1210 | mpReplace(iopt, s_m, sign, qrow); |
---|
1211 | return 0; |
---|
1212 | } |
---|
1213 | this->mpRowWeight(dr); |
---|
1214 | this->mpColWeight(dc); |
---|
1215 | sum = 0.0; |
---|
1216 | for(j=s_m; j>=0; j--) |
---|
1217 | sum += dr[j]; |
---|
1218 | r = dr[row]; |
---|
1219 | a = this->mpRowAdr(row); |
---|
1220 | for(j=s_n; j>=0; j--) |
---|
1221 | { |
---|
1222 | p = a[qcol[j]]; |
---|
1223 | if (p) |
---|
1224 | { |
---|
1225 | lp = mpPolyWeight(p); |
---|
1226 | ro = r - lp; |
---|
1227 | f1 = ro * (dc[j]-lp); |
---|
1228 | if (f1 != 0.0) |
---|
1229 | { |
---|
1230 | f2 = lp * (sum - ro - dc[j]); |
---|
1231 | f2 += f1; |
---|
1232 | } |
---|
1233 | else |
---|
1234 | f2 = lp-r-dc[j]; |
---|
1235 | if (f2 < fo) |
---|
1236 | { |
---|
1237 | fo = f2; |
---|
1238 | iopt = row; |
---|
1239 | jopt = j; |
---|
1240 | } |
---|
1241 | } |
---|
1242 | } |
---|
1243 | if (iopt < 0) |
---|
1244 | return 0; |
---|
1245 | mpReplace(iopt, s_m, sign, qrow); |
---|
1246 | mpReplace(jopt, s_n, sign, qcol); |
---|
1247 | return 1; |
---|
1248 | } |
---|
1249 | |
---|
1250 | void mp_permmatrix::mpToIntvec(intvec *v) |
---|
1251 | { |
---|
1252 | int i; |
---|
1253 | |
---|
1254 | for (i=v->rows()-1; i>=0; i--) |
---|
1255 | (*v)[i] = qcol[i]+1; |
---|
1256 | } |
---|
1257 | |
---|
1258 | void mp_permmatrix::mpRowReorder() |
---|
1259 | { |
---|
1260 | int k, i, i1, i2; |
---|
1261 | |
---|
1262 | if (a_m > a_n) |
---|
1263 | k = a_m - a_n; |
---|
1264 | else |
---|
1265 | k = 0; |
---|
1266 | for (i=a_m-1; i>=k; i--) |
---|
1267 | { |
---|
1268 | i1 = qrow[i]; |
---|
1269 | if (i1 != i) |
---|
1270 | { |
---|
1271 | this->mpRowSwap(i1, i); |
---|
1272 | i2 = 0; |
---|
1273 | while (qrow[i2] != i) i2++; |
---|
1274 | qrow[i2] = i1; |
---|
1275 | } |
---|
1276 | } |
---|
1277 | } |
---|
1278 | |
---|
1279 | void mp_permmatrix::mpColReorder() |
---|
1280 | { |
---|
1281 | int k, j, j1, j2; |
---|
1282 | |
---|
1283 | if (a_n > a_m) |
---|
1284 | k = a_n - a_m; |
---|
1285 | else |
---|
1286 | k = 0; |
---|
1287 | for (j=a_n-1; j>=k; j--) |
---|
1288 | { |
---|
1289 | j1 = qcol[j]; |
---|
1290 | if (j1 != j) |
---|
1291 | { |
---|
1292 | this->mpColSwap(j1, j); |
---|
1293 | j2 = 0; |
---|
1294 | while (qcol[j2] != j) j2++; |
---|
1295 | qcol[j2] = j1; |
---|
1296 | } |
---|
1297 | } |
---|
1298 | } |
---|
1299 | |
---|
1300 | // private |
---|
1301 | void mp_permmatrix::mpInitMat() |
---|
1302 | { |
---|
1303 | int k; |
---|
1304 | |
---|
1305 | s_m = a_m; |
---|
1306 | s_n = a_n; |
---|
1307 | piv_s = 0; |
---|
1308 | qrow = (int *)Alloc(a_m*sizeof(int)); |
---|
1309 | qcol = (int *)Alloc(a_n*sizeof(int)); |
---|
1310 | for (k=a_m-1; k>=0; k--) qrow[k] = k; |
---|
