[0d5627] | 1 | #include "config.h" |
---|
[4dfcb1] | 2 | |
---|
[9c6887] | 3 | #ifndef NOSTREAMIO |
---|
[4dfcb1] | 4 | #ifdef HAVE_CSTDIO |
---|
| 5 | #include <cstdio> |
---|
| 6 | #else |
---|
[c99b6b] | 7 | #include <stdio.h> |
---|
[4dfcb1] | 8 | #endif |
---|
| 9 | #ifdef HAVE_IOSTREAM_H |
---|
[c99b6b] | 10 | #include <iostream.h> |
---|
[4dd2c4] | 11 | #elif defined(HAVE_IOSTREAM) |
---|
[4dfcb1] | 12 | #include <iostream> |
---|
| 13 | #endif |
---|
[4dd2c4] | 14 | #endif |
---|
[c99b6b] | 15 | |
---|
[517530] | 16 | #include "cf_assert.h" |
---|
| 17 | |
---|
[fe2d4c] | 18 | #include "templates/ftmpl_functions.h" |
---|
[c99b6b] | 19 | #include "cf_defs.h" |
---|
| 20 | #include "canonicalform.h" |
---|
| 21 | #include "cf_iter.h" |
---|
| 22 | #include "cf_primes.h" |
---|
| 23 | #include "cf_algorithm.h" |
---|
| 24 | #include "algext.h" |
---|
[359d742] | 25 | #include "cf_map.h" |
---|
| 26 | #include "cf_generator.h" |
---|
[c99b6b] | 27 | |
---|
[2156ec] | 28 | #ifdef HAVE_NTL |
---|
| 29 | #include "NTLconvert.h" |
---|
| 30 | #endif |
---|
| 31 | |
---|
[fe2d4c] | 32 | /// compressing two polynomials F and G, M is used for compressing, |
---|
| 33 | /// N to reverse the compression |
---|
| 34 | static |
---|
| 35 | int myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
---|
| 36 | CFMap & N, bool topLevel) |
---|
| 37 | { |
---|
| 38 | int n= tmax (F.level(), G.level()); |
---|
| 39 | int * degsf= new int [n + 1]; |
---|
| 40 | int * degsg= new int [n + 1]; |
---|
| 41 | |
---|
| 42 | for (int i = 0; i <= n; i++) |
---|
| 43 | degsf[i]= degsg[i]= 0; |
---|
| 44 | |
---|
| 45 | degsf= degrees (F, degsf); |
---|
| 46 | degsg= degrees (G, degsg); |
---|
| 47 | |
---|
| 48 | int both_non_zero= 0; |
---|
| 49 | int f_zero= 0; |
---|
| 50 | int g_zero= 0; |
---|
| 51 | int both_zero= 0; |
---|
| 52 | |
---|
| 53 | if (topLevel) |
---|
| 54 | { |
---|
| 55 | for (int i= 1; i <= n; i++) |
---|
| 56 | { |
---|
| 57 | if (degsf[i] != 0 && degsg[i] != 0) |
---|
| 58 | { |
---|
| 59 | both_non_zero++; |
---|
| 60 | continue; |
---|
| 61 | } |
---|
| 62 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
---|
| 63 | { |
---|
| 64 | f_zero++; |
---|
| 65 | continue; |
---|
| 66 | } |
---|
| 67 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
---|
| 68 | { |
---|
| 69 | g_zero++; |
---|
| 70 | continue; |
---|
| 71 | } |
---|
| 72 | } |
---|
| 73 | |
---|
| 74 | if (both_non_zero == 0) |
---|
| 75 | { |
---|
| 76 | delete [] degsf; |
---|
| 77 | delete [] degsg; |
---|
| 78 | return 0; |
---|
| 79 | } |
---|
| 80 | |
---|
| 81 | // map Variables which do not occur in both polynomials to higher levels |
---|
| 82 | int k= 1; |
---|
| 83 | int l= 1; |
---|
| 84 | for (int i= 1; i <= n; i++) |
---|
| 85 | { |
---|
| 86 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
---|
| 87 | { |
---|
| 88 | if (k + both_non_zero != i) |
---|
| 89 | { |
---|
| 90 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
---|
| 91 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
---|
| 92 | } |
---|
| 93 | k++; |
---|
| 94 | } |
---|
| 95 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
---|
| 96 | { |
---|
| 97 | if (l + g_zero + both_non_zero != i) |
---|
| 98 | { |
---|
| 99 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
---|
| 100 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
---|
| 101 | } |
---|
| 102 | l++; |
---|
| 103 | } |
---|
| 104 | } |
---|
| 105 | |
---|
| 106 | // sort Variables x_{i} in increasing order of |
---|
| 107 | // min(deg_{x_{i}}(f),deg_{x_{i}}(g)) |
---|
| 108 | int m= tmax (F.level(), G.level()); |
---|
| 109 | int min_max_deg; |
---|
| 110 | k= both_non_zero; |
---|
| 111 | l= 0; |
---|
| 112 | int i= 1; |
---|
| 113 | while (k > 0) |
---|
| 114 | { |
---|
| 115 | if (degsf [i] != 0 && degsg [i] != 0) |
---|
| 116 | min_max_deg= tmax (degsf[i], degsg[i]); |
---|
| 117 | else |
---|
| 118 | min_max_deg= 0; |
---|
| 119 | while (min_max_deg == 0) |
---|
| 120 | { |
---|
| 121 | i++; |
---|
| 122 | min_max_deg= tmax (degsf[i], degsg[i]); |
---|
| 123 | if (degsf [i] != 0 && degsg [i] != 0) |
---|
| 124 | min_max_deg= tmax (degsf[i], degsg[i]); |
---|
| 125 | else |
---|
| 126 | min_max_deg= 0; |
---|
| 127 | } |
---|
| 128 | for (int j= i + 1; j <= m; j++) |
---|
| 129 | { |
---|
| 130 | if (tmax (degsf[j],degsg[j]) <= min_max_deg && degsf[j] != 0 && degsg [j] != 0) |
---|
| 131 | { |
---|
| 132 | min_max_deg= tmax (degsf[j], degsg[j]); |
---|
| 133 | l= j; |
---|
| 134 | } |
---|
| 135 | } |
---|
| 136 | if (l != 0) |
---|
| 137 | { |
---|
| 138 | if (l != k) |
---|
| 139 | { |
---|
| 140 | M.newpair (Variable (l), Variable(k)); |
---|
| 141 | N.newpair (Variable (k), Variable(l)); |
---|
| 142 | degsf[l]= 0; |
---|
| 143 | degsg[l]= 0; |
---|
| 144 | l= 0; |
---|
| 145 | } |
---|
| 146 | else |
---|
| 147 | { |
---|
| 148 | degsf[l]= 0; |
---|
| 149 | degsg[l]= 0; |
---|
| 150 | l= 0; |
---|
| 151 | } |
---|
| 152 | } |
---|
| 153 | else if (l == 0) |
---|
| 154 | { |
---|
| 155 | if (i != k) |
---|
| 156 | { |
---|
| 157 | M.newpair (Variable (i), Variable (k)); |
---|
| 158 | N.newpair (Variable (k), Variable (i)); |
---|
| 159 | degsf[i]= 0; |
---|
| 160 | degsg[i]= 0; |
---|
| 161 | } |
---|
| 162 | else |
---|
| 163 | { |
---|
| 164 | degsf[i]= 0; |
---|
| 165 | degsg[i]= 0; |
---|
| 166 | } |
---|
| 167 | i++; |
---|
| 168 | } |
---|
| 169 | k--; |
---|
| 170 | } |
---|
