1 | /**************************************** |
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2 | * Computer Algebra System SINGULAR * |
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3 | ****************************************/ |
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4 | /* $Id: matpol.cc,v 1.37 2000-12-31 15:14:35 obachman 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 "structs.h" |
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16 | #include "tok.h" |
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17 | #include "lists.h" |
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18 | #include "ipid.h" |
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19 | #include "kstd1.h" |
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20 | #include "polys.h" |
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21 | #include "omalloc.h" |
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22 | #include "febase.h" |
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23 | #include "numbers.h" |
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24 | #include "ideals.h" |
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25 | #include "subexpr.h" |
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26 | #include "intvec.h" |
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27 | #include "ring.h" |
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28 | #include "sparsmat.h" |
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29 | #include "matpol.h" |
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30 | |
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31 | |
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32 | omBin ip_smatrix_bin = omGetSpecBin(sizeof(ip_smatrix)); |
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33 | /*0 implementation*/ |
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34 | |
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35 | |
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36 | typedef int perm[100]; |
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37 | static void mpReplace(int j, int n, int &sign, int *perm); |
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38 | static int mpNextperm(perm * z, int max); |
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39 | static poly mpLeibnitz(matrix a); |
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40 | static poly minuscopy (poly p); |
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41 | static poly pInsert(poly p1, poly p2); |
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42 | static poly mpExdiv ( poly m, poly d); |
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43 | static poly mpSelect (poly fro, poly what); |
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44 | |
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45 | static void mpPartClean(matrix, int, int); |
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46 | static void mpFinalClean(matrix); |
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47 | static int mpPrepareRow (matrix, int, int); |
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48 | static int mpPreparePiv (matrix, int, int); |
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49 | static int mpPivBar(matrix, int, int); |
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50 | static int mpPivRow(matrix, int, int); |
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51 | static float mpPolyWeight(poly); |
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52 | static void mpSwapRow(matrix, int, int, int); |
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53 | static void mpSwapCol(matrix, int, int, int); |
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54 | static void mpElimBar(matrix, matrix, poly, int, int); |
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55 | |
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56 | /*2 |
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57 | * create a r x c zero-matrix |
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58 | */ |
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59 | matrix mpNew(int r, int c) |
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60 | { |
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61 | if (r<=0) r=1; |
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62 | if ( (((int)(INT_MAX/sizeof(poly))) / r) <= c) |
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63 | { |
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64 | Werror("internal error: creating matrix[%d][%d]",r,c); |
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65 | return NULL; |
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66 | } |
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67 | matrix rc = (matrix)omAllocBin(ip_smatrix_bin); |
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68 | rc->nrows = r; |
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69 | rc->ncols = c; |
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70 | rc->rank = r; |
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71 | if (c != 0) |
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72 | { |
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73 | int s=r*c*sizeof(poly); |
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74 | rc->m = (polyset)omAlloc0(s); |
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75 | //if (rc->m==NULL) |
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76 | //{ |
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77 | // Werror("internal error: creating matrix[%d][%d]",r,c); |
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78 | // return NULL; |
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79 | //} |
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80 | } |
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81 | return rc; |
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82 | } |
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83 | |
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84 | /*2 |
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85 | *copies matrix a to b |
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86 | */ |
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87 | matrix mpCopy (matrix a) |
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88 | { |
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89 | idTest((ideal)a); |
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90 | poly t; |
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91 | int i, m=MATROWS(a), n=MATCOLS(a); |
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92 | matrix b = mpNew(m, n); |
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93 | |
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94 | for (i=m*n-1; i>=0; i--) |
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95 | { |
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96 | t = a->m[i]; |
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97 | pNormalize(t); |
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98 | b->m[i] = pCopy(t); |
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99 | } |
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100 | b->rank=a->rank; |
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101 | return b; |
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102 | } |
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103 | |
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104 | /*2 |
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105 | * make it a p * unit matrix |
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106 | */ |
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107 | matrix mpInitP(int r, int c, poly p) |
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108 | { |
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109 | matrix rc = mpNew(r,c); |
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110 | int i=min(r,c), n = c*(i-1)+i-1, inc = c+1; |
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111 | |
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112 | pNormalize(p); |
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113 | while (n>0) |
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114 | { |
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115 | rc->m[n] = pCopy(p); |
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116 | n -= inc; |
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117 | } |
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118 | rc->m[0]=p; |
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119 | return rc; |
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120 | } |
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121 | |
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122 | /*2 |
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123 | * make it a v * unit matrix |
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124 | */ |
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125 | matrix mpInitI(int r, int c, int v) |
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126 | { |
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127 | return mpInitP(r,c,pISet(v)); |
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128 | } |
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129 | |
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130 | /*2 |
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131 | * c = f*a |
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132 | */ |
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133 | matrix mpMultI(matrix a, int f) |
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134 | { |