1311 | for (k=a_n-1; k>=0; k--) qcol[k] = k; |
---|
1312 | } |
---|
1313 | |
---|
1314 | poly * mp_permmatrix::mpRowAdr(int r) |
---|
1315 | { |
---|
1316 | return &(Xarray[a_n*qrow[r]]); |
---|
1317 | } |
---|
1318 | |
---|
1319 | poly * mp_permmatrix::mpColAdr(int c) |
---|
1320 | { |
---|
1321 | return &(Xarray[qcol[c]]); |
---|
1322 | } |
---|
1323 | |
---|
1324 | void mp_permmatrix::mpRowWeight(float *wrow) |
---|
1325 | { |
---|
1326 | poly p, *a; |
---|
1327 | int i, j; |
---|
1328 | float count; |
---|
1329 | |
---|
1330 | for (i=s_m; i>=0; i--) |
---|
1331 | { |
---|
1332 | a = this->mpRowAdr(i); |
---|
1333 | count = 0.0; |
---|
1334 | for(j=s_n; j>=0; j--) |
---|
1335 | { |
---|
1336 | p = a[qcol[j]]; |
---|
1337 | if (p) |
---|
1338 | count += mpPolyWeight(p); |
---|
1339 | } |
---|
1340 | wrow[i] = count; |
---|
1341 | } |
---|
1342 | } |
---|
1343 | |
---|
1344 | void mp_permmatrix::mpColWeight(float *wcol) |
---|
1345 | { |
---|
1346 | poly p, *a; |
---|
1347 | int i, j; |
---|
1348 | float count; |
---|
1349 | |
---|
1350 | for (j=s_n; j>=0; j--) |
---|
1351 | { |
---|
1352 | a = this->mpColAdr(j); |
---|
1353 | count = 0.0; |
---|
1354 | for(i=s_m; i>=0; i--) |
---|
1355 | { |
---|
1356 | p = a[a_n*qrow[i]]; |
---|
1357 | if (p) |
---|
1358 | count += mpPolyWeight(p); |
---|
1359 | } |
---|
1360 | wcol[j] = count; |
---|
1361 | } |
---|
1362 | } |
---|
1363 | |
---|
1364 | void mp_permmatrix::mpRowSwap(int i1, int i2) |
---|
1365 | { |
---|
1366 | poly p, *a1, *a2; |
---|
1367 | int j; |
---|
1368 | |
---|
1369 | a1 = &(Xarray[a_n*i1]); |
---|
1370 | a2 = &(Xarray[a_n*i2]); |
---|
1371 | for (j=a_n-1; j>= 0; j--) |
---|
1372 | { |
---|
1373 | p = a1[j]; |
---|
1374 | a1[j] = a2[j]; |
---|
1375 | a2[j] = p; |
---|
1376 | } |
---|
1377 | } |
---|
1378 | |
---|
1379 | void mp_permmatrix::mpColSwap(int j1, int j2) |
---|
1380 | { |
---|
1381 | poly p, *a1, *a2; |
---|
1382 | int i, k = a_n*a_m; |
---|
1383 | |
---|
1384 | a1 = &(Xarray[j1]); |
---|
1385 | a2 = &(Xarray[j2]); |
---|
1386 | for (i=0; i< k; i+=a_n) |
---|
1387 | { |
---|
1388 | p = a1[i]; |
---|
1389 | a1[i] = a2[i]; |
---|
1390 | a2[i] = p; |
---|
1391 | } |
---|
1392 | } |
---|
1393 | |
---|
1394 | int mp_permmatrix::mpGetRow() |
---|
1395 | { |
---|
1396 | return qrow[s_m]; |
---|
1397 | } |
---|
1398 | |
---|
1399 | int mp_permmatrix::mpGetCol() |
---|
1400 | { |
---|
1401 | return qcol[s_n]; |
---|
1402 | } |
---|
1403 | |
---|
1404 | /*2 |
---|
1405 | * exact division a/b, used in Bareiss algorithm |
---|
1406 | * a destroyed, b NOT destroyed |
---|
1407 | */ |
---|
1408 | static poly mpDivide(poly a, poly b) |
---|
1409 | { |
---|
1410 | number x, y, z; |
---|
1411 | poly r, tail, t, h, h0; |