| 171 | } |
---|
| 172 | else |
---|
| 173 | { |
---|
| 174 | //arrange Variables such that no gaps occur |
---|
| 175 | for (int i= 1; i <= n; i++) |
---|
| 176 | { |
---|
| 177 | if (degsf[i] == 0 && degsg[i] == 0) |
---|
| 178 | { |
---|
| 179 | both_zero++; |
---|
| 180 | continue; |
---|
| 181 | } |
---|
| 182 | else |
---|
| 183 | { |
---|
| 184 | if (both_zero != 0) |
---|
| 185 | { |
---|
| 186 | M.newpair (Variable (i), Variable (i - both_zero)); |
---|
| 187 | N.newpair (Variable (i - both_zero), Variable (i)); |
---|
| 188 | } |
---|
| 189 | } |
---|
| 190 | } |
---|
| 191 | } |
---|
| 192 | |
---|
| 193 | delete [] degsf; |
---|
| 194 | delete [] degsg; |
---|
| 195 | |
---|
| 196 | return 1; |
---|
| 197 | } |
---|
| 198 | |
---|
[ad8e1b] | 199 | void tryInvert( const CanonicalForm & F, const CanonicalForm & M, CanonicalForm & inv, bool & fail ) |
---|
| 200 | { // F, M are required to be "univariate" polynomials in an algebraic variable |
---|
| 201 | // we try to invert F modulo M |
---|
| 202 | if(F.inBaseDomain()) |
---|
| 203 | { |
---|
| 204 | if(F.isZero()) |
---|
| 205 | { |
---|
| 206 | fail = true; |
---|
| 207 | return; |
---|
| 208 | } |
---|
| 209 | inv = 1/F; |
---|
| 210 | return; |
---|
| 211 | } |
---|
| 212 | CanonicalForm b; |
---|
| 213 | Variable a = M.mvar(); |
---|
| 214 | Variable x = Variable(1); |
---|
| 215 | if(!extgcd( replacevar( F, a, x ), replacevar( M, a, x ), inv, b ).isOne()) |
---|
| 216 | fail = true; |
---|
| 217 | else |
---|
| 218 | inv = replacevar( inv, x, a ); // change back to alg var |
---|
| 219 | } |
---|
| 220 | |
---|
[a8e8b9] | 221 | void tryDivrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
| 222 | CanonicalForm& R, CanonicalForm& inv, const CanonicalForm& mipo, |
---|
| 223 | bool& fail) |
---|
| 224 | { |
---|
| 225 | if (F.inCoeffDomain()) |
---|
| 226 | { |
---|
| 227 | Q= 0; |
---|
| 228 | R= F; |
---|
| 229 | return; |
---|
| 230 | } |
---|
| 231 | |
---|
| 232 | CanonicalForm A, B; |
---|
| 233 | Variable x= F.mvar(); |
---|
| 234 | A= F; |
---|
| 235 | B= G; |
---|
| 236 | int degA= degree (A, x); |
---|
| 237 | int degB= degree (B, x); |
---|
| 238 | |
---|
| 239 | if (degA < degB) |
---|
| 240 | { |
---|
| 241 | R= A; |
---|
| 242 | Q= 0; |
---|
| 243 | return; |
---|
| 244 | } |
---|
| 245 | |
---|
| 246 | tryInvert (Lc (B), mipo, inv, fail); |
---|
| 247 | if (fail) |
---|
| 248 | return; |
---|
| 249 | |
---|
| 250 | R= A; |
---|
| 251 | Q= 0; |
---|
| 252 | CanonicalForm Qi; |
---|
| 253 | for (int i= degA -degB; i >= 0; i--) |
---|
| 254 | { |
---|
| 255 | if (degree (R, x) == i + degB) |
---|
| 256 | { |
---|
| 257 | Qi= Lc (R)*inv*power (x, i); |
---|
| 258 | Qi= reduce (Qi, mipo); |
---|
| 259 | R -= Qi*B; |
---|
| 260 | R= reduce (R, mipo); |
---|
| 261 | Q += Qi; |
---|
| 262 | } |
---|
| 263 | } |
---|
| 264 | } |
---|
| 265 | |
---|
[ad8e1b] | 266 | void tryEuclid( const CanonicalForm & A, const CanonicalForm & B, const CanonicalForm & M, CanonicalForm & result, bool & fail ) |
---|
[c99b6b] | 267 | { |
---|
| 268 | CanonicalForm P; |
---|
[ad8e1b] | 269 | if(A.inCoeffDomain()) |
---|
| 270 | { |
---|
| 271 | tryInvert( A, M, P, fail ); |
---|
| 272 | if(fail) |
---|
| 273 | return; |
---|
| 274 | result = 1; |
---|
| 275 | return; |
---|
| 276 | } |
---|
| 277 | if(B.inCoeffDomain()) |
---|
| 278 | { |
---|
| 279 | tryInvert( B, M, P, fail ); |
---|
| 280 | if(fail) |
---|
| 281 | return; |
---|
| 282 | result = 1; |
---|
| 283 | return; |
---|
| 284 | } |
---|
| 285 | // here: both not inCoeffDomain |
---|
| 286 | if( A.degree() > B.degree() ) |
---|
[c99b6b] | 287 | { |
---|
| 288 | P = A; result = B; |
---|
| 289 | } |
---|
| 290 | else |
---|
| 291 | { |
---|
| 292 | P = B; result = A; |
---|
| 293 | } |
---|
| 294 | CanonicalForm inv; |
---|
| 295 | if( result.isZero() ) |
---|
| 296 | { |
---|
| 297 | tryInvert( Lc(P), M, inv, fail ); |
---|
| 298 | if(fail) |
---|
| 299 | return; |
---|
[ad8e1b] | 300 | result = inv*P; // monify result (not reduced, yet) |
---|
[5df7d0] | 301 | result= reduce (result, M); |
---|
[c99b6b] | 302 | return; |
---|
| 303 | } |
---|
[ad8e1b] | 304 | Variable x = P.mvar(); |
---|
[a8e8b9] | 305 | CanonicalForm rem, Q; |
---|
[c99b6b] | 306 | // here: degree(P) >= degree(result) |
---|
| 307 | while(true) |
---|
| 308 | { |
---|
[a8e8b9] | 309 | tryDivrem (P, result, Q, rem, inv, M, fail); |
---|
| 310 | if (fail) |
---|
[c99b6b] | 311 | return; |
---|
| 312 | if( rem.isZero() ) |
---|
| 313 | { |
---|
[ad8e1b] | 314 | result *= inv; |
---|
[5df7d0] | 315 | result= reduce (result, M); |
---|
[c99b6b] | 316 | return; |
---|
| 317 | } |
---|
[ad8e1b] | 318 | if(result.degree(x) >= rem.degree(x)) |
---|
| 319 | { |
---|
| 320 | P = result; |
---|
| 321 | result = rem; |
---|
| 322 | } |
---|
| 323 | else |
---|
| 324 | P = rem; |
---|
[359d742] | 325 | } |
---|
[c99b6b] | 326 | } |
---|
| 327 | |
---|
| 328 | bool hasFirstAlgVar( const CanonicalForm & f, Variable & a ) |
---|
| 329 | { |
---|
| 330 | if( f.inBaseDomain() ) // f has NO alg. variable |
---|
| 331 | return false; |
---|
| 332 | if( f.level()<0 ) // f has only alg. vars, so take the first one |
---|
| 333 | { |
---|
| 334 | a = f.mvar(); |
---|
| 335 | return true; |
---|
| 336 | } |
---|
| 337 | for(CFIterator i=f; i.hasTerms(); i++) |
---|
| 338 | if( hasFirstAlgVar( i.coeff(), a )) |
---|
| 339 | return true; // 'a' is already set |
---|
| 340 | return false; |