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135 | int k, n = a->nrows, m = a->ncols; |
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136 | poly p = pISet(f); |
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137 | matrix c = mpNew(n,m); |
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138 | |
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139 | for (k=m*n-1; k>0; k--) |
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140 | c->m[k] = ppMult_qq(a->m[k], p); |
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141 | c->m[0] = pMult(pCopy(a->m[0]), p); |
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142 | return c; |
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143 | } |
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144 | |
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145 | /*2 |
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146 | * multiply a matrix 'a' by a poly 'p', destroy the args |
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147 | */ |
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148 | matrix mpMultP(matrix a, poly p) |
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149 | { |
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150 | int k, n = a->nrows, m = a->ncols; |
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151 | |
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152 | pNormalize(p); |
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153 | for (k=m*n-1; k>0; k--) |
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154 | a->m[k] = pMult(a->m[k], pCopy(p)); |
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155 | a->m[0] = pMult(a->m[0], p); |
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156 | return a; |
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157 | } |
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158 | |
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159 | matrix mpAdd(matrix a, matrix b) |
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160 | { |
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161 | int k, n = a->nrows, m = a->ncols; |
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162 | if ((n != b->nrows) || (m != b->ncols)) |
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163 | { |
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164 | /* |
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165 | * Werror("cannot add %dx%d matrix and %dx%d matrix", |
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166 | * m,n,b->cols(),b->rows()); |
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167 | */ |
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168 | return NULL; |
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169 | } |
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170 | matrix c = mpNew(n,m); |
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171 | for (k=m*n-1; k>=0; k--) |
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172 | c->m[k] = pAdd(pCopy(a->m[k]), pCopy(b->m[k])); |
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173 | return c; |
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174 | } |
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175 | |
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176 | matrix mpSub(matrix a, matrix b) |
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177 | { |
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178 | int k, n = a->nrows, m = a->ncols; |
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179 | if ((n != b->nrows) || (m != b->ncols)) |
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180 | { |
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181 | /* |
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182 | * Werror("cannot sub %dx%d matrix and %dx%d matrix", |
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183 | * m,n,b->cols(),b->rows()); |
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184 | */ |
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185 | return NULL; |
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186 | } |
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187 | matrix c = mpNew(n,m); |
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188 | for (k=m*n-1; k>=0; k--) |
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189 | c->m[k] = pSub(pCopy(a->m[k]), pCopy(b->m[k])); |
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190 | return c; |
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191 | } |
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192 | |
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193 | matrix mpMult(matrix a, matrix b) |
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194 | { |
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195 | int i, j, k; |
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196 | poly s, t, aik, bkj; |
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197 | int m = MATROWS(a); |
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198 | int p = MATCOLS(a); |
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199 | int q = MATCOLS(b); |
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200 | |
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201 | if (p!=MATROWS(b)) |
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202 | { |
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203 | /* |
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204 | * Werror("cannot multiply %dx%d matrix and %dx%d matrix", |
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205 | * m,p,b->rows(),q); |
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206 | */ |
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207 | return NULL; |
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208 | } |
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209 | matrix c = mpNew(m,q); |
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210 | |
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211 | for (i=1; i<=m; i++) |
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212 | { |
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213 | for (j=1; j<=q; j++) |
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214 | { |
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215 | t = NULL; |
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216 | for (k=1; k<=p; k++) |
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217 | { |
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218 | s = ppMult_qq(MATELEM(a,i,k), MATELEM(b,k,j)); |
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219 | t = pAdd(t,s); |
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220 | } |
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221 | pNormalize(t); |
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222 | MATELEM(c,i,j) = t; |
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223 | } |
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224 | } |
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225 | return c; |
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226 | } |
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227 | |
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228 | matrix mpTransp(matrix a) |
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229 | { |
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230 | int i, j, r = MATROWS(a), c = MATCOLS(a); |
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231 | poly *p; |
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232 | matrix b = mpNew(c,r); |
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233 | |
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234 | p = b->m; |
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235 | for (i=0; i<c; i++) |
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236 | { |
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237 | for (j=0; j<r; j++) |
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238 | { |
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239 | *p++ = pCopy(a->m[j*c+i]); |
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240 | } |
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241 | } |
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242 | return b; |
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243 | } |
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244 | |
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245 | /*2 |
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246 | *returns the trace of matrix a |
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247 | */ |
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248 | poly mpTrace ( matrix a) |
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249 | { |
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250 | int i; |
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251 | int n = (MATCOLS(a)<MATROWS(a)) ? MATCOLS(a) : MATROWS(a); |
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252 | poly t = NULL; |
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253 | |
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254 | for (i=1; i<=n; i++) |
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255 | t = pAdd(t, pCopy(MATELEM(a,i,i))); |
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256 | return t; |
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257 | } |
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258 | |
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259 | /*2 |
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260 | *returns the trace of the product of a and b |
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261 | */ |
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262 | poly TraceOfProd ( matrix a, matrix b, int n) |