---|
1412 | int i, deg; |
---|
1413 | |
---|
1414 | r = a; |
---|
1415 | x = pGetCoeff(b); |
---|
1416 | deg = pTotaldegree(b); |
---|
1417 | tail = pNext(b); |
---|
1418 | if (tail == NULL) |
---|
1419 | { |
---|
1420 | do |
---|
1421 | { |
---|
1422 | if (deg != 0) |
---|
1423 | { |
---|
1424 | for (i=pVariables; i; i--) |
---|
1425 | pSubExp(r,i, pGetExp(b,i)); |
---|
1426 | pSetm(r); |
---|
1427 | } |
---|
1428 | y = nDiv(pGetCoeff(r),x); |
---|
1429 | nNormalize(y); |
---|
1430 | pSetCoeff(r,y); |
---|
1431 | pIter(r); |
---|
1432 | } while (r != NULL); |
---|
1433 | //pTest(a); |
---|
1434 | return a; |
---|
1435 | } |
---|
1436 | h0 = pInit(); |
---|
1437 | do |
---|
1438 | { |
---|
1439 | if (deg != 0) |
---|
1440 | { |
---|
1441 | for (i=pVariables; i>0; i--) |
---|
1442 | pSubExp(r,i,pGetExp(b,i)); |
---|
1443 | pSetm(r); |
---|
1444 | } |
---|
1445 | y = nDiv(pGetCoeff(r), x); |
---|
1446 | nNormalize(y); |
---|
1447 | pSetCoeff(r,y); |
---|
1448 | t = tail; |
---|
1449 | h = h0; |
---|
1450 | do |
---|
1451 | { |
---|
1452 | h = pNext(h) = pInit(); |
---|
1453 | for (i=pVariables; i>0; i--) |
---|
1454 | pSetExp(h,i, pGetExp(r,i)+pGetExp(t,i)); |
---|
1455 | pSetm(h); |
---|
1456 | z = nMult(y, pGetCoeff(t)); |
---|
1457 | pSetCoeff0(h,nNeg(z)); |
---|
1458 | pIter(t); |
---|
1459 | } while (t != NULL); |
---|
1460 | pNext(h) = NULL; |
---|
1461 | r = pNext(r) = pAdd(pNext(r),pNext(h0)); |
---|
1462 | } while (r!=NULL); |
---|
1463 | pFree1(h0); |
---|
1464 | //pTest(a); |
---|
1465 | return a; |
---|
1466 | } |
---|
1467 | |
---|
1468 | /* |
---|
1469 | * perform replacement for pivot strategy in Bareiss algorithm |
---|
1470 | * change sign of determinant |
---|
1471 | */ |
---|
1472 | static void mpReplace(int j, int n, int &sign, int *perm) |
---|
1473 | { |
---|
1474 | int k; |
---|
1475 | |
---|
1476 | if (j != n) |
---|
1477 | { |
---|
1478 | k = perm[n]; |
---|
1479 | perm[n] = perm[j]; |
---|
1480 | perm[j] = k; |
---|
1481 | sign = -sign; |
---|
1482 | } |
---|
1483 | } |
---|
1484 | |
---|
1485 | /* |
---|
1486 | * weigth of a polynomial, for pivot strategy |
---|
1487 | * modify this for characteristic 0 !!! |
---|
1488 | */ |
---|
1489 | static float mpPolyWeight(poly p) |
---|
1490 | { |
---|
1491 | int i; |
---|
1492 | float res; |
---|
1493 | |
---|
1494 | if (pNext(p) == NULL) |
---|
1495 | { |
---|
1496 | res = (float)nSize(pGetCoeff(p)); |
---|
1497 | if (pTotaldegree(p) != 0) res += 1.0; |
---|
1498 | } |
---|
1499 | else |
---|
1500 | { |
---|
1501 | i = 0; |
---|
1502 | res = 0.0; |
---|
1503 | do |
---|
1504 | { |
---|
1505 | i++; |
---|
1506 | res += (float)nSize(pGetCoeff(p)); |
---|
1507 | pIter(p); |
---|
1508 | } |