---|
| 341 | } |
---|
| 342 | |
---|
[ad8e1b] | 343 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ); |
---|
| 344 | int * leadDeg(const CanonicalForm & f, int *degs); |
---|
| 345 | bool isLess(int *a, int *b, int lower, int upper); |
---|
| 346 | bool isEqual(int *a, int *b, int lower, int upper); |
---|
| 347 | CanonicalForm firstLC(const CanonicalForm & f); |
---|
| 348 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
---|
| 349 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ); |
---|
| 350 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ); |
---|
| 351 | static void tryDivide( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, CanonicalForm & result, bool & divides, bool & fail ); |
---|
[359d742] | 352 | |
---|
[5df7d0] | 353 | static inline CanonicalForm |
---|
| 354 | tryNewtonInterp (const CanonicalForm alpha, const CanonicalForm u, |
---|
| 355 | const CanonicalForm newtonPoly, const CanonicalForm oldInterPoly, |
---|
| 356 | const Variable & x, const CanonicalForm& M, bool& fail) |
---|
| 357 | { |
---|
| 358 | CanonicalForm interPoly; |
---|
| 359 | |
---|
| 360 | CanonicalForm inv; |
---|
| 361 | tryInvert (newtonPoly (alpha, x), M, inv, fail); |
---|
| 362 | if (fail) |
---|
| 363 | return 0; |
---|
| 364 | |
---|
| 365 | interPoly= oldInterPoly+reduce ((u - oldInterPoly (alpha, x))*inv*newtonPoly, M); |
---|
| 366 | return interPoly; |
---|
| 367 | } |
---|
[359d742] | 368 | |
---|
[fe2d4c] | 369 | void tryBrownGCD( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, bool & fail, bool topLevel ) |
---|
[ad8e1b] | 370 | { // assume F,G are multivariate polys over Z/p(a) for big prime p, M "univariate" polynomial in an algebraic variable |
---|
| 371 | // M is assumed to be monic |
---|
[359d742] | 372 | if(F.isZero()) |
---|
| 373 | { |
---|
| 374 | if(G.isZero()) |
---|
| 375 | { |
---|
| 376 | result = G; // G is zero |
---|
| 377 | return; |
---|
| 378 | } |
---|
| 379 | if(G.inCoeffDomain()) |
---|
| 380 | { |
---|
| 381 | tryInvert(G,M,result,fail); |
---|
[ad8e1b] | 382 | if(fail) |
---|
| 383 | return; |
---|
| 384 | result = 1; |
---|
[359d742] | 385 | return; |
---|
| 386 | } |
---|
| 387 | // try to make G monic modulo M |
---|
| 388 | CanonicalForm inv; |
---|
| 389 | tryInvert(Lc(G),M,inv,fail); |
---|
| 390 | if(fail) |
---|
| 391 | return; |
---|
| 392 | result = inv*G; |
---|
[5df7d0] | 393 | result= reduce (result, M); |
---|
[359d742] | 394 | return; |
---|
| 395 | } |
---|
| 396 | if(G.isZero()) // F is non-zero |
---|
| 397 | { |
---|
| 398 | if(F.inCoeffDomain()) |
---|
| 399 | { |
---|
| 400 | tryInvert(F,M,result,fail); |
---|
[ad8e1b] | 401 | if(fail) |
---|
| 402 | return; |
---|
| 403 | result = 1; |
---|
[359d742] | 404 | return; |
---|
| 405 | } |
---|
| 406 | // try to make F monic modulo M |
---|
| 407 | CanonicalForm inv; |
---|
| 408 | tryInvert(Lc(F),M,inv,fail); |
---|
| 409 | if(fail) |
---|
| 410 | return; |
---|
| 411 | result = inv*F; |
---|
[5df7d0] | 412 | result= reduce (result, M); |
---|
[359d742] | 413 | return; |
---|
| 414 | } |
---|
[ad8e1b] | 415 | // here: F,G both nonzero |
---|
[359d742] | 416 | if(F.inCoeffDomain()) |
---|
| 417 | { |
---|
| 418 | tryInvert(F,M,result,fail); |
---|
[ad8e1b] | 419 | if(fail) |
---|
| 420 | return; |
---|
| 421 | result = 1; |
---|
[359d742] | 422 | return; |
---|
| 423 | } |
---|
| 424 | if(G.inCoeffDomain()) |
---|
| 425 | { |
---|
| 426 | tryInvert(G,M,result,fail); |
---|
[ad8e1b] | 427 | if(fail) |
---|
| 428 | return; |
---|
| 429 | result = 1; |
---|
[359d742] | 430 | return; |
---|
| 431 | } |
---|
| 432 | CFMap MM,NN; |
---|
[fe2d4c] | 433 | int lev= myCompress (F, G, MM, NN, topLevel); |
---|
| 434 | if (lev == 0) |
---|
| 435 | { |
---|
| 436 | result= 1; |
---|
| 437 | return; |
---|
| 438 | } |
---|
[359d742] | 439 | CanonicalForm f=MM(F); |
---|
| 440 | CanonicalForm g=MM(G); |
---|
[ad8e1b] | 441 | // here: f,g are compressed |
---|
[359d742] | 442 | // compute largest variable in f or g (least one is Variable(1)) |
---|
| 443 | int mv = f.level(); |
---|
| 444 | if(g.level() > mv) |
---|
| 445 | mv = g.level(); |
---|
| 446 | // here: mv is level of the largest variable in f, g |
---|
| 447 | if(mv == 1) // f,g univariate |
---|
| 448 | { |
---|
| 449 | tryEuclid(f,g,M,result,fail); |
---|
| 450 | if(fail) |
---|
| 451 | return; |
---|
[5df7d0] | 452 | result= NN (reduce (result, M)); // do not forget to map back |
---|
[359d742] | 453 | return; |
---|
| 454 | } |
---|
| 455 | // here: mv > 1 |
---|
[ad8e1b] | 456 | CanonicalForm cf = tryvcontent(f, Variable(2), M, fail); // cf is univariate poly in var(1) |
---|
| 457 | if(fail) |
---|
| 458 | return; |
---|
| 459 | CanonicalForm cg = tryvcontent(g, Variable(2), M, fail); |
---|
| 460 | if(fail) |
---|
| 461 | return; |
---|
[359d742] | 462 | CanonicalForm c; |
---|
| 463 | tryEuclid(cf,cg,M,c,fail); |
---|
| 464 | if(fail) |
---|
| 465 | return; |
---|
[ad8e1b] | 466 | // f /= cf |
---|
[13f494] | 467 | f.tryDiv (cf, M, fail); |
---|
[ad8e1b] | 468 | if(fail) |
---|
| 469 | return; |
---|
| 470 | // g /= cg |
---|
[13f494] | 471 | g.tryDiv (cg, M, fail); |
---|
[ad8e1b] | 472 | if(fail) |
---|
| 473 | return; |
---|
[359d742] | 474 | if(f.inCoeffDomain()) |
---|
| 475 | { |
---|
| 476 | tryInvert(f,M,result,fail); |
---|
| 477 | if(fail) |
---|
| 478 | return; |
---|
[ad8e1b] | 479 | result = NN(c); |
---|
[359d742] | 480 | return; |