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263 | { |
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264 | int i, j; |
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265 | poly p, t = NULL; |
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266 | |
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267 | for (i=1; i<=n; i++) |
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268 | { |
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269 | for (j=1; j<=n; j++) |
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270 | { |
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271 | p = ppMult_qq(MATELEM(a,i,j), MATELEM(b,j,i)); |
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272 | t = pAdd(t, p); |
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273 | } |
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274 | } |
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275 | return t; |
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276 | } |
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277 | |
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278 | /* |
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279 | * C++ classes for Bareiss algorithm |
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280 | */ |
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281 | class row_col_weight |
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282 | { |
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283 | private: |
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284 | int ym, yn; |
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285 | public: |
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286 | float *wrow, *wcol; |
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287 | row_col_weight() : ym(0) {} |
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288 | row_col_weight(int, int); |
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289 | ~row_col_weight(); |
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290 | }; |
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291 | |
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292 | /*2 |
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293 | * a submatrix M of a matrix X[m,n]: |
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294 | * 0 <= i < s_m <= a_m |
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295 | * 0 <= j < s_n <= a_n |
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296 | * M = ( Xarray[qrow[i],qcol[j]] ) |
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297 | * if a_m = a_n and s_m = s_n |
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298 | * det(X) = sign*div^(s_m-1)*det(M) |
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299 | * resticted pivot for elimination |
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300 | * 0 <= j < piv_s |
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301 | */ |
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302 | class mp_permmatrix |
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303 | { |
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304 | private: |
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305 | int a_m, a_n, s_m, s_n, sign, piv_s; |
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306 | int *qrow, *qcol; |
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307 | poly *Xarray; |
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308 | void mpInitMat(); |
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309 | poly * mpRowAdr(int); |
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310 | poly * mpColAdr(int); |
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311 | void mpRowWeight(float *); |
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312 | void mpColWeight(float *); |
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313 | void mpRowSwap(int, int); |
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314 | void mpColSwap(int, int); |
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315 | public: |
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316 | mp_permmatrix() : a_m(0) {} |
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317 | mp_permmatrix(matrix); |
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318 | mp_permmatrix(mp_permmatrix *); |
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319 | ~mp_permmatrix(); |
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320 | int mpGetRow(); |
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321 | int mpGetCol(); |
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322 | int mpGetRdim(); |
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323 | int mpGetCdim(); |
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324 | int mpGetSign(); |
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325 | void mpSetSearch(int s); |
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326 | void mpSaveArray(); |
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327 | poly mpGetElem(int, int); |
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328 | void mpSetElem(poly, int, int); |
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329 | void mpDelElem(int, int); |
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330 | void mpElimBareiss(poly); |
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331 | int mpPivotBareiss(row_col_weight *); |
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332 | int mpPivotRow(row_col_weight *, int); |
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333 | void mpToIntvec(intvec *); |
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334 | void mpRowReorder(); |
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335 | void mpColReorder(); |
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336 | }; |
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337 | |
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338 | #ifndef SIZE_OF_SYSTEM_PAGE |
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339 | #define SIZE_OF_SYSTEM_PAGE 4096 |
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340 | #endif |
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341 | /*2 |
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342 | * entries of a are minors and go to result (only if not in R) |
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343 | */ |
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344 | void mpMinorToResult(ideal result, int &elems, matrix a, int r, int c, |
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345 | ideal R) |
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346 | { |
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347 | poly *q1; |
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348 | int e=IDELEMS(result); |
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349 | int i,j; |
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350 | |
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351 | if (R != NULL) |
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352 | { |
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353 | for (i=r-1;i>=0;i--) |
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354 | { |
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355 | q1 = &(a->m)[i*a->ncols]; |
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356 | for (j=c-1;j>=0;j--) |
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357 | { |
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358 | if (q1[j]!=NULL) q1[j] = kNF(R,currQuotient,q1[j]); |
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359 | } |
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360 | } |
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361 | } |
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362 | for (i=r-1;i>=0;i--) |
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363 | { |
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364 | q1 = &(a->m)[i*a->ncols]; |
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365 | for (j=c-1;j>=0;j--) |
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366 | { |
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367 | if (q1[j]!=NULL) |
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368 | { |
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369 | if (elems>=e) |
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370 | { |
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371 | if(e<SIZE_OF_SYSTEM_PAGE) |
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372 | { |
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373 | pEnlargeSet(&(result->m),e,e); |
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374 | e += e; |
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375 | } |
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376 | else |
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377 | { |
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378 | pEnlargeSet(&(result->m),e,SIZE_OF_SYSTEM_PAGE); |
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379 | e += SIZE_OF_SYSTEM_PAGE; |
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380 | } |
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381 | IDELEMS(result) =e; |
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382 | } |
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383 | result->m[elems] = q1[j]; |
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384 | q1[j] = NULL; |
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385 | elems++; |
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386 | } |
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387 | } |
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388 | } |