---|
1509 | while (p); |
---|
1510 | res += (float)i; |
---|
1511 | } |
---|
1512 | return res; |
---|
1513 | } |
---|
1514 | |
---|
1515 | static int mpNextperm(perm * z, int max) |
---|
1516 | { |
---|
1517 | int s, i, k, t; |
---|
1518 | s = max; |
---|
1519 | do |
---|
1520 | { |
---|
1521 | s--; |
---|
1522 | } |
---|
1523 | while ((s > 0) && ((*z)[s] >= (*z)[s+1])); |
---|
1524 | if (s==0) |
---|
1525 | return 0; |
---|
1526 | do |
---|
1527 | { |
---|
1528 | (*z)[s]++; |
---|
1529 | k = 0; |
---|
1530 | do |
---|
1531 | { |
---|
1532 | k++; |
---|
1533 | } |
---|
1534 | while (((*z)[k] != (*z)[s]) && (k!=s)); |
---|
1535 | } |
---|
1536 | while (k < s); |
---|
1537 | for (i=s+1; i <= max; i++) |
---|
1538 | { |
---|
1539 | (*z)[i]=0; |
---|
1540 | do |
---|
1541 | { |
---|
1542 | (*z)[i]++; |
---|
1543 | k=0; |
---|
1544 | do |
---|
1545 | { |
---|
1546 | k++; |
---|
1547 | } |
---|
1548 | while (((*z)[k] != (*z)[i]) && (k != i)); |
---|
1549 | } |
---|
1550 | while (k < i); |
---|
1551 | } |
---|
1552 | s = max+1; |
---|
1553 | do |
---|
1554 | { |
---|
1555 | s--; |
---|
1556 | } |
---|
1557 | while ((s > 0) && ((*z)[s] > (*z)[s+1])); |
---|
1558 | t = 1; |
---|
1559 | for (i=1; i<max; i++) |
---|
1560 | for (k=i+1; k<=max; k++) |
---|
1561 | if ((*z)[k] < (*z)[i]) |
---|
1562 | t = -t; |
---|
1563 | (*z)[0] = t; |
---|
1564 | return s; |
---|
1565 | } |
---|
1566 | |
---|
1567 | static poly mpLeibnitz(matrix a) |
---|
1568 | { |
---|
1569 | int i, e, n; |
---|
1570 | poly p, d; |
---|
1571 | perm z; |
---|
1572 | |
---|
1573 | n = MATROWS(a); |
---|
1574 | memset(&z,0,(n+2)*sizeof(int)); |
---|
1575 | p = pOne(); |
---|
1576 | for (i=1; i <= n; i++) |
---|
1577 | p = pMult(p, pCopy(MATELEM(a, i, i))); |
---|
1578 | d = p; |
---|
1579 | for (i=1; i<= n; i++) |
---|
1580 | z[i] = i; |
---|
1581 | z[0]=1; |
---|
1582 | e = 1; |
---|
1583 | if (n!=1) |
---|
1584 | { |
---|
1585 | while (e) |
---|
1586 | { |
---|
1587 | e = mpNextperm((perm *)&z, n); |
---|
1588 | p = pOne(); |
---|
1589 | for (i = 1; i <= n; i++) |
---|
1590 | p = pMult(p, pCopy(MATELEM(a, i, z[i]))); |
---|
1591 | if (z[0] > 0) |
---|
1592 | d = pAdd(d, p); |
---|
1593 | else |
---|
1594 | d = pSub(d, p); |
---|
1595 | } |
---|
1596 | } |
---|
1597 | return d; |
---|
1598 | } |
---|
1599 | |
---|
1600 | static poly minuscopy (poly p) |
---|
1601 | { |
---|
1602 | poly w; |
---|
1603 | number e; |
---|
1604 | e = nInit(-1); |
---|
1605 | w = pCopy(p); |
---|
1606 | pMultN(w, e); |
---|
1607 | nDelete(&e); |
---|
1608 | return w; |
---|
1609 | } |
---|
1610 | |
---|
1611 | /*2 |
---|
1612 | * insert a monomial into a list, avoid duplicates |
---|
1613 | * arguments are destroyed |
---|