---|
| 481 | } |
---|
| 482 | if(g.inCoeffDomain()) |
---|
| 483 | { |
---|
| 484 | tryInvert(g,M,result,fail); |
---|
| 485 | if(fail) |
---|
| 486 | return; |
---|
[ad8e1b] | 487 | result = NN(c); |
---|
[359d742] | 488 | return; |
---|
| 489 | } |
---|
| 490 | int *L = new int[mv+1]; // L is addressed by i from 2 to mv |
---|
| 491 | int *N = new int[mv+1]; |
---|
| 492 | for(int i=2; i<=mv; i++) |
---|
| 493 | L[i] = N[i] = 0; |
---|
| 494 | L = leadDeg(f, L); |
---|
| 495 | N = leadDeg(g, N); |
---|
| 496 | CanonicalForm gamma; |
---|
| 497 | tryEuclid( firstLC(f), firstLC(g), M, gamma, fail ); |
---|
| 498 | if(fail) |
---|
| 499 | return; |
---|
[ad8e1b] | 500 | for(int i=2; i<=mv; i++) // entries at i=0,1 not visited |
---|
[359d742] | 501 | if(N[i] < L[i]) |
---|
| 502 | L[i] = N[i]; |
---|
| 503 | // L is now upper bound for degrees of gcd |
---|
| 504 | int *dg_im = new int[mv+1]; // for the degree vector of the image we don't need any entry at i=1 |
---|
| 505 | for(int i=2; i<=mv; i++) |
---|
| 506 | dg_im[i] = 0; // initialize |
---|
| 507 | CanonicalForm gamma_image, m=1; |
---|
| 508 | CanonicalForm gm=0; |
---|
[5df7d0] | 509 | CanonicalForm g_image, alpha, gnew; |
---|
[359d742] | 510 | FFGenerator gen = FFGenerator(); |
---|
[6f08f3] | 511 | Variable x= Variable (1); |
---|
[13f494] | 512 | bool divides= true; |
---|
[359d742] | 513 | for(FFGenerator gen = FFGenerator(); gen.hasItems(); gen.next()) |
---|
| 514 | { |
---|
| 515 | alpha = gen.item(); |
---|
[6f08f3] | 516 | gamma_image = reduce(gamma(alpha, x),M); // plug in alpha for var(1) |
---|
[359d742] | 517 | if(gamma_image.isZero()) // skip lc-bad points var(1)-alpha |
---|
| 518 | continue; |
---|
[6f08f3] | 519 | tryBrownGCD( f(alpha, x), g(alpha, x), M, g_image, fail, false ); // recursive call with one var less |
---|
[359d742] | 520 | if(fail) |
---|
| 521 | return; |
---|
[ad8e1b] | 522 | g_image = reduce(g_image, M); |
---|
[359d742] | 523 | if(g_image.inCoeffDomain()) // early termination |
---|
| 524 | { |
---|
| 525 | tryInvert(g_image,M,result,fail); |
---|
| 526 | if(fail) |
---|
| 527 | return; |
---|
| 528 | result = NN(c); |
---|
| 529 | return; |
---|
| 530 | } |
---|
| 531 | for(int i=2; i<=mv; i++) |
---|
| 532 | dg_im[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
| 533 | dg_im = leadDeg(g_image, dg_im); // dg_im cannot be NIL-pointer |
---|
| 534 | if(isEqual(dg_im, L, 2, mv)) |
---|
| 535 | { |
---|
[5df7d0] | 536 | CanonicalForm inv; |
---|
| 537 | tryInvert (firstLC (g_image), M, inv, fail); |
---|
| 538 | if (fail) |
---|
| 539 | return; |
---|
| 540 | g_image *= inv; |
---|
[359d742] | 541 | g_image *= gamma_image; // multiply by multiple of image lc(gcd) |
---|
[5df7d0] | 542 | g_image= reduce (g_image, M); |
---|
| 543 | gnew= tryNewtonInterp (alpha, g_image, m, gm, x, M, fail); |
---|
[359d742] | 544 | // gnew = gm mod m |
---|
| 545 | // gnew = g_image mod var(1)-alpha |
---|
| 546 | // mnew = m * (var(1)-alpha) |
---|
| 547 | if(fail) |
---|
| 548 | return; |
---|
[5df7d0] | 549 | m *= (x - alpha); |
---|
[359d742] | 550 | if(gnew == gm) // gnew did not change |
---|
| 551 | { |
---|
[ad8e1b] | 552 | cf = tryvcontent(gm, Variable(2), M, fail); |
---|
| 553 | if(fail) |
---|
| 554 | return; |
---|
| 555 | divides = true; |
---|
[13f494] | 556 | g_image= gm; |
---|
| 557 | g_image.tryDiv (cf, M, fail); |
---|
[ad8e1b] | 558 | if(fail) |
---|
| 559 | return; |
---|
[13f494] | 560 | divides= tryFdivides (g_image,f, M, fail); // trial division (f) |
---|
[ad8e1b] | 561 | if(fail) |
---|
[359d742] | 562 | return; |
---|
[ad8e1b] | 563 | if(divides) |
---|
| 564 | { |
---|
[13f494] | 565 | bool divides2= tryFdivides (g_image,g, M, fail); // trial division (g) |
---|
[ad8e1b] | 566 | if(fail) |
---|
| 567 | return; |
---|
[13f494] | 568 | if(divides2) |
---|
[ad8e1b] | 569 | { |
---|
[5df7d0] | 570 | result = NN(reduce (c*g_image, M)); |
---|
[ad8e1b] | 571 | return; |
---|
| 572 | } |
---|
[359d742] | 573 | } |
---|
| 574 | } |
---|
| 575 | gm = gnew; |
---|
| 576 | continue; |
---|
| 577 | } |
---|
| 578 | |
---|
| 579 | if(isLess(L, dg_im, 2, mv)) // dg_im > L --> current point unlucky |
---|
| 580 | continue; |
---|
| 581 | |
---|
[ad8e1b] | 582 | // here: isLess(dg_im, L, 2, mv) --> all previous points were unlucky |
---|
| 583 | m = CanonicalForm(1); // reset |
---|
| 584 | gm = 0; // reset |
---|
| 585 | for(int i=2; i<=mv; i++) // tighten bound |
---|
| 586 | L[i] = dg_im[i]; |
---|
[359d742] | 587 | } |
---|
| 588 | // we are out of evaluation points |
---|
| 589 | fail = true; |
---|
| 590 | } |
---|
| 591 | |
---|
[2156ec] | 592 | static CanonicalForm |
---|
| 593 | myicontent ( const CanonicalForm & f, const CanonicalForm & c ) |
---|
| 594 | { |
---|
[517530] | 595 | #ifdef HAVE_NTL |
---|
[2156ec] | 596 | if (f.isOne() || c.isOne()) |
---|
| 597 | return 1; |
---|
| 598 | if ( f.inBaseDomain() && c.inBaseDomain()) |
---|
| 599 | { |
---|
| 600 | if (c.isZero()) return abs(f); |
---|
| 601 | return bgcd( f, c ); |
---|
| 602 | } |
---|
| 603 | else if ( (f.inCoeffDomain() && c.inCoeffDomain()) || |
---|
| 604 | (f.inCoeffDomain() && c.inBaseDomain()) || |
---|
| 605 | (f.inBaseDomain() && c.inCoeffDomain())) |
---|
| 606 | { |
---|
| 607 | if (c.isZero()) return abs (f); |
---|
| 608 | ZZX NTLf= convertFacCF2NTLZZX (f); |
---|
| 609 | ZZX NTLc= convertFacCF2NTLZZX (c); |
---|
| 610 | NTLc= GCD (NTLc, NTLf); |
---|