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389 | } |
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390 | |
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391 | /*2 |
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392 | * produces recursively the ideal of all arxar-minors of a |
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393 | */ |
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394 | void mpRecMin(int ar,ideal result,int &elems,matrix a,int lr,int lc, |
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395 | poly barDiv, ideal R) |
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396 | { |
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397 | int k; |
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398 | int kr=lr-1,kc=lc-1; |
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399 | matrix nextLevel=mpNew(kr,kc); |
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400 | |
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401 | loop |
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402 | { |
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403 | /*--- look for an optimal row and bring it to last position ------------*/ |
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404 | if(mpPrepareRow(a,lr,lc)==0) break; |
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405 | /*--- now take all pivotŽs from the last row ------------*/ |
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406 | k = lc; |
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407 | loop |
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408 | { |
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409 | if(mpPreparePiv(a,lr,k)==0) break; |
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410 | mpElimBar(a,nextLevel,barDiv,lr,k); |
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411 | k--; |
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412 | if (ar>1) |
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413 | { |
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414 | mpRecMin(ar-1,result,elems,nextLevel,kr,k,a->m[kr*a->ncols+k],R); |
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415 | mpPartClean(nextLevel,kr,k); |
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416 | } |
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417 | else mpMinorToResult(result,elems,nextLevel,kr,k,R); |
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418 | if (ar>k-1) break; |
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419 | } |
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420 | if (ar>=kr) break; |
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421 | /*--- now we have to take out the last row...------------*/ |
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422 | lr = kr; |
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423 | kr--; |
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424 | } |
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425 | mpFinalClean(nextLevel); |
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426 | } |
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427 | |
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428 | /*2 |
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429 | *returns the determinant of the matrix m; |
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430 | *uses Bareiss algorithm |
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431 | */ |
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432 | poly mpDetBareiss (matrix a) |
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433 | { |
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434 | int s; |
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435 | poly div, res; |
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436 | if (MATROWS(a) != MATCOLS(a)) |
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437 | { |
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438 | Werror("det of %d x %d matrix",MATROWS(a),MATCOLS(a)); |
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439 | return NULL; |
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440 | } |
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441 | matrix c = mpCopy(a); |
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442 | mp_permmatrix *Bareiss = new mp_permmatrix(c); |
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443 | row_col_weight w(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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444 | |
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445 | /* Bareiss */ |
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446 | div = NULL; |
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447 | while(Bareiss->mpPivotBareiss(&w)) |
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448 | { |
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449 | Bareiss->mpElimBareiss(div); |
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450 | div = Bareiss->mpGetElem(Bareiss->mpGetRdim(), Bareiss->mpGetCdim()); |
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451 | } |
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452 | Bareiss->mpRowReorder(); |
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453 | Bareiss->mpColReorder(); |
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454 | Bareiss->mpSaveArray(); |
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455 | s = Bareiss->mpGetSign(); |
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456 | delete Bareiss; |
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457 | |
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458 | /* result */ |
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459 | res = MATELEM(c,1,1); |
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460 | MATELEM(c,1,1) = NULL; |
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461 | idDelete((ideal *)&c); |
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462 | if (s < 0) |
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463 | res = pNeg(res); |
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464 | return res; |
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465 | } |
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466 | |
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467 | /*2 |
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468 | *returns the determinant of the matrix m; |
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469 | *uses Newtons formulea for symmetric functions |
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470 | */ |
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471 | poly mpDet (matrix m) |
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472 | { |
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473 | int i,j,k,n; |
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474 | poly p,q; |
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475 | matrix a, s; |
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476 | matrix ma[100]; |
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477 | number c=NULL, d=NULL, ONE=NULL; |
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478 | |
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479 | n = MATROWS(m); |
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480 | if (n != MATCOLS(m)) |
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481 | { |
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482 | Werror("det of %d x %d matrix",n,MATCOLS(m)); |
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483 | return NULL; |
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484 | } |
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485 | k=rChar(); |
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486 | if (((k > 0) && (k <= n)) |
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487 | #ifdef SRING |
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488 | || (pSRING) |
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489 | #endif |
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490 | ) |
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491 | return mpLeibnitz(m); |
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492 | ONE = nInit(1); |
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493 | ma[1]=mpCopy(m); |
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494 | k = (n+1) / 2; |
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495 | s = mpNew(1, n); |
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496 | MATELEM(s,1,1) = mpTrace(m); |
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497 | for (i=2; i<=k; i++) |
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498 | { |
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499 | //ma[i] = mpNew(n,n); |
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500 | ma[i]=mpMult(ma[i-1], ma[1]); |
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501 | MATELEM(s,1,i) = mpTrace(ma[i]); |
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502 | pTest(MATELEM(s,1,i)); |
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503 | } |
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504 | for (i=k+1; i<=n; i++) |
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505 | { |
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506 | MATELEM(s,1,i) = TraceOfProd(ma[i / 2], ma[(i+1) / 2], n); |
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507 | pTest(MATELEM(s,1,i)); |
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508 | } |
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509 | for (i=1; i<=k; i++) |
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510 | idDelete((ideal *)&(ma[i])); |