1614 | */ |
---|
1615 | static poly pInsert(poly p1, poly p2) |
---|
1616 | { |
---|
1617 | poly a1, p, a2, a; |
---|
1618 | int c; |
---|
1619 | |
---|
1620 | if (p1==NULL) return p2; |
---|
1621 | if (p2==NULL) return p1; |
---|
1622 | a1 = p1; |
---|
1623 | a2 = p2; |
---|
1624 | a = p = pOne(); |
---|
1625 | loop |
---|
1626 | { |
---|
1627 | c = pComp(a1, a2); |
---|
1628 | if (c == 1) |
---|
1629 | { |
---|
1630 | a = pNext(a) = a1; |
---|
1631 | pIter(a1); |
---|
1632 | if (a1==NULL) |
---|
1633 | { |
---|
1634 | pNext(a) = a2; |
---|
1635 | break; |
---|
1636 | } |
---|
1637 | } |
---|
1638 | else if (c == -1) |
---|
1639 | { |
---|
1640 | a = pNext(a) = a2; |
---|
1641 | pIter(a2); |
---|
1642 | if (a2==NULL) |
---|
1643 | { |
---|
1644 | pNext(a) = a1; |
---|
1645 | break; |
---|
1646 | } |
---|
1647 | } |
---|
1648 | else |
---|
1649 | { |
---|
1650 | pDelete1(&a2); |
---|
1651 | a = pNext(a) = a1; |
---|
1652 | pIter(a1); |
---|
1653 | if (a1==NULL) |
---|
1654 | { |
---|
1655 | pNext(a) = a2; |
---|
1656 | break; |
---|
1657 | } |
---|
1658 | else if (a2==NULL) |
---|
1659 | { |
---|
1660 | pNext(a) = a1; |
---|
1661 | break; |
---|
1662 | } |
---|
1663 | } |
---|
1664 | } |
---|
1665 | pDelete1(&p); |
---|
1666 | return p; |
---|
1667 | } |
---|
1668 | |
---|
1669 | /*2 |
---|
1670 | *if what == xy the result is the list of all different power products |
---|
1671 | * x^i*y^j (i, j >= 0) that appear in fro |
---|
1672 | */ |
---|
1673 | static poly mpSelect (poly fro, poly what) |
---|
1674 | { |
---|
1675 | int i; |
---|
1676 | poly h, res; |
---|
1677 | res = NULL; |
---|
1678 | while (fro!=NULL) |
---|
1679 | { |
---|
1680 | h = pOne(); |
---|
1681 | for (i=1; i<=pVariables; i++) |
---|
1682 | pSetExp(h,i, pGetExp(fro,i) * pGetExp(what, i)); |
---|
1683 | pSetComp(h, pGetComp(fro)); |
---|
1684 | pSetm(h); |
---|
1685 | res = pInsert(h, res); |
---|
1686 | fro = fro->next; |
---|
1687 | } |
---|
1688 | return res; |
---|
1689 | } |
---|
1690 | |
---|
1691 | /*2 |
---|
1692 | *exact divisor: let d == x^i*y^j, m is thought to have only one term; |
---|
1693 | * return m/d iff d divides m, and no x^k*y^l (k>i or l>j) divides m |
---|
1694 | */ |
---|
1695 | static poly mpExdiv ( poly m, poly d) |
---|
1696 | { |
---|
1697 | int i; |
---|
1698 | poly h = pHead(m); |
---|
1699 | for (i=1; i<=pVariables; i++) |
---|
1700 | { |
---|
1701 | if (pGetExp(d,i) > 0) |
---|
1702 | { |
---|
1703 | if (pGetExp(d,i) != pGetExp(h,i)) |
---|
1704 | { |
---|
1705 | pDelete(&h); |
---|
1706 | return NULL; |
---|
1707 | } |
---|
1708 | pSetExp(h,i,0); |
---|
1709 | } |
---|
1710 | } |
---|
1711 | pSetm(h); |
---|
1712 | return h; |
---|
1713 | } |
---|
1714 | |
---|