1 | /**************************************** |
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2 | * Computer Algebra System SINGULAR * |
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3 | ****************************************/ |
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4 | /* $Id$ */ |
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5 | /* |
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6 | * ABSTRACT: numbers modulo n |
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7 | */ |
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8 | |
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9 | #include <string.h> |
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10 | #include <kernel/mod2.h> |
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11 | #include <omalloc/mylimits.h> |
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12 | #include <kernel/structs.h> |
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13 | #include <kernel/febase.h> |
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14 | #include <omalloc/omalloc.h> |
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15 | #include <kernel/numbers.h> |
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16 | #include <kernel/longrat.h> |
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17 | #include <kernel/mpr_complex.h> |
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18 | #include <kernel/ring.h> |
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19 | #include <kernel/rmodulon.h> |
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20 | #include <kernel/si_gmp.h> |
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21 | |
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22 | #ifdef HAVE_RINGS |
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23 | extern omBin gmp_nrz_bin; |
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24 | |
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25 | int_number nrnMinusOne = NULL; |
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26 | unsigned long nrnExponent = 0; |
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27 | |
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28 | /* |
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29 | * create a number from int |
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30 | */ |
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31 | number nrnInit (int i, const ring r) |
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32 | { |
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33 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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34 | mpz_init_set_si(erg, i); |
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35 | mpz_mod(erg, erg, r->nrnModul); |
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36 | return (number) erg; |
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37 | } |
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38 | |
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39 | void nrnDelete(number *a, const ring r) |
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40 | { |
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41 | if (*a == NULL) return; |
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42 | mpz_clear((int_number) *a); |
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43 | omFreeBin((ADDRESS) *a, gmp_nrz_bin); |
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44 | *a = NULL; |
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45 | } |
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46 | |
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47 | number nrnCopy(number a) |
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48 | { |
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49 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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50 | mpz_init_set(erg, (int_number) a); |
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51 | return (number) erg; |
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52 | } |
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53 | |
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54 | number cfrnCopy(number a, const ring r) |
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55 | { |
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56 | return nrnCopy(a); |
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57 | } |
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58 | |
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59 | int nrnSize(number a) |
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60 | { |
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61 | if (a == NULL) return 0; |
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62 | return sizeof(mpz_t); |
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63 | } |
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64 | |
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65 | /* |
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66 | * convert a number to int |
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67 | */ |
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68 | int nrnInt(number &n, const ring r) |
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69 | { |
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70 | return (int) mpz_get_si( (int_number) n); |
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71 | } |
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72 | |
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73 | /* |
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74 | * Multiply two numbers |
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75 | */ |
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76 | number nrnMult (number a, number b) |
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77 | { |
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78 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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79 | mpz_init(erg); |
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80 | mpz_mul(erg, (int_number) a, (int_number) b); |
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81 | mpz_mod(erg, erg, currRing->nrnModul); |
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82 | return (number) erg; |
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83 | } |
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84 | |
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85 | void nrnPower (number a, int i, number * result) |
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86 | { |
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87 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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88 | mpz_init(erg); |
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89 | mpz_powm_ui(erg, (int_number) a, i, currRing->nrnModul); |
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90 | *result = (number) erg; |
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91 | } |