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511 | /* the array s contains the traces of the powers of the matrix m, |
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512 | * these are the power sums of the eigenvalues of m */ |
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513 | a = mpNew(1,n); |
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514 | MATELEM(a,1,1) = minuscopy(MATELEM(s,1,1)); |
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515 | for (i=2; i<=n; i++) |
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516 | { |
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517 | p = pCopy(MATELEM(s,1,i)); |
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518 | for (j=i-1; j>=1; j--) |
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519 | { |
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520 | q = ppMult_qq(MATELEM(s,1,j), MATELEM(a,1,i-j)); |
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521 | pTest(q); |
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522 | p = pAdd(p,q); |
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523 | } |
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524 | // c= -1/i |
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525 | d = nInit(-(int)i); |
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526 | c = nDiv(ONE, d); |
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527 | nDelete(&d); |
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528 | |
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529 | pMult_nn(p, c); |
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530 | pTest(p); |
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531 | MATELEM(a,1,i) = p; |
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532 | nDelete(&c); |
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533 | } |
---|
534 | /* the array a contains the elementary symmetric functions of the |
---|
535 | * eigenvalues of m */ |
---|
536 | for (i=1; i<=n-1; i++) |
---|
537 | { |
---|
538 | //pDelete(&(MATELEM(a,1,i))); |
---|
539 | pDelete(&(MATELEM(s,1,i))); |
---|
540 | } |
---|
541 | pDelete(&(MATELEM(s,1,n))); |
---|
542 | /* up to a sign, the determinant is the n-th elementary symmetric function */ |
---|
543 | if ((n/2)*2 < n) |
---|
544 | { |
---|
545 | d = nInit(-1); |
---|
546 | pMult_nn(MATELEM(a,1,n), d); |
---|
547 | nDelete(&d); |
---|
548 | } |
---|
549 | nDelete(&ONE); |
---|
550 | idDelete((ideal *)&s); |
---|
551 | poly result=MATELEM(a,1,n); |
---|
552 | MATELEM(a,1,n)=NULL; |
---|
553 | idDelete((ideal *)&a); |
---|
554 | return result; |
---|
555 | } |
---|
556 | |
---|
557 | /*2 |
---|
558 | * compute all ar-minors of the matrix a |
---|
559 | */ |
---|
560 | matrix mpWedge(matrix a, int ar) |
---|
561 | { |
---|
562 | int i,j,k,l; |
---|
563 | int *rowchoise,*colchoise; |
---|
564 | BOOLEAN rowch,colch; |
---|
565 | matrix result; |
---|
566 | matrix tmp; |
---|
567 | poly p; |
---|
568 | |
---|
569 | i = binom(a->nrows,ar); |
---|
570 | j = binom(a->ncols,ar); |
---|
571 | |
---|
572 | rowchoise=(int *)omAlloc(ar*sizeof(int)); |
---|
573 | colchoise=(int *)omAlloc(ar*sizeof(int)); |
---|
574 | result =mpNew(i,j); |
---|
575 | tmp=mpNew(ar,ar); |
---|
576 | l = 1; /* k,l:the index in result*/ |
---|
577 | idInitChoise(ar,1,a->nrows,&rowch,rowchoise); |
---|
578 | while (!rowch) |
---|
579 | { |
---|
580 | k=1; |
---|
581 | idInitChoise(ar,1,a->ncols,&colch,colchoise); |
---|
582 | while (!colch) |
---|
583 | { |
---|
584 | for (i=1; i<=ar; i++) |
---|
585 | { |
---|
586 | for (j=1; j<=ar; j++) |
---|
587 | { |
---|
588 | MATELEM(tmp,i,j) = MATELEM(a,rowchoise[i-1],colchoise[j-1]); |
---|
589 | } |
---|
590 | } |
---|
591 | p = mpDetBareiss(tmp); |
---|
592 | if ((k+l) & 1) p=pNeg(p); |
---|
593 | MATELEM(result,l,k) = p; |
---|
594 | k++; |
---|
595 | idGetNextChoise(ar,a->ncols,&colch,colchoise); |
---|
596 | } |
---|
597 | idGetNextChoise(ar,a->nrows,&rowch,rowchoise); |
---|
598 | l++; |
---|
599 | } |
---|
600 | /*delete the matrix tmp*/ |
---|
601 | for (i=1; i<=ar; i++) |
---|
602 | { |
---|
603 | for (j=1; j<=ar; j++) MATELEM(tmp,i,j) = NULL; |
---|
604 | } |
---|
605 | idDelete((ideal *) &tmp); |
---|
606 | return (result); |
---|
607 | } |
---|
608 | |
---|
609 | /*2 |
---|
610 | * compute the jacobi matrix of an ideal |
---|
611 | */ |
---|
612 | BOOLEAN mpJacobi(leftv res,leftv a) |
---|
613 | { |
---|
614 | int i,j; |
---|
615 | matrix result; |
---|
616 | ideal id=(ideal)a->Data(); |
---|
617 | |
---|
618 | result =mpNew(IDELEMS(id),pVariables); |
---|
619 | for (i=1; i<=IDELEMS(id); i++) |
---|
620 | { |
---|
621 | for (j=1; j<=pVariables; j++) |
---|
622 | { |
---|
623 | MATELEM(result,i,j) = pDiff(id->m[i-1],j); |
---|
624 | } |
---|
625 | } |
---|
626 | res->data=(char *)result; |
---|
627 | return FALSE; |
---|
628 | } |
---|
629 | |
---|
630 | /*2 |
---|
631 | * returns the Koszul-matrix of degree d of a vectorspace with dimension n |
---|
632 | * uses the first n entrees of id, if id <> NULL |
---|
633 | */ |
---|
634 | BOOLEAN mpKoszul(leftv res,leftv b/*in*/, leftv c/*ip*/,leftv id) |
---|
635 | { |
---|
636 | int n=(int)b->Data(); |
---|
637 | int d=(int)c->Data(); |
---|
638 | int k,l,sign,row,col; |
---|
639 | matrix result; |
---|
640 | ideal temp; |
---|
641 | BOOLEAN bo; |
---|
642 | poly p; |
---|
643 | |
---|
644 | if ((d>n) || (d<1) || (n<1)) |
---|
645 | { |
---|
646 | res->data=(char *)mpNew(1,1); |
---|
647 | return FALSE; |
---|
648 | } |
---|
649 | int *choise = (int*)omAlloc(d*sizeof(int)); |
---|
650 | if (id==NULL) |
---|
651 | temp=idMaxIdeal(1); |
---|
652 | else |
---|
653 | temp=(ideal)id->Data(); |
---|
654 | |
---|
655 | k = binom(n,d); |
---|
656 | l = k*d; |
---|
657 | l /= n-d+1; |
---|
658 | result =mpNew(l,k); |
---|
659 | col = 1; |
---|
660 | idInitChoise(d,1,n,&bo,choise); |
---|
661 | while (!bo) |
---|
662 | { |
---|
663 | sign = 1; |
---|
664 | for (l=1;l<=d;l++) |
---|
665 | { |
---|
666 | if (choise[l-1]<=IDELEMS(temp)) |
---|
667 | { |
---|
668 | p = pCopy(temp->m[choise[l-1]-1]); |
---|
669 | if (sign == -1) p = pNeg(p); |
---|
670 | sign *= -1; |
---|
671 | row = idGetNumberOfChoise(l-1,d,1,n,choise); |
---|
672 | MATELEM(result,row,col) = p; |
---|
673 | } |
---|
674 | } |
---|
675 | col++; |
---|
676 | idGetNextChoise(d,n,&bo,choise); |
---|
677 | } |
---|
678 | if (id==NULL) idDelete(&temp); |
---|
679 | |
---|
680 | res->data=(char *)result; |
---|
681 | return FALSE; |
---|
682 | } |
---|
683 | |
---|
684 | ///*2 |
---|
685 | //*homogenize all elements of matrix (not the matrix itself) |
---|
686 | //*/ |
---|
687 | //matrix mpHomogen(matrix a, int v) |
---|
688 | //{ |
---|
689 | // int i,j; |
---|
690 | // poly p; |
---|
691 | // |
---|
692 | // for (i=1;i<=MATROWS(a);i++) |
---|
693 | // { |
---|
694 | // for (j=1;j<=MATCOLS(a);j++) |
---|
695 | // { |
---|
696 | // p=pHomogen(MATELEM(a,i,j),v); |
---|
697 | // pDelete(&(MATELEM(a,i,j))); |
---|
698 | // MATELEM(a,i,j)=p; |
---|
699 | // } |
---|
700 | // } |
---|
701 | // return a; |
---|
702 | //} |
---|
703 | |
---|
704 | /*2 |
---|
705 | * corresponds to Maple's coeffs: |
---|
706 | * var has to be the number of a variable |
---|
707 | */ |
---|
708 | matrix mpCoeffs (ideal I, int var) |
---|
709 | { |
---|
710 | poly h,f; |
---|
711 | int l, i, c, m=0; |
---|
712 | matrix co; |
---|
713 | /* look for maximal power m of x_var in I */ |
---|
714 | for (i=IDELEMS(I)-1; i>=0; i--) |
---|
715 | { |
---|
716 | f=I->m[i]; |
---|
717 | while (f!=NULL) |
---|
718 | { |
---|
719 | l=pGetExp(f,var); |
---|
720 | if (l>m) m=l; |
---|
721 | pIter(f); |
---|
722 | } |
---|
723 | } |
---|
724 | co=mpNew((m+1)*I->rank,IDELEMS(I)); |
---|
725 | /* divide each monomial by a power of x_var, |
---|
726 | * remember the power in l and the component in c*/ |
---|
727 | for (i=IDELEMS(I)-1; i>=0; i--) |
---|
728 | { |
---|
729 | f=I->m[i]; |
---|
730 | while (f!=NULL) |
---|
731 | { |
---|
732 | l=pGetExp(f,var); |
---|
733 | pSetExp(f,var,0); |
---|
734 | c=max(pGetComp(f),1); |
---|
735 | pSetComp(f,0); |
---|
736 | pSetm(f); |
---|
737 | /* now add the resulting monomial to co*/ |
---|
738 | h=pNext(f); |
---|
739 | pNext(f)=NULL; |
---|
740 | //MATELEM(co,c*(m+1)-l,i+1) |
---|
741 | // =pAdd(MATELEM(co,c*(m+1)-l,i+1),f); |
---|
742 | MATELEM(co,(c-1)*(m+1)+l+1,i+1) |
---|
743 | =pAdd(MATELEM(co,(c-1)*(m+1)+l+1,i+1),f); |
---|
744 | /* iterate f*/ |
---|
745 | f=h; |
---|
746 | } |
---|
747 | } |
---|
748 | return co; |
---|
749 | } |
---|
750 | |
---|
751 | /*2 |
---|
752 | * given the result c of mpCoeffs(ideal/module i, var) |
---|
753 | * i of rank r |
---|
754 | * build the matrix of the corresponding monomials in m |
---|
755 | */ |
---|
756 | void mpMonomials(matrix c, int r, int var, matrix m) |
---|
757 | { |
---|
758 | /* clear contents of m*/ |
---|
759 | int k,l; |
---|
760 | for (k=MATROWS(m);k>0;k--) |
---|
761 | { |
---|
762 | for(l=MATCOLS(m);l>0;l--) |
---|
763 | { |
---|
764 | pDelete(&MATELEM(m,k,l)); |
---|
765 | } |
---|
766 | } |
---|
767 | omfreeSize((ADDRESS)m->m,MATROWS(m)*MATCOLS(m)*sizeof(poly)); |
---|
768 | /* allocate monoms in the right size r x MATROWS(c)*/ |
---|
769 | m->m=(polyset)omAlloc0(r*MATROWS(c)*sizeof(poly)); |
---|
770 | MATROWS(m)=r; |
---|
771 | MATCOLS(m)=MATROWS(c); |
---|
772 | m->rank=r; |
---|
773 | /* the maximal power p of x_var: MATCOLS(m)=r*(p+1) */ |
---|
774 | int p=MATCOLS(m)/r-1; |
---|
775 | /* fill in the powers of x_var=h*/ |
---|
776 | poly h=pOne(); |
---|
777 | for(k=r;k>0; k--) |
---|
778 | { |
---|
779 | MATELEM(m,k,k*(p+1))=pOne(); |
---|
780 | } |
---|
781 | for(l=p;l>0; l--) |
---|
782 | { |
---|
783 | pSetExp(h,var,l); |
---|
784 | pSetm(h); |
---|
785 | for(k=r;k>0; k--) |
---|
786 | { |
---|
787 | MATELEM(m,k,k*(p+1)-l)=pCopy(h); |