| 611 | if (f.inCoeffDomain()) |
---|
| 612 | return convertNTLZZX2CF(NTLc,f.mvar()); |
---|
| 613 | else |
---|
| 614 | return convertNTLZZX2CF(NTLc,c.mvar()); |
---|
| 615 | } |
---|
| 616 | else |
---|
| 617 | { |
---|
| 618 | CanonicalForm g = c; |
---|
| 619 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
---|
| 620 | g = myicontent( i.coeff(), g ); |
---|
| 621 | return g; |
---|
| 622 | } |
---|
[517530] | 623 | #else |
---|
| 624 | return 1; |
---|
[2156ec] | 625 | #endif |
---|
[517530] | 626 | } |
---|
[2156ec] | 627 | |
---|
| 628 | CanonicalForm |
---|
| 629 | myicontent ( const CanonicalForm & f ) |
---|
| 630 | { |
---|
| 631 | #ifdef HAVE_NTL |
---|
| 632 | return myicontent( f, 0 ); |
---|
| 633 | #else |
---|
| 634 | return 1; |
---|
| 635 | #endif |
---|
| 636 | } |
---|
| 637 | |
---|
[ad8e1b] | 638 | CanonicalForm QGCD( const CanonicalForm & F, const CanonicalForm & G ) |
---|
| 639 | { // f,g in Q(a)[x1,...,xn] |
---|
| 640 | if(F.isZero()) |
---|
| 641 | { |
---|
| 642 | if(G.isZero()) |
---|
| 643 | return G; // G is zero |
---|
| 644 | if(G.inCoeffDomain()) |
---|
| 645 | return CanonicalForm(1); |
---|
[1682691] | 646 | CanonicalForm lcinv= 1/Lc (G); |
---|
| 647 | return G*lcinv; // return monic G |
---|
[ad8e1b] | 648 | } |
---|
| 649 | if(G.isZero()) // F is non-zero |
---|
| 650 | { |
---|
| 651 | if(F.inCoeffDomain()) |
---|
| 652 | return CanonicalForm(1); |
---|
[1682691] | 653 | CanonicalForm lcinv= 1/Lc (F); |
---|
| 654 | return F*lcinv; // return monic F |
---|
[ad8e1b] | 655 | } |
---|
| 656 | if(F.inCoeffDomain() || G.inCoeffDomain()) |
---|
| 657 | return CanonicalForm(1); |
---|
| 658 | // here: both NOT inCoeffDomain |
---|
| 659 | CanonicalForm f, g, tmp, M, q, D, Dp, cl, newq, mipo; |
---|
| 660 | int p, i; |
---|
| 661 | int *bound, *other; // degree vectors |
---|
| 662 | bool fail; |
---|
[713bdb] | 663 | bool off_rational=!isOn(SW_RATIONAL); |
---|
[ad8e1b] | 664 | On( SW_RATIONAL ); // needed by bCommonDen |
---|
| 665 | f = F * bCommonDen(F); |
---|
| 666 | g = G * bCommonDen(G); |
---|
[2156ec] | 667 | CanonicalForm contf= myicontent (f); |
---|
| 668 | CanonicalForm contg= myicontent (g); |
---|
| 669 | f /= contf; |
---|
| 670 | g /= contg; |
---|
| 671 | CanonicalForm gcdcfcg= myicontent (contf, contg); |
---|
[ad8e1b] | 672 | Variable a, b; |
---|
| 673 | if(hasFirstAlgVar(f,a)) |
---|
| 674 | { |
---|
| 675 | if(hasFirstAlgVar(g,b)) |
---|
| 676 | { |
---|
| 677 | if(b.level() > a.level()) |
---|
| 678 | a = b; |
---|
| 679 | } |
---|
| 680 | } |
---|
| 681 | else |
---|
| 682 | { |
---|
| 683 | if(!hasFirstAlgVar(g,a))// both not in extension |
---|
| 684 | { |
---|
| 685 | Off( SW_RATIONAL ); |
---|
| 686 | Off( SW_USE_QGCD ); |
---|
[2156ec] | 687 | tmp = gcdcfcg*gcd( f, g ); |
---|
[ad8e1b] | 688 | On( SW_USE_QGCD ); |
---|
[713bdb] | 689 | if (off_rational) Off(SW_RATIONAL); |
---|
[ad8e1b] | 690 | return tmp; |
---|
| 691 | } |
---|
| 692 | } |
---|
| 693 | // here: a is the biggest alg. var in f and g AND some of f,g is in extension |
---|
| 694 | // (in the sequel b is used to swap alg/poly vars) |
---|
| 695 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
| 696 | tmp = getMipo(a); |
---|
| 697 | M = tmp * bCommonDen(tmp); |
---|
| 698 | // here: f, g in Z[a][x1,...,xn], M in Z[a] not necessarily monic |
---|
| 699 | Off( SW_RATIONAL ); // needed by mod |
---|
| 700 | // calculate upper bound for degree vector of gcd |
---|
| 701 | int mv = f.level(); i = g.level(); |
---|
| 702 | if(i > mv) |
---|
| 703 | mv = i; |
---|
| 704 | // here: mv is level of the largest variable in f, g |
---|
| 705 | b = Variable(mv+1); |
---|
| 706 | bound = new int[mv+1]; // 'bound' could be indexed from 0 to mv, but we will only use from 1 to mv |
---|
| 707 | other = new int[mv+1]; |
---|
| 708 | for(int i=1; i<=mv; i++) // initialize 'bound', 'other' with zeros |
---|
| 709 | bound[i] = other[i] = 0; |
---|
| 710 | bound = leadDeg(f,bound); // 'bound' is set the leading degree vector of f |
---|
| 711 | other = leadDeg(g,other); |
---|
| 712 | for(int i=1; i<=mv; i++) // entry at i=0 not visited |
---|
| 713 | if(other[i] < bound[i]) |
---|
| 714 | bound[i] = other[i]; |
---|
| 715 | // now 'bound' is the smaller vector |
---|
| 716 | cl = lc(M) * lc(f) * lc(g); |
---|
| 717 | q = 1; |
---|
| 718 | D = 0; |
---|
[fe2d4c] | 719 | CanonicalForm test= 0; |
---|
| 720 | bool equal= false; |
---|
[ad8e1b] | 721 | for( i=cf_getNumBigPrimes()-1; i>-1; i-- ) |
---|
| 722 | { |
---|
| 723 | p = cf_getBigPrime(i); |
---|
| 724 | if( mod( cl, p ).isZero() ) // skip lc-bad primes |
---|
| 725 | continue; |
---|
| 726 | fail = false; |
---|
| 727 | setCharacteristic(p); |
---|
| 728 | mipo = mapinto(M); |
---|
| 729 | mipo /= mipo.lc(); |
---|
| 730 | // here: mipo is monic |
---|
| 731 | tryBrownGCD( mapinto(f), mapinto(g), mipo, Dp, fail ); |
---|
| 732 | if( fail ) // mipo splits in char p |
---|
| 733 | continue; |
---|
| 734 | if( Dp.inCoeffDomain() ) // early termination |
---|
| 735 | { |
---|
| 736 | tryInvert(Dp,mipo,tmp,fail); // check if zero divisor |
---|
| 737 | if(fail) |
---|
| 738 | continue; |
---|
| 739 | setReduce(a,true); |
---|
[713bdb] | 740 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[0a7d0ca] | 741 | setCharacteristic(0); |
---|
[2156ec] | 742 | return gcdcfcg; |
---|
[ad8e1b] | 743 | } |
---|
[0a7d0ca] | 744 | setCharacteristic(0); |
---|