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92 | |
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93 | number nrnAdd (number a, number b) |
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94 | { |
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95 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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96 | mpz_init(erg); |
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97 | mpz_add(erg, (int_number) a, (int_number) b); |
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98 | mpz_mod(erg, erg, currRing->nrnModul); |
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99 | return (number) erg; |
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100 | } |
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101 | |
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102 | number nrnSub (number a, number b) |
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103 | { |
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104 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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105 | mpz_init(erg); |
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106 | mpz_sub(erg, (int_number) a, (int_number) b); |
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107 | mpz_mod(erg, erg, currRing->nrnModul); |
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108 | return (number) erg; |
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109 | } |
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110 | |
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111 | number nrnNeg (number c) |
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112 | { |
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113 | // nNeg inplace !!! |
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114 | mpz_sub((int_number) c, currRing->nrnModul, (int_number) c); |
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115 | return c; |
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116 | } |
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117 | |
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118 | number nrnInvers (number c) |
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119 | { |
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120 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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121 | mpz_init(erg); |
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122 | mpz_invert(erg, (int_number) c, currRing->nrnModul); |
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123 | return (number) erg; |
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124 | } |
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125 | |
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126 | /* |
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127 | * Give the smallest non unit k, such that a * x = k = b * y has a solution |
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128 | * TODO: lcm(gcd,gcd) besser als gcd(lcm) ? |
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129 | */ |
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130 | number nrnLcm (number a,number b,ring r) |
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131 | { |
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132 | number erg = nrnGcd(NULL, a, r); |
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133 | number tmp = nrnGcd(NULL, b, r); |
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134 | mpz_lcm((int_number) erg, (int_number) erg, (int_number) tmp); |
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135 | nrnDelete(&tmp, NULL); |
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136 | return (number) erg; |
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137 | } |
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138 | |
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139 | /* |
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140 | * Give the largest non unit k, such that a = x * k, b = y * k has |
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141 | * a solution. |
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142 | */ |
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143 | number nrnGcd (number a,number b,ring r) |
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144 | { |
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145 | if ((a == NULL) && (b == NULL)) return nrnInit(0,r); |
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146 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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147 | mpz_init_set(erg, r->nrnModul); |
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148 | if (a != NULL) mpz_gcd(erg, erg, (int_number) a); |
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149 | if (b != NULL) mpz_gcd(erg, erg, (int_number) b); |
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150 | return (number) erg; |
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151 | } |
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152 | |
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153 | /* Not needed any more, but may have room for improvement |
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154 | number nrnGcd3 (number a,number b, number c,ring r) |
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155 | { |
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156 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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157 | mpz_init(erg); |
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158 | if (a == NULL) a = (number) r->nrnModul; |
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159 | if (b == NULL) b = (number) r->nrnModul; |
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160 | if (c == NULL) c = (number) r->nrnModul; |
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161 | mpz_gcd(erg, (int_number) a, (int_number) b); |
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162 | mpz_gcd(erg, erg, (int_number) c); |
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163 | mpz_gcd(erg, erg, r->nrnModul); |
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164 | return (number) erg; |
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165 | } |
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166 | */ |
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167 | |
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168 | /* |
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169 | * Give the largest non unit k, such that a = x * k, b = y * k has |
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170 | * a solution and r, s, s.t. k = s*a + t*b |
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171 | */ |
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172 | number nrnExtGcd (number a, number b, number *s, number *t) |
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173 | { |