---|
788 | } |
---|
789 | } |
---|
790 | pDelete(&h); |
---|
791 | } |
---|
792 | |
---|
793 | matrix mpCoeffProc (poly f, poly vars) |
---|
794 | { |
---|
795 | poly sel, h; |
---|
796 | int l, i; |
---|
797 | int pos_of_1 = -1; |
---|
798 | matrix co; |
---|
799 | |
---|
800 | if (f==NULL) |
---|
801 | { |
---|
802 | co = mpNew(2, 1); |
---|
803 | MATELEM(co,1,1) = pOne(); |
---|
804 | MATELEM(co,2,1) = NULL; |
---|
805 | return co; |
---|
806 | } |
---|
807 | sel = mpSelect(f, vars); |
---|
808 | l = pLength(sel); |
---|
809 | co = mpNew(2, l); |
---|
810 | if (pOrdSgn==-1) |
---|
811 | { |
---|
812 | for (i=l; i>=1; i--) |
---|
813 | { |
---|
814 | h = sel; |
---|
815 | pIter(sel); |
---|
816 | pNext(h)=NULL; |
---|
817 | MATELEM(co,1,i) = h; |
---|
818 | MATELEM(co,2,i) = NULL; |
---|
819 | if (pIsConstant(h)) pos_of_1 = i; |
---|
820 | } |
---|
821 | } |
---|
822 | else |
---|
823 | { |
---|
824 | for (i=1; i<=l; i++) |
---|
825 | { |
---|
826 | h = sel; |
---|
827 | pIter(sel); |
---|
828 | pNext(h)=NULL; |
---|
829 | MATELEM(co,1,i) = h; |
---|
830 | MATELEM(co,2,i) = NULL; |
---|
831 | if (pIsConstant(h)) pos_of_1 = i; |
---|
832 | } |
---|
833 | } |
---|
834 | while (f!=NULL) |
---|
835 | { |
---|
836 | i = 1; |
---|
837 | loop |
---|
838 | { |
---|
839 | if (i!=pos_of_1) |
---|
840 | { |
---|
841 | h = mpExdiv(f, MATELEM(co,1,i)); |
---|
842 | if (h!=NULL) |
---|
843 | { |
---|
844 | MATELEM(co,2,i) = pAdd(MATELEM(co,2,i), h); |
---|
845 | break; |
---|
846 | } |
---|
847 | } |
---|
848 | if (i == l) |
---|
849 | { |
---|
850 | // check monom 1 last: |
---|
851 | if (pos_of_1 != -1) |
---|
852 | { |
---|
853 | h = mpExdiv(f, MATELEM(co,1,pos_of_1)); |
---|
854 | if (h!=NULL) |
---|
855 | { |
---|
856 | MATELEM(co,2,pos_of_1) = pAdd(MATELEM(co,2,pos_of_1), h); |
---|
857 | } |
---|
858 | } |
---|
859 | break; |
---|
860 | } |
---|
861 | i ++; |
---|
862 | } |
---|
863 | pIter(f); |
---|
864 | } |
---|
865 | return co; |
---|
866 | } |
---|
867 | |
---|
868 | /*2 |
---|
869 | *exact divisor: let d == x^i*y^j, m is thought to have only one term; |
---|
870 | * return m/d iff d divides m, and no x^k*y^l (k>i or l>j) divides m |
---|
871 | */ |
---|
872 | static poly mpExdiv ( poly m, poly d) |
---|
873 | { |
---|
874 | int i; |
---|
875 | poly h = pHead(m); |
---|
876 | for (i=1; i<=pVariables; i++) |
---|
877 | { |
---|
878 | if (pGetExp(d,i) > 0) |
---|
879 | { |
---|
880 | if (pGetExp(d,i) != pGetExp(h,i)) |
---|
881 | { |
---|
882 | pDelete(&h); |
---|
883 | return NULL; |
---|
884 | } |
---|
885 | pSetExp(h,i,0); |
---|
886 | } |
---|
887 | } |
---|
888 | pSetm(h); |
---|
889 | return h; |
---|
890 | } |
---|
891 | |
---|
892 | void mpCoef2(poly v, poly mon, matrix *c, matrix *m) |
---|
893 | { |
---|
894 | polyset s; |
---|
895 | poly p; |
---|
896 | int sl,i,j; |
---|
897 | int l=0; |
---|
898 | poly sel=mpSelect(v,mon); |
---|
899 | |
---|
900 | pVec2Polys(sel,&s,&sl); |
---|
901 | for (i=0; i<sl; i++) |
---|
902 | l=max(l,pLength(s[i])); |
---|
903 | *c=mpNew(sl,l); |
---|
904 | *m=mpNew(sl,l); |
---|
905 | poly h; |
---|
906 | int isConst; |
---|
907 | for (j=1; j<=sl;j++) |
---|
908 | { |
---|
909 | p=s[j-1]; |
---|
910 | if (pIsConstant(p)) /*p != NULL */ |
---|
911 | { |
---|
912 | isConst=-1; |
---|
913 | i=l; |
---|
914 | } |
---|
915 | else |
---|
916 | { |
---|
917 | isConst=1; |
---|
918 | i=1; |
---|
919 | } |
---|
920 | while(p!=NULL) |
---|
921 | { |
---|
922 | h = pHead(p); |
---|
923 | MATELEM(*m,j,i) = h; |
---|
924 | i+=isConst; |
---|
925 | p = p->next; |
---|
926 | } |
---|
927 | } |
---|
928 | while (v!=NULL) |
---|
929 | { |
---|
930 | i = 1; |
---|
931 | j = pGetComp(v); |
---|
932 | loop |
---|
933 | { |
---|
934 | poly mp=MATELEM(*m,j,i); |
---|
935 | if (mp!=NULL) |
---|
936 | { |
---|
937 | h = mpExdiv(v, mp /*MATELEM(*m,j,i)*/); |
---|
938 | if (h!=NULL) |
---|
939 | { |
---|
940 | pSetComp(h,0); |
---|
941 | MATELEM(*c,j,i) = pAdd(MATELEM(*c,j,i), h); |
---|
942 | break; |
---|
943 | } |
---|
944 | } |
---|
945 | if (i < l) |
---|
946 | i++; |
---|
947 | else |
---|
948 | break; |
---|
949 | } |
---|
950 | v = v->next; |
---|
951 | } |
---|
952 | } |
---|
953 | |
---|
954 | |
---|
955 | BOOLEAN mpEqual(matrix a, matrix b) |
---|
956 | { |
---|
957 | if ((MATCOLS(a)!=MATCOLS(b)) || (MATROWS(a)!=MATROWS(b))) |
---|
958 | return FALSE; |
---|
959 | int i=MATCOLS(a)*MATROWS(b)-1; |
---|
960 | while (i>=0) |
---|
961 | { |
---|
962 | if (a->m[i]==NULL) |
---|
963 | { |
---|
964 | if (b->m[i]!=NULL) return FALSE; |
---|
965 | } |
---|
966 | else |
---|
967 | if (pCmp(a->m[i],b->m[i])!=0) return FALSE; |
---|
968 | i--; |
---|
969 | } |
---|
970 | i=MATCOLS(a)*MATROWS(b)-1; |
---|
971 | while (i>=0) |
---|
972 | { |
---|
973 | if (pSub(pCopy(a->m[i]),pCopy(b->m[i]))!=NULL) return FALSE; |
---|
974 | i--; |
---|
975 | } |
---|
976 | return TRUE; |
---|
977 | } |
---|
978 | |
---|
979 | /* --------------- internal stuff ------------------- */ |
---|
980 | |
---|
981 | row_col_weight::row_col_weight(int i, int j) |
---|
982 | { |
---|
983 | ym = i; |
---|
984 | yn = j; |
---|
985 | wrow = (float *)omAlloc(i*sizeof(float)); |
---|
986 | wcol = (float *)omAlloc(j*sizeof(float)); |
---|
987 | } |
---|
988 | |
---|
989 | row_col_weight::~row_col_weight() |
---|
990 | { |
---|
991 | if (ym!=0) |
---|
992 | { |
---|
993 | omFreeSize((ADDRESS)wcol, yn*sizeof(float)); |
---|
994 | omFreeSize((ADDRESS)wrow, ym*sizeof(float)); |
---|
995 | } |
---|
996 | } |
---|
997 | |
---|
998 | mp_permmatrix::mp_permmatrix(matrix A) : sign(1) |
---|
999 | { |
---|
1000 | a_m = A->nrows; |
---|
1001 | a_n = A->ncols; |
---|
1002 | this->mpInitMat(); |
---|
1003 | Xarray = A->m; |
---|
1004 | } |
---|
1005 | |
---|
1006 | mp_permmatrix::mp_permmatrix(mp_permmatrix *M) |
---|
1007 | { |
---|
1008 | poly p, *athis, *aM; |
---|
1009 | int i, j; |
---|
1010 | |
---|
1011 | a_m = M->s_m; |
---|
1012 | a_n = M->s_n; |
---|
1013 | sign = M->sign; |
---|
1014 | this->mpInitMat(); |
---|
1015 | Xarray = (poly *)omAlloc0(a_m*a_n*sizeof(poly)); |
---|
1016 | for (i=a_m-1; i>=0; i--) |
---|
1017 | { |
---|
1018 | athis = this->mpRowAdr(i); |
---|
1019 | aM = M->mpRowAdr(i); |
---|
1020 | for (j=a_n-1; j>=0; j--) |
---|
1021 | { |
---|
1022 | p = aM[M->qcol[j]]; |
---|
1023 | if (p) |
---|
1024 | { |
---|
1025 | athis[j] = pCopy(p); |
---|
1026 | } |
---|
1027 | } |
---|
1028 | } |
---|
1029 | } |
---|
1030 | |
---|
1031 | mp_permmatrix::~mp_permmatrix() |
---|
1032 | { |
---|
1033 | int k; |
---|
1034 | |
---|
1035 | if (a_m != 0) |
---|
1036 | { |
---|
1037 | omFreeSize((ADDRESS)qrow,a_m*sizeof(int)); |
---|
1038 | omFreeSize((ADDRESS)qcol,a_n*sizeof(int)); |
---|
1039 | if (Xarray != NULL) |
---|
1040 | { |
---|
1041 | for (k=a_m*a_n-1; k>=0; k--) |
---|
1042 | pDelete(&Xarray[k]); |
---|
1043 | omFreeSize((ADDRESS)Xarray,a_m*a_n*sizeof(poly)); |
---|
1044 | } |
---|
1045 | } |
---|
1046 | } |
---|
1047 | |
---|
1048 | int mp_permmatrix::mpGetRdim() { return s_m; } |
---|
1049 | |
---|
1050 | int mp_permmatrix::mpGetCdim() { return s_n; } |
---|
1051 | |
---|
1052 | int mp_permmatrix::mpGetSign() { return sign; } |
---|
1053 | |
---|
1054 | void mp_permmatrix::mpSetSearch(int s) { piv_s = s; } |
---|
1055 | |
---|
1056 | void mp_permmatrix::mpSaveArray() { Xarray = NULL; } |
---|
1057 | |
---|
1058 | poly mp_permmatrix::mpGetElem(int r, int c) |
---|
1059 | { |
---|
1060 | return Xarray[a_n*qrow[r]+qcol[c]]; |
---|
1061 | } |
---|
1062 | |
---|
1063 | void mp_permmatrix::mpSetElem(poly p, int r, int c) |
---|
1064 | { |
---|
1065 | Xarray[a_n*qrow[r]+qcol[c]] = p; |
---|
1066 | } |
---|
1067 | |
---|
1068 | void mp_permmatrix::mpDelElem(int r, int c) |
---|
1069 | { |
---|
1070 | pDelete(&Xarray[a_n*qrow[r]+qcol[c]]); |
---|
1071 | } |
---|
1072 | |
---|
1073 | /* |
---|
1074 | * the Bareiss-type elimination with division by div (div != NULL) |
---|
1075 | */ |
---|
1076 | void mp_permmatrix::mpElimBareiss(poly div) |
---|
1077 | { |
---|
1078 | poly piv, elim, q1, q2, *ap, *a; |
---|
1079 | int i, j, jj; |
---|
1080 | |
---|
1081 | ap = this->mpRowAdr(s_m); |
---|
1082 | piv = ap[qcol[s_n]]; |
---|
1083 | for(i=s_m-1; i>=0; i--) |
---|
1084 | { |
---|
1085 | a = this->mpRowAdr(i); |
---|
1086 | elim = a[qcol[s_n]]; |
---|
1087 | if (elim != NULL) |
---|
1088 | { |
---|
1089 | elim = pNeg(elim); |
---|
1090 | for (j=s_n-1; j>=0; j--) |
---|
1091 | { |
---|
1092 | q2 = NULL; |
---|
1093 | jj = qcol[j]; |
---|
1094 | if (ap[jj] != NULL) |
---|
1095 | { |
---|
1096 | q2 = SM_MULT(ap[jj], elim, div); |
---|
1097 | if (a[jj] != NULL) |
---|
1098 | { |
---|
1099 | q1 = SM_MULT(a[jj], piv, div); |
---|
1100 | pDelete(&a[jj]); |
---|
1101 | q2 = pAdd(q2, q1); |
---|
1102 | } |
---|
1103 | } |
---|
1104 | else if (a[jj] != NULL) |
---|
1105 | { |
---|
1106 | q2 = SM_MULT(a[jj], piv, div); |
---|
1107 | } |
---|
1108 | if ((q2!=NULL) && div) |
---|
1109 | SM_DIV(q2, div); |
---|
1110 | a[jj] = q2; |
---|
1111 | } |
---|
1112 | pDelete(&a[qcol[s_n]]); |
---|
1113 | } |
---|
1114 | else |
---|
1115 | { |
---|
1116 | for (j=s_n-1; j>=0; j--) |
---|
1117 | { |
---|
1118 | jj = qcol[j]; |
---|
1119 | if (a[jj] != NULL) |
---|
1120 | { |
---|
1121 | q2 = SM_MULT(a[jj], piv, div); |
---|
1122 | pDelete(&a[jj]); |
---|
1123 | if (div) |
---|
1124 | SM_DIV(q2, div); |
---|
1125 | a[jj] = q2; |
---|
1126 | } |
---|
1127 | } |
---|
1128 | } |
---|
1129 | } |
---|
1130 | } |
---|
1131 | |
---|
1132 | /*2 |
---|
1133 | * pivot strategy for Bareiss algorithm |
---|
1134 | */ |
---|
1135 | int mp_permmatrix::mpPivotBareiss(row_col_weight *C) |
---|
1136 | { |
---|
1137 | poly p, *a; |
---|
1138 | int i, j, iopt, jopt; |
---|
1139 | float sum, f1, f2, fo, r, ro, lp; |
---|
1140 | float *dr = C->wrow, *dc = C->wcol; |
---|
1141 | |
---|