[ad8e1b] | 745 | // here: Dp NOT inCoeffDomain |
---|
| 746 | for(int i=1; i<=mv; i++) |
---|
| 747 | other[i] = 0; // reset (this is necessary, because some entries may not be updated by call to leadDeg) |
---|
| 748 | other = leadDeg(Dp,other); |
---|
[806c18] | 749 | |
---|
[ad8e1b] | 750 | if(isEqual(bound, other, 1, mv)) // equal |
---|
| 751 | { |
---|
| 752 | chineseRemainder( D, q, replacevar( mapinto(Dp), a, b ), p, tmp, newq ); |
---|
| 753 | // tmp = Dp mod p |
---|
| 754 | // tmp = D mod q |
---|
| 755 | // newq = p*q |
---|
| 756 | q = newq; |
---|
| 757 | if( D != tmp ) |
---|
| 758 | D = tmp; |
---|
| 759 | On( SW_RATIONAL ); |
---|
| 760 | tmp = replacevar( Farey( D, q ), b, a ); // Farey and switch back to alg var |
---|
| 761 | setReduce(a,true); // reduce expressions modulo mipo |
---|
| 762 | On( SW_RATIONAL ); // needed by fdivides |
---|
[fe2d4c] | 763 | if (test != tmp) |
---|
| 764 | test= tmp; |
---|
| 765 | else |
---|
| 766 | equal= true; // modular image did not add any new information |
---|
| 767 | if(equal && fdivides( tmp, f ) && fdivides( tmp, g )) // trial division |
---|
[ad8e1b] | 768 | { |
---|
| 769 | Off( SW_RATIONAL ); |
---|
| 770 | setReduce(a,true); |
---|
[713bdb] | 771 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[2156ec] | 772 | return tmp*gcdcfcg; |
---|
[ad8e1b] | 773 | } |
---|
| 774 | Off( SW_RATIONAL ); |
---|
| 775 | setReduce(a,false); // do not reduce expressions modulo mipo |
---|
| 776 | continue; |
---|
| 777 | } |
---|
| 778 | if( isLess(bound, other, 1, mv) ) // current prime unlucky |
---|
| 779 | continue; |
---|
| 780 | // here: isLess(other, bound, 1, mv) ) ==> all previous primes unlucky |
---|
| 781 | q = p; |
---|
| 782 | D = replacevar( mapinto(Dp), a, b ); // shortcut CRA // shortcut CRA |
---|
| 783 | for(int i=1; i<=mv; i++) // tighten bound |
---|
| 784 | bound[i] = other[i]; |
---|
| 785 | } |
---|
| 786 | // hopefully, we never reach this point |
---|
| 787 | setReduce(a,true); |
---|
| 788 | Off( SW_USE_QGCD ); |
---|
[2156ec] | 789 | D = gcdcfcg*gcd( f, g ); |
---|
[ad8e1b] | 790 | On( SW_USE_QGCD ); |
---|
[713bdb] | 791 | if (off_rational) Off(SW_RATIONAL); else On(SW_RATIONAL); |
---|
[ad8e1b] | 792 | return D; |
---|
| 793 | } |
---|
| 794 | |
---|
| 795 | |
---|
| 796 | int * leadDeg(const CanonicalForm & f, int *degs) |
---|
| 797 | { // leading degree vector w.r.t. lex. monomial order x(i+1) > x(i) |
---|
| 798 | // if f is in a coeff domain, the zero pointer is returned |
---|
| 799 | // 'a' should point to an array of sufficient size level(f)+1 |
---|
| 800 | if(f.inCoeffDomain()) |
---|
| 801 | return 0; |
---|
| 802 | CanonicalForm tmp = f; |
---|
| 803 | do |
---|
| 804 | { |
---|
| 805 | degs[tmp.level()] = tmp.degree(); |
---|
| 806 | tmp = LC(tmp); |
---|
| 807 | } |
---|
| 808 | while(!tmp.inCoeffDomain()); |
---|
| 809 | return degs; |
---|
| 810 | } |
---|
| 811 | |
---|
| 812 | |
---|
| 813 | bool isLess(int *a, int *b, int lower, int upper) |
---|
| 814 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
---|
| 815 | for(int i=upper; i>=lower; i--) |
---|
| 816 | if(a[i] == b[i]) |
---|
| 817 | continue; |
---|
| 818 | else |
---|
| 819 | return a[i] < b[i]; |
---|
| 820 | return true; |
---|
| 821 | } |
---|
| 822 | |
---|
| 823 | |
---|
| 824 | bool isEqual(int *a, int *b, int lower, int upper) |
---|
| 825 | { // compares the degree vectors a,b on the specified part. Note: x(i+1) > x(i) |
---|
| 826 | for(int i=lower; i<=upper; i++) |
---|
| 827 | if(a[i] != b[i]) |
---|
| 828 | return false; |
---|
| 829 | return true; |
---|
| 830 | } |
---|
| 831 | |
---|
| 832 | |
---|
| 833 | CanonicalForm firstLC(const CanonicalForm & f) |
---|
| 834 | { // returns the leading coefficient (LC) of level <= 1 |
---|
| 835 | CanonicalForm ret = f; |
---|
| 836 | while(ret.level() > 1) |
---|
| 837 | ret = LC(ret); |
---|
| 838 | return ret; |
---|
| 839 | } |
---|
| 840 | |
---|
| 841 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, const CanonicalForm & M, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
---|
| 842 | { // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
---|
| 843 | // F and G must have the same level AND level > 0 |
---|
| 844 | // we try to calculate gcd(F,G) = s*F + t*G |
---|
| 845 | // if a zero divisor is encontered, 'fail' is set to one |
---|
| 846 | // M is assumed to be monic |
---|
| 847 | CanonicalForm P; |
---|
| 848 | if(F.inCoeffDomain()) |
---|
| 849 | { |
---|
| 850 | tryInvert( F, M, P, fail ); |
---|
| 851 | if(fail) |
---|
| 852 | return; |
---|
| 853 | result = 1; |
---|
| 854 | s = P; t = 0; |
---|
| 855 | return; |
---|
| 856 | } |
---|
| 857 | if(G.inCoeffDomain()) |
---|
| 858 | { |
---|
| 859 | tryInvert( G, M, P, fail ); |
---|
| 860 | if(fail) |
---|
| 861 | return; |
---|
| 862 | result = 1; |
---|
| 863 | s = 0; t = P; |
---|
| 864 | return; |
---|
| 865 | } |
---|
| 866 | // here: both not inCoeffDomain |
---|
| 867 | CanonicalForm inv, rem, tmp, u, v, q, sum=0; |
---|
| 868 | if( F.degree() > G.degree() ) |
---|
| 869 | { |
---|
| 870 | P = F; result = G; s=v=0; t=u=1; |
---|
| 871 | } |
---|
| 872 | else |
---|
| 873 | { |
---|
| 874 | P = G; result = F; s=v=1; t=u=0; |
---|
| 875 | } |
---|
| 876 | Variable x = P.mvar(); |
---|
| 877 | // here: degree(P) >= degree(result) |
---|
| 878 | while(true) |
---|
| 879 | { |
---|
[fe2d4c] | 880 | tryDivrem (P, result, q, rem, inv, M, fail); |
---|