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174 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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175 | int_number bs = (int_number) omAllocBin(gmp_nrz_bin); |
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176 | int_number bt = (int_number) omAllocBin(gmp_nrz_bin); |
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177 | mpz_init(erg); |
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178 | mpz_init(bs); |
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179 | mpz_init(bt); |
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180 | mpz_gcdext(erg, bs, bt, (int_number) a, (int_number) b); |
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181 | mpz_mod(bs, bs, currRing->nrnModul); |
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182 | mpz_mod(bt, bt, currRing->nrnModul); |
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183 | *s = (number) bs; |
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184 | *t = (number) bt; |
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185 | return (number) erg; |
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186 | } |
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187 | |
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188 | BOOLEAN nrnIsZero (number a) |
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189 | { |
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190 | #ifdef LDEBUG |
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191 | if (a == NULL) return FALSE; |
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192 | #endif |
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193 | return 0 == mpz_cmpabs_ui((int_number) a, 0); |
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194 | } |
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195 | |
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196 | BOOLEAN nrnIsOne (number a) |
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197 | { |
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198 | #ifdef LDEBUG |
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199 | if (a == NULL) return FALSE; |
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200 | #endif |
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201 | return 0 == mpz_cmp_si((int_number) a, 1); |
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202 | } |
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203 | |
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204 | BOOLEAN nrnIsMOne (number a) |
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205 | { |
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206 | #ifdef LDEBUG |
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207 | if (a == NULL) return FALSE; |
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208 | #endif |
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209 | return 0 == mpz_cmp((int_number) a, nrnMinusOne); |
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210 | } |
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211 | |
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212 | BOOLEAN nrnEqual (number a,number b) |
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213 | { |
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214 | return 0 == mpz_cmp((int_number) a, (int_number) b); |
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215 | } |
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216 | |
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217 | BOOLEAN nrnGreater (number a,number b) |
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218 | { |
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219 | return 0 < mpz_cmp((int_number) a, (int_number) b); |
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220 | } |
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221 | |
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222 | BOOLEAN nrnGreaterZero (number k) |
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223 | { |
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224 | return 0 < mpz_cmp_si((int_number) k, 0); |
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225 | } |
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226 | |
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227 | BOOLEAN nrnIsUnit (number a) |
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228 | { |
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229 | number tmp = nrnGcd(a, (number) currRing->nrnModul, currRing); |
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230 | bool res = nrnIsOne(tmp); |
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231 | nrnDelete(&tmp, NULL); |
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232 | return res; |
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233 | } |
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234 | |
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235 | number nrnGetUnit (number k) |
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236 | { |
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237 | if (mpz_divisible_p(currRing->nrnModul, (int_number) k)) return nrnInit(1,currRing); |
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238 | |
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239 | int_number unit = (int_number) nrnGcd(k, 0, currRing); |
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240 | mpz_tdiv_q(unit, (int_number) k, unit); |
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241 | int_number gcd = (int_number) nrnGcd((number) unit, 0, currRing); |
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242 | if (!nrnIsOne((number) gcd)) |
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243 | { |
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244 | int_number ctmp; |
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245 | // tmp := unit^2 |
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246 | int_number tmp = (int_number) nrnMult((number) unit,(number) unit); |
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247 | // gcd_new := gcd(tmp, 0) |
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248 | int_number gcd_new = (int_number) nrnGcd((number) tmp, 0, currRing); |
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249 | while (!nrnEqual((number) gcd_new,(number) gcd)) |
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250 | { |
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251 | // gcd := gcd_new |
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252 | ctmp = gcd; |
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253 | gcd = gcd_new; |
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254 | gcd_new = ctmp; |
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255 | // tmp := tmp * unit |
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256 | mpz_mul(tmp, tmp, unit); |
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257 | mpz_mod(tmp, tmp, currRing->nrnModul); |
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258 | // gcd_new := gcd(tmp, 0) |
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259 | mpz_gcd(gcd_new, tmp, currRing->nrnModul); |