1142 | fo = 1.0e20; |
---|
1143 | ro = 0.0; |
---|
1144 | iopt = jopt = -1; |
---|
1145 | |
---|
1146 | s_n--; |
---|
1147 | s_m--; |
---|
1148 | if (s_m == 0) |
---|
1149 | return 0; |
---|
1150 | if (s_n == 0) |
---|
1151 | { |
---|
1152 | for(i=s_m; i>=0; i--) |
---|
1153 | { |
---|
1154 | p = this->mpRowAdr(i)[qcol[0]]; |
---|
1155 | if (p) |
---|
1156 | { |
---|
1157 | f1 = mpPolyWeight(p); |
---|
1158 | if (f1 < fo) |
---|
1159 | { |
---|
1160 | fo = f1; |
---|
1161 | if (iopt >= 0) |
---|
1162 | pDelete(&(this->mpRowAdr(iopt)[qcol[0]])); |
---|
1163 | iopt = i; |
---|
1164 | } |
---|
1165 | else |
---|
1166 | pDelete(&(this->mpRowAdr(i)[qcol[0]])); |
---|
1167 | } |
---|
1168 | } |
---|
1169 | if (iopt >= 0) |
---|
1170 | mpReplace(iopt, s_m, sign, qrow); |
---|
1171 | return 0; |
---|
1172 | } |
---|
1173 | this->mpRowWeight(dr); |
---|
1174 | this->mpColWeight(dc); |
---|
1175 | sum = 0.0; |
---|
1176 | for(i=s_m; i>=0; i--) |
---|
1177 | sum += dr[i]; |
---|
1178 | for(i=s_m; i>=0; i--) |
---|
1179 | { |
---|
1180 | r = dr[i]; |
---|
1181 | a = this->mpRowAdr(i); |
---|
1182 | for(j=s_n; j>=0; j--) |
---|
1183 | { |
---|
1184 | p = a[qcol[j]]; |
---|
1185 | if (p) |
---|
1186 | { |
---|
1187 | lp = mpPolyWeight(p); |
---|
1188 | ro = r - lp; |
---|
1189 | f1 = ro * (dc[j]-lp); |
---|
1190 | if (f1 != 0.0) |
---|
1191 | { |
---|
1192 | f2 = lp * (sum - ro - dc[j]); |
---|
1193 | f2 += f1; |
---|
1194 | } |
---|
1195 | else |
---|
1196 | f2 = lp-r-dc[j]; |
---|
1197 | if (f2 < fo) |
---|
1198 | { |
---|
1199 | fo = f2; |
---|
1200 | iopt = i; |
---|
1201 | jopt = j; |
---|
1202 | } |
---|
1203 | } |
---|
1204 | } |
---|
1205 | } |
---|
1206 | if (iopt < 0) |
---|
1207 | return 0; |
---|
1208 | mpReplace(iopt, s_m, sign, qrow); |
---|
1209 | mpReplace(jopt, s_n, sign, qcol); |
---|
1210 | return 1; |
---|
1211 | } |
---|
1212 | |
---|
1213 | /*2 |
---|
1214 | * pivot strategy for Bareiss algorithm with defined row |
---|
1215 | */ |
---|
1216 | int mp_permmatrix::mpPivotRow(row_col_weight *C, int row) |
---|
1217 | { |
---|
1218 | poly p, *a; |
---|
1219 | int j, iopt, jopt; |
---|
1220 | float sum, f1, f2, fo, r, ro, lp; |
---|
1221 | float *dr = C->wrow, *dc = C->wcol; |
---|
1222 | |
---|
1223 | fo = 1.0e20; |
---|
1224 | ro = 0.0; |
---|
1225 | iopt = jopt = -1; |
---|
1226 | |
---|
1227 | s_n--; |
---|
1228 | s_m--; |
---|
1229 | if (s_m == 0) |
---|
1230 | return 0; |
---|
1231 | if (s_n == 0) |
---|
1232 | { |
---|
1233 | p = this->mpRowAdr(row)[qcol[0]]; |
---|
1234 | if (p) |
---|
1235 | { |
---|
1236 | f1 = mpPolyWeight(p); |
---|
1237 | if (f1 < fo) |
---|
1238 | { |
---|
1239 | fo = f1; |
---|
1240 | if (iopt >= 0) |
---|
1241 | pDelete(&(this->mpRowAdr(iopt)[qcol[0]])); |
---|
1242 | iopt = row; |
---|
1243 | } |
---|
1244 | else |
---|
1245 | pDelete(&(this->mpRowAdr(row)[qcol[0]])); |
---|
1246 | } |
---|
1247 | if (iopt >= 0) |
---|
1248 | mpReplace(iopt, s_m, sign, qrow); |
---|
1249 | return 0; |
---|
1250 | } |
---|
1251 | this->mpRowWeight(dr); |
---|
1252 | this->mpColWeight(dc); |
---|
1253 | sum = 0.0; |
---|
1254 | for(j=s_m; j>=0; j--) |
---|
1255 | sum += dr[j]; |
---|
1256 | r = dr[row]; |
---|
1257 | a = this->mpRowAdr(row); |
---|
1258 | for(j=s_n; j>=0; j--) |
---|
1259 | { |
---|
1260 | p = a[qcol[j]]; |
---|
1261 | if (p) |
---|
1262 | { |
---|
1263 | lp = mpPolyWeight(p); |
---|
1264 | ro = r - lp; |
---|
1265 | f1 = ro * (dc[j]-lp); |
---|
1266 | if (f1 != 0.0) |
---|
1267 | { |
---|
1268 | f2 = lp * (sum - ro - dc[j]); |
---|
1269 | f2 += f1; |
---|
1270 | } |
---|
1271 | else |
---|
1272 | f2 = lp-r-dc[j]; |
---|
1273 | if (f2 < fo) |
---|
1274 | { |
---|
1275 | fo = f2; |
---|
1276 | iopt = row; |
---|
1277 | jopt = j; |
---|
1278 | } |
---|
1279 | } |
---|
1280 | } |
---|
1281 | if (iopt < 0) |
---|
1282 | return 0; |
---|
1283 | mpReplace(iopt, s_m, sign, qrow); |
---|
1284 | mpReplace(jopt, s_n, sign, qcol); |
---|
1285 | return 1; |
---|
1286 | } |
---|
1287 | |
---|
1288 | void mp_permmatrix::mpToIntvec(intvec *v) |
---|
1289 | { |
---|
1290 | int i; |
---|
1291 | |
---|
1292 | for (i=v->rows()-1; i>=0; i--) |
---|
1293 | (*v)[i] = qcol[i]+1; |
---|
1294 | } |
---|
1295 | |
---|
1296 | void mp_permmatrix::mpRowReorder() |
---|
1297 | { |
---|
1298 | int k, i, i1, i2; |
---|
1299 | |
---|
1300 | if (a_m > a_n) |
---|
1301 | k = a_m - a_n; |
---|
1302 | else |
---|
1303 | k = 0; |
---|
1304 | for (i=a_m-1; i>=k; i--) |
---|
1305 | { |
---|
1306 | i1 = qrow[i]; |
---|
1307 | if (i1 != i) |
---|
1308 | { |
---|
1309 | this->mpRowSwap(i1, i); |
---|
1310 | i2 = 0; |
---|
1311 | while (qrow[i2] != i) i2++; |
---|
1312 | qrow[i2] = i1; |
---|
1313 | } |
---|
1314 | } |
---|
1315 | } |
---|
1316 | |
---|
1317 | void mp_permmatrix::mpColReorder() |
---|
1318 | { |
---|
1319 | int k, j, j1, j2; |
---|
1320 | |
---|
1321 | if (a_n > a_m) |
---|
1322 | k = a_n - a_m; |
---|
1323 | else |
---|
1324 | k = 0; |
---|
1325 | for (j=a_n-1; j>=k; j--) |
---|
1326 | { |
---|
1327 | j1 = qcol[j]; |
---|
1328 | if (j1 != j) |
---|
1329 | { |
---|
1330 | this->mpColSwap(j1, j); |
---|
1331 | j2 = 0; |
---|
1332 | while (qcol[j2] != j) j2++; |
---|
1333 | qcol[j2] = j1; |
---|
1334 | } |
---|
1335 | } |
---|
1336 | } |
---|
1337 | |
---|
1338 | // private |
---|
1339 | void mp_permmatrix::mpInitMat() |
---|
1340 | { |
---|
1341 | int k; |
---|
1342 | |
---|
1343 | s_m = a_m; |
---|
1344 | s_n = a_n; |
---|
1345 | piv_s = 0; |
---|
1346 | qrow = (int *)omAlloc(a_m*sizeof(int)); |
---|
1347 | qcol = (int *)omAlloc(a_n*sizeof(int)); |
---|
1348 | for (k=a_m-1; k>=0; k--) qrow[k] = k; |
---|
1349 | for (k=a_n-1; k>=0; k--) qcol[k] = k; |
---|
1350 | } |
---|
1351 | |
---|
1352 | poly * mp_permmatrix::mpRowAdr(int r) |
---|
1353 | { |
---|
1354 | return &(Xarray[a_n*qrow[r]]); |
---|
1355 | } |
---|
1356 | |
---|
1357 | poly * mp_permmatrix::mpColAdr(int c) |
---|
1358 | { |
---|
1359 | return &(Xarray[qcol[c]]); |
---|
1360 | } |
---|
1361 | |
---|
1362 | void mp_permmatrix::mpRowWeight(float *wrow) |
---|
1363 | { |
---|
1364 | poly p, *a; |
---|
1365 | int i, j; |
---|
1366 | float count; |
---|
1367 | |
---|
1368 | for (i=s_m; i>=0; i--) |
---|
1369 | { |
---|
1370 | a = this->mpRowAdr(i); |
---|
1371 | count = 0.0; |
---|
1372 | for(j=s_n; j>=0; j--) |
---|
1373 | { |
---|
1374 | p = a[qcol[j]]; |
---|
1375 | if (p) |
---|
1376 | count += mpPolyWeight(p); |
---|
1377 | } |
---|
1378 | wrow[i] = count; |
---|
1379 | } |
---|
1380 | } |
---|
1381 | |
---|
1382 | void mp_permmatrix::mpColWeight(float *wcol) |
---|
1383 | { |
---|
1384 | poly p, *a; |
---|
1385 | int i, j; |
---|
1386 | float count; |
---|
1387 | |
---|
1388 | for (j=s_n; j>=0; j--) |
---|
1389 | { |
---|
1390 | a = this->mpColAdr(j); |
---|
1391 | count = 0.0; |
---|
1392 | for(i=s_m; i>=0; i--) |
---|
1393 | { |
---|
1394 | p = a[a_n*qrow[i]]; |
---|
1395 | if (p) |
---|
1396 | count += mpPolyWeight(p); |
---|
1397 | } |
---|
1398 | wcol[j] = count; |
---|
1399 | } |
---|
1400 | } |
---|
1401 | |
---|
1402 | void mp_permmatrix::mpRowSwap(int i1, int i2) |
---|
1403 | { |
---|
1404 | poly p, *a1, *a2; |
---|
1405 | int j; |
---|
1406 | |
---|
1407 | a1 = &(Xarray[a_n*i1]); |
---|
1408 | a2 = &(Xarray[a_n*i2]); |
---|
1409 | for (j=a_n-1; j>= 0; j--) |
---|
1410 | { |
---|
1411 | p = a1[j]; |
---|
1412 | a1[j] = a2[j]; |
---|
1413 | a2[j] = p; |
---|
1414 | } |
---|
1415 | } |
---|
1416 | |
---|
1417 | void mp_permmatrix::mpColSwap(int j1, int j2) |
---|
1418 | { |
---|
1419 | poly p, *a1, *a2; |
---|
1420 | int i, k = a_n*a_m; |
---|
1421 | |
---|
1422 | a1 = &(Xarray[j1]); |
---|
1423 | a2 = &(Xarray[j2]); |
---|
1424 | for (i=0; i< k; i+=a_n) |
---|
1425 | { |
---|
1426 | p = a1[i]; |
---|
1427 | a1[i] = a2[i]; |
---|
1428 | a2[i] = p; |
---|
1429 | } |
---|
1430 | } |
---|
1431 | |
---|
1432 | int mp_permmatrix::mpGetRow() |
---|
1433 | { |
---|
1434 | return qrow[s_m]; |
---|
1435 | } |
---|
1436 | |
---|
1437 | int mp_permmatrix::mpGetCol() |
---|
1438 | { |
---|
1439 | return qcol[s_n]; |
---|
1440 | } |
---|
1441 | |
---|
1442 | /* |
---|
1443 | * perform replacement for pivot strategy in Bareiss algorithm |
---|
1444 | * change sign of determinant |
---|
1445 | */ |
---|
1446 | static void mpReplace(int j, int n, int &sign, int *perm) |
---|
1447 | { |
---|
1448 | int k; |
---|
1449 | |
---|
1450 | if (j != n) |
---|
1451 | { |
---|
1452 | k = perm[n]; |
---|
1453 | perm[n] = perm[j]; |
---|
1454 | perm[j] = k; |
---|
1455 | sign = -sign; |
---|
1456 | } |
---|
1457 | } |
---|
1458 | |
---|
1459 | static int mpNextperm(perm * z, int max) |
---|
1460 | { |
---|
1461 | int s, i, k, t; |
---|
1462 | s = max; |
---|
1463 | do |
---|
1464 | { |
---|
1465 | s--; |
---|
1466 | } |
---|
1467 | while ((s > 0) && ((*z)[s] >= (*z)[s+1])); |
---|
1468 | if (s==0) |
---|
1469 | return 0; |
---|
1470 | do |
---|
1471 | { |
---|
1472 | (*z)[s]++; |
---|
1473 | k = 0; |
---|
1474 | do |
---|
1475 | { |
---|
1476 | k++; |
---|
1477 | } |
---|
1478 | while (((*z)[k] != (*z)[s]) && (k!=s)); |
---|
1479 | } |
---|
1480 | while (k < s); |
---|
1481 | for (i=s+1; i <= max; i++) |
---|
1482 | { |
---|
1483 | (*z)[i]=0; |
---|
1484 | do |
---|
1485 | { |
---|
1486 | (*z)[i]++; |
---|
1487 | k=0; |
---|
1488 | do |
---|
1489 | { |
---|
1490 | k++; |
---|
1491 | } |
---|
1492 | while (((*z)[k] != (*z)[i]) && (k != i)); |
---|
1493 | } |
---|
1494 | while (k < i); |
---|
1495 | } |
---|
1496 | s = max+1; |
---|
1497 | do |
---|
1498 | { |
---|
1499 | s--; |
---|
1500 | } |
---|
1501 | while ((s > 0) && ((*z)[s] > (*z)[s+1])); |
---|
1502 | t = 1; |
---|
1503 | for (i=1; i<max; i++) |
---|
1504 | for (k=i+1; k<=max; k++) |
---|
1505 | if ((*z)[k] < (*z)[i]) |
---|
1506 | t = -t; |
---|
1507 | (*z)[0] = t; |
---|
1508 | return s; |
---|
1509 | } |
---|
1510 | |