[ad8e1b] | 881 | if(fail) |
---|
| 882 | return; |
---|
| 883 | if( rem.isZero() ) |
---|
| 884 | { |
---|
| 885 | s*=inv; |
---|
[4a05ed] | 886 | s= reduce (s, M); |
---|
[ad8e1b] | 887 | t*=inv; |
---|
[4a05ed] | 888 | t= reduce (t, M); |
---|
[ad8e1b] | 889 | result *= inv; // monify result |
---|
[4a05ed] | 890 | result= reduce (result, M); |
---|
[ad8e1b] | 891 | return; |
---|
| 892 | } |
---|
| 893 | sum += q; |
---|
| 894 | if(result.degree(x) >= rem.degree(x)) |
---|
| 895 | { |
---|
| 896 | P=result; |
---|
| 897 | result=rem; |
---|
| 898 | tmp=u-sum*s; |
---|
| 899 | u=s; |
---|
| 900 | s=tmp; |
---|
| 901 | tmp=v-sum*t; |
---|
| 902 | v=t; |
---|
| 903 | t=tmp; |
---|
| 904 | sum = 0; // reset |
---|
| 905 | } |
---|
| 906 | else |
---|
| 907 | P = rem; |
---|
| 908 | } |
---|
| 909 | } |
---|
| 910 | |
---|
| 911 | |
---|
| 912 | static CanonicalForm trycontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
| 913 | { // as 'content', but takes care of zero divisors |
---|
| 914 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
---|
| 915 | Variable y = f.mvar(); |
---|
| 916 | if ( y == x ) |
---|
| 917 | return trycf_content( f, 0, M, fail ); |
---|
| 918 | if ( y < x ) |
---|
| 919 | return f; |
---|
| 920 | return swapvar( trycontent( swapvar( f, y, x ), y, M, fail ), y, x ); |
---|
| 921 | } |
---|
| 922 | |
---|
| 923 | |
---|
| 924 | static CanonicalForm tryvcontent ( const CanonicalForm & f, const Variable & x, const CanonicalForm & M, bool & fail ) |
---|
| 925 | { // as vcontent, but takes care of zero divisors |
---|
| 926 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
---|
| 927 | if ( f.mvar() <= x ) |
---|
| 928 | return trycontent( f, x, M, fail ); |
---|
| 929 | CFIterator i; |
---|
| 930 | CanonicalForm d = 0, e, ret; |
---|
| 931 | for ( i = f; i.hasTerms() && ! d.isOne() && ! fail; i++ ) |
---|
| 932 | { |
---|
| 933 | e = tryvcontent( i.coeff(), x, M, fail ); |
---|
| 934 | if(fail) |
---|
| 935 | break; |
---|
| 936 | tryBrownGCD( d, e, M, ret, fail ); |
---|
| 937 | d = ret; |
---|
| 938 | } |
---|
| 939 | return d; |
---|
| 940 | } |
---|
| 941 | |
---|
| 942 | |
---|
| 943 | static CanonicalForm trycf_content ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, bool & fail ) |
---|
| 944 | { // as cf_content, but takes care of zero divisors |
---|
| 945 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
| 946 | { |
---|
| 947 | CFIterator i = f; |
---|
| 948 | CanonicalForm tmp = g, result; |
---|
| 949 | while ( i.hasTerms() && ! tmp.isOne() && ! fail ) |
---|
| 950 | { |
---|
| 951 | tryBrownGCD( i.coeff(), tmp, M, result, fail ); |
---|
| 952 | tmp = result; |
---|
| 953 | i++; |
---|
| 954 | } |
---|
| 955 | return result; |
---|
| 956 | } |
---|
| 957 | return abs( f ); |
---|
| 958 | } |
---|
| 959 | |
---|
| 960 | |
---|
| 961 | static void tryDivide( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm & M, CanonicalForm & result, bool & divides, bool & fail ) |
---|
| 962 | { // M "univariate" monic polynomial |
---|
| 963 | // f, g polynomials with coeffs modulo M. |
---|
| 964 | // if f is divisible by g, 'divides' is set to 1 and 'result' == f/g mod M coefficientwise. |
---|
| 965 | // 'fail' is set to 1, iff a zero divisor is encountered. |
---|
| 966 | // divides==1 implies fail==0 |
---|
| 967 | // required: getReduce(M.mvar())==0 |
---|
| 968 | if(g.inBaseDomain()) |
---|
| 969 | { |
---|
| 970 | result = f/g; |
---|
| 971 | divides = true; |
---|
| 972 | return; |
---|
| 973 | } |
---|
| 974 | if(g.inCoeffDomain()) |
---|
| 975 | { |
---|
| 976 | tryInvert(g,M,result,fail); |
---|
| 977 | if(fail) |
---|
| 978 | return; |
---|
| 979 | result = reduce(f*result, M); |
---|
| 980 | divides = true; |
---|
| 981 | return; |
---|
| 982 | } |
---|
| 983 | // here: g NOT inCoeffDomain |
---|
| 984 | Variable x = g.mvar(); |
---|
| 985 | if(f.degree(x) < g.degree(x)) |
---|
| 986 | { |
---|
| 987 | divides = false; |
---|
| 988 | return; |
---|
| 989 | } |
---|
| 990 | // here: f.degree(x) > 0 and f.degree(x) >= g.degree(x) |
---|
| 991 | CanonicalForm F = f; |
---|
| 992 | CanonicalForm q, leadG = LC(g); |
---|
| 993 | result = 0; |
---|
| 994 | while(!F.isZero()) |
---|
| 995 | { |
---|
| 996 | tryDivide(F.LC(x),leadG,M,q,divides,fail); |
---|
| 997 | if(fail || !divides) |
---|
| 998 | return; |
---|
| 999 | if(F.degree(x)<g.degree(x)) |
---|
| 1000 | { |
---|
| 1001 | divides = false; |
---|
| 1002 | return; |
---|
| 1003 | } |
---|
| 1004 | q *= power(x,F.degree(x)-g.degree(x)); |
---|
| 1005 | result += q; |
---|
| 1006 | F = reduce(F-q*g, M); |
---|
| 1007 | } |
---|
| 1008 | result = reduce(result, M); |
---|
| 1009 | divides = true; |
---|
| 1010 | } |
---|
| 1011 | |
---|
[4a05ed] | 1012 | void tryExtgcd( const CanonicalForm & F, const CanonicalForm & G, CanonicalForm & result, CanonicalForm & s, CanonicalForm & t, bool & fail ) |
---|
| 1013 | { |
---|
| 1014 | // F, G are univariate polynomials (i.e. they have exactly one polynomial variable) |
---|
| 1015 | // F and G must have the same level AND level > 0 |
---|
| 1016 | // we try to calculate gcd(f,g) = s*F + t*G |
---|
| 1017 | // if a zero divisor is encontered, 'fail' is set to one |
---|
| 1018 | Variable a, b; |
---|
| 1019 | if( !hasFirstAlgVar(F,a) && !hasFirstAlgVar(G,b) ) // note lazy evaluation |
---|
| 1020 | { |
---|
| 1021 | result = extgcd( F, G, s, t ); // no zero divisors possible |
---|