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260 | } |
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261 | // unit := unit + nrnModul / gcd_new |
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262 | mpz_tdiv_q(tmp, currRing->nrnModul, gcd_new); |
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263 | mpz_add(unit, unit, tmp); |
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264 | mpz_mod(unit, unit, currRing->nrnModul); |
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265 | nrnDelete((number*) &gcd_new, NULL); |
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266 | nrnDelete((number*) &tmp, NULL); |
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267 | } |
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268 | nrnDelete((number*) &gcd, NULL); |
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269 | return (number) unit; |
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270 | } |
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271 | |
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272 | BOOLEAN nrnDivBy (number a,number b) |
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273 | { |
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274 | if (a == NULL) |
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275 | return mpz_divisible_p(currRing->nrnModul, (int_number) b); |
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276 | else |
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277 | return mpz_divisible_p((int_number) a, (int_number) b); |
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278 | /* |
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279 | number bs = nrnGcd(a, b, currRing); |
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280 | mpz_tdiv_q((int_number) bs, (int_number) b, (int_number) bs); |
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281 | bool res = nrnIsUnit(bs); |
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282 | nrnDelete(&bs, NULL); |
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283 | return res; |
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284 | */ |
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285 | } |
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286 | |
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287 | int nrnDivComp(number a, number b) |
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288 | { |
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289 | if (nrnEqual(a, b)) return 2; |
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290 | if (mpz_divisible_p((int_number) a, (int_number) b)) return -1; |
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291 | if (mpz_divisible_p((int_number) b, (int_number) a)) return 1; |
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292 | return 0; |
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293 | } |
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294 | |
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295 | number nrnDiv (number a,number b) |
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296 | { |
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297 | if (a == NULL) a = (number) currRing->nrnModul; |
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298 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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299 | mpz_init(erg); |
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300 | if (mpz_divisible_p((int_number) a, (int_number) b)) |
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301 | { |
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302 | mpz_divexact(erg, (int_number) a, (int_number) b); |
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303 | return (number) erg; |
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304 | } |
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305 | else |
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306 | { |
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307 | int_number gcd = (int_number) nrnGcd(a, b, currRing); |
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308 | mpz_divexact(erg, (int_number) b, gcd); |
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309 | if (!nrnIsUnit((number) erg)) |
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310 | { |
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311 | WerrorS("Division not possible, even by cancelling zero divisors."); |
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312 | WerrorS("Result is integer division without remainder."); |
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313 | mpz_tdiv_q(erg, (int_number) a, (int_number) b); |
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314 | nrnDelete((number*) &gcd, NULL); |
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315 | return (number) erg; |
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316 | } |
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317 | // a / gcd(a,b) * [b / gcd (a,b)]^(-1) |
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318 | int_number tmp = (int_number) nrnInvers((number) erg); |
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319 | mpz_divexact(erg, (int_number) a, gcd); |
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320 | mpz_mul(erg, erg, tmp); |
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321 | nrnDelete((number*) &gcd, NULL); |
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322 | nrnDelete((number*) &tmp, NULL); |
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323 | mpz_mod(erg, erg, currRing->nrnModul); |
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324 | return (number) erg; |
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325 | } |
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326 | } |
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327 | |
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328 | number nrnMod (number a, number b) |
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329 | { |
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330 | /* |
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331 | We need to return the number r which is uniquely determined by the |
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332 | following two properties: |
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333 | (1) 0 <= r < |b| (with respect to '<' and '<=' performed in Z x Z) |
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334 | (2) There exists some k in the integers Z such that a = k * b + r. |
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335 | Consider g := gcd(n, |b|). Note that then |b|/g is a unit in Z/n. |
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336 | Now, there are three cases: |
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337 | (a) g = 1 |
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338 | Then |b| is a unit in Z/n, i.e. |b| (and also b) divides a. |
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339 | Thus r = 0. |
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340 | (b) g <> 1 and g divides a |