---|
1511 | static poly mpLeibnitz(matrix a) |
---|
1512 | { |
---|
1513 | int i, e, n; |
---|
1514 | poly p, d; |
---|
1515 | perm z; |
---|
1516 | |
---|
1517 | n = MATROWS(a); |
---|
1518 | memset(&z,0,(n+2)*sizeof(int)); |
---|
1519 | p = pOne(); |
---|
1520 | for (i=1; i <= n; i++) |
---|
1521 | p = pMult(p, pCopy(MATELEM(a, i, i))); |
---|
1522 | d = p; |
---|
1523 | for (i=1; i<= n; i++) |
---|
1524 | z[i] = i; |
---|
1525 | z[0]=1; |
---|
1526 | e = 1; |
---|
1527 | if (n!=1) |
---|
1528 | { |
---|
1529 | while (e) |
---|
1530 | { |
---|
1531 | e = mpNextperm((perm *)&z, n); |
---|
1532 | p = pOne(); |
---|
1533 | for (i = 1; i <= n; i++) |
---|
1534 | p = pMult(p, pCopy(MATELEM(a, i, z[i]))); |
---|
1535 | if (z[0] > 0) |
---|
1536 | d = pAdd(d, p); |
---|
1537 | else |
---|
1538 | d = pSub(d, p); |
---|
1539 | } |
---|
1540 | } |
---|
1541 | return d; |
---|
1542 | } |
---|
1543 | |
---|
1544 | static poly minuscopy (poly p) |
---|
1545 | { |
---|
1546 | poly w; |
---|
1547 | number e; |
---|
1548 | e = nInit(-1); |
---|
1549 | w = pCopy(p); |
---|
1550 | pMult_nn(w, e); |
---|
1551 | nDelete(&e); |
---|
1552 | return w; |
---|
1553 | } |
---|
1554 | |
---|
1555 | /*2 |
---|
1556 | * insert a monomial into a list, avoid duplicates |
---|
1557 | * arguments are destroyed |
---|
1558 | */ |
---|
1559 | static poly pInsert(poly p1, poly p2) |
---|
1560 | { |
---|
1561 | poly a1, p, a2, a; |
---|
1562 | int c; |
---|
1563 | |
---|
1564 | if (p1==NULL) return p2; |
---|
1565 | if (p2==NULL) return p1; |
---|
1566 | a1 = p1; |
---|
1567 | a2 = p2; |
---|
1568 | a = p = pOne(); |
---|
1569 | loop |
---|
1570 | { |
---|
1571 | c = pCmp(a1, a2); |
---|
1572 | if (c == 1) |
---|
1573 | { |
---|
1574 | a = pNext(a) = a1; |
---|
1575 | pIter(a1); |
---|
1576 | if (a1==NULL) |
---|
1577 | { |
---|
1578 | pNext(a) = a2; |
---|
1579 | break; |
---|
1580 | } |
---|
1581 | } |
---|
1582 | else if (c == -1) |
---|
1583 | { |
---|
1584 | a = pNext(a) = a2; |
---|
1585 | pIter(a2); |
---|
1586 | if (a2==NULL) |
---|
1587 | { |
---|
1588 | pNext(a) = a1; |
---|
1589 | break; |
---|
1590 | } |
---|
1591 | } |
---|
1592 | else |
---|
1593 | { |
---|
1594 | pDeleteLm(&a2); |
---|
1595 | a = pNext(a) = a1; |
---|
1596 | pIter(a1); |
---|
1597 | if (a1==NULL) |
---|
1598 | { |
---|
1599 | pNext(a) = a2; |
---|
1600 | break; |
---|
1601 | } |
---|
1602 | else if (a2==NULL) |
---|
1603 | { |
---|
1604 | pNext(a) = a1; |
---|
1605 | break; |
---|
1606 | } |
---|
1607 | } |
---|
1608 | } |
---|
1609 | pDeleteLm(&p); |
---|
1610 | return p; |
---|
1611 | } |
---|
1612 | |
---|
1613 | /*2 |
---|
1614 | *if what == xy the result is the list of all different power products |
---|
1615 | * x^i*y^j (i, j >= 0) that appear in fro |
---|
1616 | */ |
---|
1617 | static poly mpSelect (poly fro, poly what) |
---|
1618 | { |
---|
1619 | int i; |
---|
1620 | poly h, res; |
---|
1621 | res = NULL; |
---|
1622 | while (fro!=NULL) |
---|
1623 | { |
---|
1624 | h = pOne(); |
---|
1625 | for (i=1; i<=pVariables; i++) |
---|
1626 | pSetExp(h,i, pGetExp(fro,i) * pGetExp(what, i)); |
---|
1627 | pSetComp(h, pGetComp(fro)); |
---|
1628 | pSetm(h); |
---|
1629 | res = pInsert(h, res); |
---|
1630 | fro = fro->next; |
---|
1631 | } |
---|
1632 | return res; |
---|
1633 | } |
---|
1634 | |
---|
1635 | /* |
---|
1636 | *static void ppp(matrix a) |
---|
1637 | *{ |
---|
1638 | * int j,i,r=a->nrows,c=a->ncols; |
---|
1639 | * for(j=1;j<=r;j++) |
---|
1640 | * { |
---|
1641 | * for(i=1;i<=c;i++) |
---|
1642 | * { |
---|
1643 | * if(MATELEM(a,j,i)!=NULL) Print("X"); |
---|
1644 | * else Print("0"); |
---|
1645 | * } |
---|
1646 | * Print("\n"); |
---|
1647 | * } |
---|
1648 | *} |
---|
1649 | */ |
---|
1650 | |
---|
1651 | static void mpPartClean(matrix a, int lr, int lc) |
---|
1652 | { |
---|
1653 | poly *q1; |
---|
1654 | int i,j; |
---|
1655 | |
---|
1656 | for (i=lr-1;i>=0;i--) |
---|
1657 | { |
---|
1658 | q1 = &(a->m)[i*a->ncols]; |
---|
1659 | for (j=lc-1;j>=0;j--) if(q1[j]) pDelete(&q1[j]); |
---|
1660 | } |
---|
1661 | } |
---|
1662 | |
---|
1663 | static void mpFinalClean(matrix a) |
---|
1664 | { |
---|
1665 | omFreeSize((ADDRESS)a->m,a->nrows*a->ncols*sizeof(poly)); |
---|
1666 | omFreeBin((ADDRESS)a, ip_smatrix_bin); |
---|
1667 | } |
---|
1668 | |
---|
1669 | /*2 |
---|
1670 | * prepare one step of 'Bareiss' algorithm |
---|
1671 | * for application in minor |
---|
1672 | */ |
---|
1673 | static int mpPrepareRow (matrix a, int lr, int lc) |
---|
1674 | { |
---|
1675 | int r; |
---|
1676 | |
---|
1677 | r = mpPivBar(a,lr,lc); |
---|
1678 | if(r==0) return 0; |
---|
1679 | if(r<lr) mpSwapRow(a, r, lr, lc); |
---|
1680 | return 1; |
---|
1681 | } |
---|
1682 | |
---|
1683 | /*2 |
---|
1684 | * prepare one step of 'Bareiss' algorithm |
---|
1685 | * for application in minor |
---|
1686 | */ |
---|
1687 | static int mpPreparePiv (matrix a, int lr, int lc) |
---|
1688 | { |
---|
1689 | int c; |
---|
1690 | |
---|
1691 | c = mpPivRow(a, lr, lc); |
---|
1692 | if(c==0) return 0; |
---|
1693 | if(c<lc) mpSwapCol(a, c, lr, lc); |
---|
1694 | return 1; |
---|
1695 | } |
---|
1696 | |
---|
1697 | /* |
---|
1698 | * find best row |
---|
1699 | */ |
---|
1700 | static int mpPivBar(matrix a, int lr, int lc) |
---|
1701 | { |
---|
1702 | float f1, f2; |
---|
1703 | poly *q1; |
---|
1704 | int i,j,io; |
---|
1705 | |
---|
1706 | io = -1; |
---|
1707 | f1 = 1.0e30; |
---|
1708 | for (i=lr-1;i>=0;i--) |
---|
1709 | { |
---|
1710 | q1 = &(a->m)[i*a->ncols]; |
---|
1711 | f2 = 0.0; |
---|
1712 | for (j=lc-1;j>=0;j--) |
---|
1713 | { |
---|
1714 | if (q1[j]!=NULL) |
---|
1715 | f2 += mpPolyWeight(q1[j]); |
---|
1716 | } |
---|
1717 | if ((f2!=0.0) && (f2<f1)) |
---|
1718 | { |
---|
1719 | f1 = f2; |
---|
1720 | io = i; |
---|
1721 | } |
---|
1722 | } |
---|
1723 | if (io<0) return 0; |
---|
1724 | else return io+1; |
---|
1725 | } |
---|
1726 | |
---|
1727 | /* |
---|
1728 | * find pivot in the last row |
---|
1729 | */ |
---|
1730 | static int mpPivRow(matrix a, int lr, int lc) |
---|
1731 | { |
---|
1732 | float f1, f2; |
---|
1733 | poly *q1; |
---|
1734 | int j,jo; |
---|
1735 | |
---|
1736 | jo = -1; |
---|
1737 | f1 = 1.0e30; |
---|
1738 | q1 = &(a->m)[(lr-1)*a->ncols]; |
---|
1739 | for (j=lc-1;j>=0;j--) |
---|
1740 | { |
---|
1741 | if (q1[j]!=NULL) |
---|
1742 | { |
---|
1743 | f2 = mpPolyWeight(q1[j]); |
---|
1744 | if (f2<f1) |
---|
1745 | { |
---|
1746 | f1 = f2; |
---|
1747 | jo = j; |
---|
1748 | } |
---|
1749 | } |
---|
1750 | } |
---|
1751 | if (jo<0) return 0; |
---|
1752 | else return jo+1; |
---|
1753 | } |
---|
1754 | |
---|
1755 | /* |
---|
1756 | * weigth of a polynomial, for pivot strategy |
---|
1757 | */ |
---|
1758 | static float mpPolyWeight(poly p) |
---|
1759 | { |
---|
1760 | int i; |
---|
1761 | float res; |
---|
1762 | |
---|
1763 | if (pNext(p) == NULL) |
---|
1764 | { |
---|
1765 | res = (float)nSize(pGetCoeff(p)); |
---|
1766 | for (i=pVariables;i>0;i--) |
---|
1767 | { |
---|
1768 | if(pGetExp(p,i)!=0) |
---|
1769 | { |
---|
1770 | res += 2.0; |
---|
1771 | break; |
---|
1772 | } |
---|
1773 | } |
---|
1774 | } |
---|
1775 | else |
---|
1776 | { |
---|
1777 | res = 0.0; |
---|
1778 | do |
---|
1779 | { |
---|
1780 | res += (float)nSize(pGetCoeff(p))+2.0; |
---|
1781 | pIter(p); |
---|
1782 | } |
---|
1783 | while (p); |
---|
1784 | } |
---|
1785 | return res; |
---|
1786 | } |
---|
1787 | |
---|
1788 | static void mpSwapRow(matrix a, int pos, int lr, int lc) |
---|
1789 | { |
---|
1790 | poly sw; |
---|
1791 | int j; |
---|
1792 | polyset a2 = a->m, a1 = &a2[a->ncols*(pos-1)]; |
---|
1793 | |
---|
1794 | a2 = &a2[a->ncols*(lr-1)]; |
---|
1795 | for (j=lc-1; j>=0; j--) |
---|
1796 | { |
---|
1797 | sw = a1[j]; |
---|
1798 | a1[j] = a2[j]; |
---|
1799 | a2[j] = sw; |
---|
1800 | } |
---|
1801 | } |
---|
1802 | |
---|
1803 | static void mpSwapCol(matrix a, int pos, int lr, int lc) |
---|
1804 | { |
---|
1805 | poly sw; |
---|
1806 | int j; |
---|
1807 | polyset a2 = a->m, a1 = &a2[pos-1]; |
---|
1808 | |
---|
1809 | a2 = &a2[lc-1]; |
---|
1810 | for (j=a->ncols*(lr-1); j>=0; j-=a->ncols) |
---|
1811 | { |
---|
1812 | sw = a1[j]; |
---|
1813 | a1[j] = a2[j]; |
---|
1814 | a2[j] = sw; |
---|
1815 | } |
---|
1816 | } |
---|
1817 | |
---|
1818 | static void mpElimBar(matrix a0, matrix re, poly div, int lr, int lc) |
---|
1819 | { |
---|
1820 | int r=lr-1, c=lc-1; |
---|
1821 | poly *b = a0->m, *x = re->m; |
---|
1822 | poly piv, elim, q1, q2, *ap, *a, *q; |
---|
1823 | int i, j; |
---|
1824 | |
---|
1825 | ap = &b[r*a0->ncols]; |
---|
1826 | piv = ap[c]; |
---|
1827 | for(j=c-1; j>=0; j--) |
---|
1828 | if (ap[j] != NULL) ap[j] = pNeg(ap[j]); |
---|
1829 | for(i=r-1; i>=0; i--) |
---|
1830 | { |
---|
1831 | a = &b[i*a0->ncols]; |
---|
1832 | q = &x[i*re->ncols]; |
---|
1833 | if (a[c] != NULL) |
---|
1834 | { |
---|
1835 | elim = a[c]; |
---|
1836 | for (j=c-1; j>=0; j--) |
---|
1837 | { |
---|
1838 | q1 = NULL; |
---|
1839 | if (a[j] != NULL) |
---|
1840 | { |
---|
1841 | q1 = SM_MULT(a[j], piv, div); |
---|
1842 | if (ap[j] != NULL) |
---|
1843 | { |
---|
1844 | q2 = SM_MULT(ap[j], elim, div); |
---|
1845 | q1 = pAdd(q1,q2); |
---|
1846 | } |
---|
1847 | } |
---|
1848 | else if (ap[j] != NULL) |
---|
1849 | q1 = SM_MULT(ap[j], elim, div); |
---|
1850 | if (q1 != NULL) |
---|
1851 | { |
---|
1852 | if (div) |
---|
1853 | SM_DIV(q1, div); |
---|
1854 | q[j] = q1; |
---|
1855 | } |
---|
1856 | } |
---|
1857 | } |
---|
1858 | else |
---|
1859 | { |
---|
1860 | for (j=c-1; j>=0; j--) |
---|
1861 | { |
---|
1862 | if (a[j] != NULL) |
---|
1863 | { |
---|
1864 | q1 = SM_MULT(a[j], piv, div); |
---|
1865 | if (div) |
---|
1866 | SM_DIV(q1, div); |
---|
1867 | q[j] = q1; |
---|
1868 | } |
---|
1869 | } |
---|
1870 | } |
---|
1871 | } |
---|
1872 | } |
---|