| 1022 | return; |
---|
| 1023 | } |
---|
| 1024 | if( b.level() > a.level() ) |
---|
| 1025 | a = b; |
---|
| 1026 | // here: a is the biggest alg. var in F and G |
---|
| 1027 | CanonicalForm M = getMipo(a); |
---|
| 1028 | CanonicalForm P; |
---|
| 1029 | if( degree(F) > degree(G) ) |
---|
| 1030 | { |
---|
| 1031 | P=F; result=G; s=0; t=1; |
---|
| 1032 | } |
---|
| 1033 | else |
---|
| 1034 | { |
---|
| 1035 | P=G; result=F; s=1; t=0; |
---|
| 1036 | } |
---|
| 1037 | CanonicalForm inv, rem, q, u, v; |
---|
| 1038 | // here: degree(P) >= degree(result) |
---|
| 1039 | while(true) |
---|
| 1040 | { |
---|
| 1041 | tryInvert( Lc(result), M, inv, fail ); |
---|
| 1042 | if(fail) |
---|
| 1043 | return; |
---|
| 1044 | // here: Lc(result) is invertible modulo M |
---|
| 1045 | q = Lc(P)*inv * power( P.mvar(), degree(P)-degree(result) ); |
---|
| 1046 | rem = P - q*result; |
---|
| 1047 | // here: s*F + t*G = result |
---|
| 1048 | if( rem.isZero() ) |
---|
| 1049 | { |
---|
| 1050 | s*=inv; |
---|
| 1051 | t*=inv; |
---|
| 1052 | result *= inv; // monify result |
---|
| 1053 | return; |
---|
| 1054 | } |
---|
| 1055 | P=result; |
---|
| 1056 | result=rem; |
---|
| 1057 | rem=u-q*s; |
---|
| 1058 | u=s; |
---|
| 1059 | s=rem; |
---|
| 1060 | rem=v-q*t; |
---|
| 1061 | v=t; |
---|
| 1062 | t=rem; |
---|
| 1063 | } |
---|
| 1064 | } |
---|
| 1065 | |
---|
[359d742] | 1066 | void tryCRA( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew, bool & fail ) |
---|
| 1067 | { // polys of level <= 1 are considered coefficients. q1,q2 are assumed to be coprime |
---|
| 1068 | // xnew = x1 mod q1 (coefficientwise in the above sense) |
---|
| 1069 | // xnew = x2 mod q2 |
---|
| 1070 | // qnew = q1*q2 |
---|
| 1071 | CanonicalForm tmp; |
---|
| 1072 | if(x1.level() <= 1 && x2.level() <= 1) // base case |
---|
| 1073 | { |
---|
| 1074 | tryExtgcd(q1,q2,tmp,xnew,qnew,fail); |
---|
| 1075 | if(fail) |
---|
| 1076 | return; |
---|
| 1077 | xnew = x1 + (x2-x1) * xnew * q1; |
---|
| 1078 | qnew = q1*q2; |
---|
| 1079 | xnew = mod(xnew,qnew); |
---|
| 1080 | return; |
---|
| 1081 | } |
---|
| 1082 | CanonicalForm tmp2; |
---|
| 1083 | xnew = 0; |
---|
| 1084 | qnew = q1 * q2; |
---|
| 1085 | // here: x1.level() > 1 || x2.level() > 1 |
---|
| 1086 | if(x1.level() > x2.level()) |
---|
| 1087 | { |
---|
| 1088 | for(CFIterator i=x1; i.hasTerms(); i++) |
---|
| 1089 | { |
---|
| 1090 | if(i.exp() == 0) // const. term |
---|
| 1091 | { |
---|
| 1092 | tryCRA(i.coeff(),q1,x2,q2,tmp,tmp2,fail); |
---|
| 1093 | if(fail) |
---|
| 1094 | return; |
---|
| 1095 | xnew += tmp; |
---|
| 1096 | } |
---|
| 1097 | else |
---|
| 1098 | { |
---|
| 1099 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1100 | if(fail) |
---|
| 1101 | return; |
---|
| 1102 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1103 | } |
---|
| 1104 | } |
---|
| 1105 | return; |
---|
| 1106 | } |
---|
| 1107 | // here: x1.level() <= x2.level() && ( x1.level() > 1 || x2.level() > 1 ) |
---|
| 1108 | if(x2.level() > x1.level()) |
---|
| 1109 | { |
---|
| 1110 | for(CFIterator j=x2; j.hasTerms(); j++) |
---|
| 1111 | { |
---|
| 1112 | if(j.exp() == 0) // const. term |
---|
| 1113 | { |
---|
| 1114 | tryCRA(x1,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1115 | if(fail) |
---|
| 1116 | return; |
---|
| 1117 | xnew += tmp; |
---|
| 1118 | } |
---|
| 1119 | else |
---|
| 1120 | { |
---|
| 1121 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1122 | if(fail) |
---|
| 1123 | return; |
---|
| 1124 | xnew += tmp * power(x2.mvar(),j.exp()); |
---|
| 1125 | } |
---|
| 1126 | } |
---|
| 1127 | return; |
---|
| 1128 | } |
---|
| 1129 | // here: x1.level() == x2.level() && x1.level() > 1 && x2.level() > 1 |
---|
| 1130 | CFIterator i = x1; |
---|
| 1131 | CFIterator j = x2; |
---|
| 1132 | while(i.hasTerms() || j.hasTerms()) |
---|
| 1133 | { |
---|
| 1134 | if(i.hasTerms()) |
---|
| 1135 | { |
---|
| 1136 | if(j.hasTerms()) |
---|
| 1137 | { |
---|
| 1138 | if(i.exp() == j.exp()) |
---|
| 1139 | { |
---|
| 1140 | tryCRA(i.coeff(),q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1141 | if(fail) |
---|
| 1142 | return; |
---|
| 1143 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1144 | i++; j++; |
---|
| 1145 | } |
---|
| 1146 | else |
---|
| 1147 | { |
---|
| 1148 | if(i.exp() < j.exp()) |
---|
| 1149 | { |
---|
| 1150 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1151 | if(fail) |
---|
| 1152 | return; |
---|
| 1153 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1154 | i++; |
---|
| 1155 | } |
---|
| 1156 | else // i.exp() > j.exp() |
---|
| 1157 | { |
---|
| 1158 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1159 | if(fail) |
---|
| 1160 | return; |
---|
| 1161 | xnew += tmp * power(x1.mvar(),j.exp()); |
---|
| 1162 | j++; |
---|
| 1163 | } |
---|
| 1164 | } |
---|
| 1165 | } |
---|
| 1166 | else // j is out of terms |
---|
| 1167 | { |
---|
| 1168 | tryCRA(i.coeff(),q1,0,q2,tmp,tmp2,fail); |
---|
| 1169 | if(fail) |
---|
| 1170 | return; |
---|
| 1171 | xnew += tmp * power(x1.mvar(),i.exp()); |
---|
| 1172 | i++; |
---|
| 1173 | } |
---|
| 1174 | } |
---|
| 1175 | else // i is out of terms |
---|
| 1176 | { |
---|
| 1177 | tryCRA(0,q1,j.coeff(),q2,tmp,tmp2,fail); |
---|
| 1178 | if(fail) |
---|
| 1179 | return; |
---|
| 1180 | xnew += tmp * power(x1.mvar(),j.exp()); |
---|
| 1181 | j++; |
---|
| 1182 | } |
---|
| 1183 | } |
---|
| 1184 | } |
---|
| 1185 | |
---|