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341 | Then a = (a/g) * (|b|/g)^(-1) * b (up to sign), i.e. again r = 0. |
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342 | (c) g <> 1 and g does not divide a |
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343 | Then denote the division with remainder of a by g as this: |
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344 | a = s * g + t. Then t = a - s * g = a - s * (|b|/g)^(-1) * |b| |
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345 | fulfills (1) and (2), i.e. r := t is the correct result. Hence |
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346 | in this third case, r is the remainder of division of a by g in Z. |
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347 | Remark: according to mpz_mod: a,b are always non-negative |
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348 | */ |
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349 | int_number g = (int_number) omAllocBin(gmp_nrz_bin); |
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350 | int_number r = (int_number) omAllocBin(gmp_nrz_bin); |
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351 | mpz_init(g); |
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352 | mpz_init_set_si(r,(long)0); |
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353 | mpz_gcd(g, (int_number) currRing->nrnModul, (int_number)b); // g is now as above |
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354 | if (mpz_cmp_si(g, (long)1) != 0) mpz_mod(r, (int_number)a, g); // the case g <> 1 |
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355 | mpz_clear(g); |
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356 | omFreeBin(g, gmp_nrz_bin); |
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357 | return (number)r; |
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358 | } |
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359 | |
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360 | number nrnIntDiv (number a,number b) |
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361 | { |
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362 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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363 | mpz_init(erg); |
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364 | if (a == NULL) a = (number) currRing->nrnModul; |
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365 | mpz_tdiv_q(erg, (int_number) a, (int_number) b); |
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366 | return (number) erg; |
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367 | } |
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368 | |
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369 | /* |
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370 | * Helper function for computing the module |
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371 | */ |
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372 | |
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373 | int_number nrnMapCoef = NULL; |
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374 | |
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375 | number nrnMapModN(number from) |
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376 | { |
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377 | return nrnMult(from, (number) nrnMapCoef); |
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378 | } |
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379 | |
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380 | number nrnMap2toM(number from) |
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381 | { |
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382 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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383 | mpz_init(erg); |
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384 | mpz_mul_ui(erg, nrnMapCoef, (NATNUMBER) from); |
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385 | mpz_mod(erg, erg, currRing->nrnModul); |
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386 | return (number) erg; |
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387 | } |
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388 | |
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389 | number nrnMapZp(number from) |
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390 | { |
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391 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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392 | mpz_init(erg); |
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393 | mpz_mul_si(erg, nrnMapCoef, (NATNUMBER) from); |
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394 | mpz_mod(erg, erg, currRing->nrnModul); |
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395 | return (number) erg; |
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396 | } |
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397 | |
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398 | number nrnMapGMP(number from) |
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399 | { |
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400 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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401 | mpz_init(erg); |
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402 | mpz_mod(erg, (int_number) from, currRing->nrnModul); |
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403 | return (number) erg; |
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404 | } |
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405 | |
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406 | number nrnMapQ(number from) |
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407 | { |
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408 | int_number erg = (int_number) omAllocBin(gmp_nrz_bin); |
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409 | mpz_init(erg); |
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410 | nlGMP(from, (number) erg); |
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411 | mpz_mod(erg, erg, currRing->nrnModul); |
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412 | return (number) erg; |
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413 | } |
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414 | |
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415 | nMapFunc nrnSetMap(const ring src, const ring dst) |
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416 | { |
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417 | /* dst = currRing */ |
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418 | if (rField_is_Ring_Z(src)) |
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419 | { |
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420 | return nrnMapGMP; |
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421 | } |
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422 | if (rField_is_Q(src)) |
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423 | { |
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424 | return nrnMapQ; |
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425 | } |
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426 | // Some type of Z/n ring / field |
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427 | if (rField_is_Ring_ModN(src) || rField_is_Ring_PtoM(src) || rField_is_Ring_2toM(src) || rField_is_Zp(src)) |
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428 | { |
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429 | if ( (src->ringtype > 0) |
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430 | && (mpz_cmp(src->ringflaga, dst->ringflaga) == 0) |
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431 | && (src->ringflagb == dst->ringflagb)) return nrnMapGMP; |
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432 | else |
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433 | { |
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434 | int_number nrnMapModul = (int_number) omAllocBin(gmp_nrz_bin); |
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435 | // Computing the n of Z/n |
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436 | if (rField_is_Zp(src)) |
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437 | { |
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438 | mpz_init_set_si(nrnMapModul, src->ch); |
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439 | } |
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440 | else |
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441 | { |
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442 | mpz_init(nrnMapModul); |
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443 | mpz_set(nrnMapModul, src->ringflaga); |
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444 | mpz_pow_ui(nrnMapModul, nrnMapModul, src->ringflagb); |
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445 | } |
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446 | // nrnMapCoef = 1 in dst if dst is a subring of src |
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447 | // nrnMapCoef = 0 in dst / src if src is a subring of dst |
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448 | if (nrnMapCoef == NULL) |
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449 | { |
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450 | nrnMapCoef = (int_number) omAllocBin(gmp_nrz_bin); |
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451 | mpz_init(nrnMapCoef); |
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452 | } |
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453 | if (mpz_divisible_p(nrnMapModul, currRing->nrnModul)) |
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454 | { |
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455 | mpz_set_si(nrnMapCoef, 1); |
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456 | } |
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457 | else |
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458 | if (nrnDivBy(NULL, (number) nrnMapModul)) |
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459 | { |
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460 | mpz_divexact(nrnMapCoef, currRing->nrnModul, nrnMapModul); |
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461 | int_number tmp = currRing->nrnModul; |
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462 | currRing->nrnModul = nrnMapModul; |
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463 | if (!nrnIsUnit((number) nrnMapCoef)) |
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464 | { |
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465 | currRing->nrnModul = tmp; |
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466 | nrnDelete((number*) &nrnMapModul, currRing); |
---|
467 | return NULL; |
---|
468 | } |
---|
469 | int_number inv = (int_number) nrnInvers((number) nrnMapCoef); |
---|
470 | currRing->nrnModul = tmp; |
---|
471 | mpz_mul(nrnMapCoef, nrnMapCoef, inv); |
---|
472 | mpz_mod(nrnMapCoef, nrnMapCoef, currRing->nrnModul); |
---|
473 | nrnDelete((number*) &inv, currRing); |
---|
474 | } |
---|
475 | else |
---|
476 | { |
---|
477 | nrnDelete((number*) &nrnMapModul, currRing); |
---|
478 | return NULL; |
---|
479 | } |
---|
480 | nrnDelete((number*) &nrnMapModul, currRing); |
---|
481 | if (rField_is_Ring_2toM(src)) |
---|
482 | return nrnMap2toM; |
---|
483 | else if (rField_is_Zp(src)) |
---|
484 | return nrnMapZp; |
---|
485 | else |
---|
486 | return nrnMapModN; |
---|
487 | } |
---|
488 | } |
---|
489 | return NULL; // default |
---|
490 | } |
---|
491 | |
---|
492 | /* |
---|
493 | * set the exponent (allocate and init tables) (TODO) |
---|
494 | */ |
---|
495 | |
---|
496 | void nrnSetExp(int m, ring r) |
---|
497 | { |
---|
498 | if ((r->nrnModul != NULL) && (mpz_cmp(r->nrnModul, r->ringflaga) == 0) && (nrnExponent == r->ringflagb)) return; |
---|
499 | |
---|
500 | nrnExponent = r->ringflagb; |
---|
501 | if (r->nrnModul == NULL) |
---|
502 | { |
---|
503 | r->nrnModul = (int_number) omAllocBin(gmp_nrz_bin); |
---|
504 | mpz_init(r->nrnModul); |
---|
505 | nrnMinusOne = (int_number) omAllocBin(gmp_nrz_bin); |
---|
506 | mpz_init(nrnMinusOne); |
---|
507 | } |
---|
508 | mpz_set(r->nrnModul, r->ringflaga); |
---|
509 | mpz_pow_ui(r->nrnModul, r->nrnModul, nrnExponent); |
---|
510 | mpz_sub_ui(nrnMinusOne, r->nrnModul, 1); |
---|
511 | } |
---|
512 | |
---|
513 | void nrnInitExp(int m, ring r) |
---|
514 | { |
---|
515 | nrnSetExp(m, r); |
---|
516 | |
---|
517 | if (mpz_cmp_ui(r->nrnModul,2) <= 0) |
---|
518 | { |
---|
519 | WarnS("nrnInitExp failed"); |
---|
520 | } |
---|
521 | } |
---|
522 | |
---|
523 | #ifdef LDEBUG |
---|
524 | BOOLEAN nrnDBTest (number a, const char *f, const int l) |
---|
525 | { |
---|
526 | if (a==NULL) return TRUE; |
---|
527 | if ( (mpz_cmp_si((int_number) a, 0) < 0) || (mpz_cmp((int_number) a, currRing->nrnModul) > 0) ) |
---|
528 | { |
---|
529 | return FALSE; |
---|
530 | } |
---|
531 | return TRUE; |
---|
532 | } |
---|
533 | #endif |
---|
534 | |
---|
535 | /*2 |
---|
536 | * extracts a long integer from s, returns the rest (COPY FROM longrat0.cc) |
---|
537 | */ |
---|
538 | static const char * nlCPEatLongC(char *s, mpz_ptr i) |
---|
539 | { |
---|
540 | const char * start=s; |
---|
541 | if (!(*s >= '0' && *s <= '9')) |
---|
542 | { |
---|
543 | mpz_init_set_si(i, 1); |
---|
544 | return s; |
---|
545 | } |
---|
546 | mpz_init(i); |
---|
547 | while (*s >= '0' && *s <= '9') s++; |
---|
548 | if (*s=='\0') |
---|
549 | { |
---|
550 | mpz_set_str(i,start,10); |
---|
551 | } |
---|
552 | else |
---|
553 | { |
---|
554 | char c=*s; |
---|
555 | *s='\0'; |
---|
556 | mpz_set_str(i,start,10); |
---|
557 | *s=c; |
---|
558 | } |
---|
559 | return s; |
---|
560 | } |
---|
561 | |
---|
562 | const char * nrnRead (const char *s, number *a) |
---|
563 | { |
---|
564 | int_number z = (int_number) omAllocBin(gmp_nrz_bin); |
---|
565 | { |
---|
566 | s = nlCPEatLongC((char *)s, z); |
---|
567 | } |
---|
568 | mpz_mod(z, z, currRing->nrnModul); |
---|
569 | *a = (number) z; |
---|
570 | return s; |
---|
571 | } |
---|
572 | #endif |
---|