[e4fe2b] | 1 | #include "config.h" |
---|
[9c3d69] | 2 | #include "canonicalform.h" |
---|
[e4fe2b] | 3 | |
---|
[0c4a34b] | 4 | #ifdef HAVE_BIFAC |
---|
[e4fe2b] | 5 | |
---|
| 6 | # ifndef NOSTREAMIO |
---|
| 7 | # include<fstream> |
---|
| 8 | # endif |
---|
| 9 | |
---|
| 10 | # include <sys/timeb.h> |
---|
[6ead9d] | 11 | |
---|
| 12 | |
---|
| 13 | |
---|
| 14 | static void |
---|
| 15 | fillVarsRec ( const CanonicalForm & f, int * vars ) |
---|
| 16 | { |
---|
| 17 | int n; |
---|
| 18 | if ( (n = f.level()) > 0 ) |
---|
| 19 | { |
---|
| 20 | vars[n] = 1; |
---|
| 21 | CFIterator i; |
---|
| 22 | for ( i = f; i.hasTerms(); ++i ) |
---|
| 23 | fillVarsRec( i.coeff(), vars ); |
---|
| 24 | } |
---|
| 25 | } |
---|
| 26 | |
---|
| 27 | int ExtensionLevel(); |
---|
| 28 | void Reduce( bool); |
---|
| 29 | |
---|
| 30 | CanonicalForm MYGCD( const CanonicalForm& f, const CanonicalForm& g); |
---|
| 31 | |
---|
| 32 | |
---|
| 33 | CanonicalForm MyContent( const CanonicalForm& F, const Variable& x) |
---|
| 34 | { |
---|
| 35 | CanonicalForm r,t; |
---|
| 36 | CanonicalForm g=F; |
---|
| 37 | CanonicalForm one=1; |
---|
| 38 | |
---|
| 39 | if( F.isZero() ) return 0; |
---|
[806c18] | 40 | if( F.inBaseDomain() ) return F; |
---|
[6ead9d] | 41 | |
---|
[806c18] | 42 | if( level(F) < 0 ) return 1; |
---|
[6ead9d] | 43 | |
---|
| 44 | r = LC( F, x); |
---|
| 45 | |
---|
| 46 | g = g - power(x,degree(g,x))*r; |
---|
| 47 | |
---|
| 48 | while( g.isZero() != 1 && r!= 1 && r!=-1 ) |
---|
| 49 | { |
---|
| 50 | t = LC(g, x); |
---|
[806c18] | 51 | if( t == 1 || t == -1 ) return 1; |
---|
[6ead9d] | 52 | r = MYGCD( r, t); |
---|
[806c18] | 53 | if( r == 1 ) return 1; |
---|
| 54 | g = g - power(x,degree(g,x))*t; |
---|
[6ead9d] | 55 | } |
---|
| 56 | return r; |
---|
| 57 | } |
---|
| 58 | |
---|
| 59 | void CurrentExtension() |
---|
| 60 | { |
---|
| 61 | Variable x('x'); |
---|
| 62 | int i; |
---|
| 63 | #ifndef NOSTREAMIO |
---|
| 64 | cout << "Current Extension: "<<endl; |
---|
| 65 | #endif |
---|
| 66 | for (i = ExtensionLevel();i>0;i--) |
---|
| 67 | { |
---|
| 68 | Variable l(-i); |
---|
| 69 | #ifndef NOSTREAMIO |
---|
| 70 | cout << "Variable: "<<l<<" Level: "<<l.level()<<" Minimal Polynom: "<<getMipo(l,x)<<endl; |
---|
| 71 | #endif |
---|
| 72 | } |
---|
| 73 | } |
---|
| 74 | |
---|
| 75 | /*Liefert den ggt aller numerischen Koeffizienten einer Canonischen Form */ |
---|
| 76 | |
---|
| 77 | CanonicalForm MyNum(const CanonicalForm & a) |
---|
| 78 | { |
---|
| 79 | bool bruch = isOn(SW_RATIONAL); |
---|
| 80 | Off (SW_RATIONAL); |
---|
| 81 | |
---|
| 82 | CanonicalForm dummy =0; |
---|
| 83 | CanonicalForm dummy2; |
---|
| 84 | |
---|
| 85 | CFIterator F =a; |
---|
| 86 | |
---|
| 87 | for ( ; F.hasTerms(); F++) |
---|
| 88 | { |
---|
| 89 | if (F.coeff().inBaseDomain()) |
---|
| 90 | { |
---|
| 91 | dummy2 = F.coeff().num(); |
---|
| 92 | if (dummy == 0) |
---|
| 93 | { |
---|
| 94 | dummy = dummy2; |
---|
| 95 | } |
---|
| 96 | else |
---|
| 97 | { |
---|
| 98 | dummy = gcd(dummy, dummy2); |
---|
| 99 | } |
---|
| 100 | } |
---|
| 101 | else |
---|
| 102 | { |
---|
| 103 | dummy2 = MyNum(F.coeff()); |
---|
| 104 | if (dummy == 0) |
---|
| 105 | { |
---|
| 106 | dummy = dummy2; |
---|
| 107 | } |
---|
| 108 | else |
---|
| 109 | { |
---|
| 110 | dummy = gcd(dummy, dummy2); |
---|
| 111 | } |
---|
| 112 | } |
---|
| 113 | } |
---|
| 114 | if (bruch) |
---|
| 115 | On (SW_RATIONAL); |
---|
| 116 | else |
---|
| 117 | Off(SW_RATIONAL); |
---|
| 118 | return dummy; |
---|
| 119 | } |
---|
| 120 | |
---|
| 121 | /* Liefert den kgV aller Nenner der Koeffizenten einer Canonischen Form */ |
---|
| 122 | |
---|
| 123 | CanonicalForm MyDen(const CanonicalForm & a) |
---|
| 124 | { |
---|
| 125 | bool bruch = isOn(SW_RATIONAL); |
---|
| 126 | Off (SW_RATIONAL); |
---|
| 127 | |
---|
| 128 | CanonicalForm dummy(1); |
---|
| 129 | CanonicalForm dummy2; |
---|
| 130 | |
---|
| 131 | CFIterator F =a; |
---|
| 132 | |
---|
| 133 | for ( ; F.hasTerms(); F++) |
---|
| 134 | { |
---|
| 135 | if (F.coeff().inBaseDomain()) |
---|
| 136 | { |
---|
| 137 | dummy2 = gcd(dummy,F.coeff().den()); |
---|
| 138 | dummy = dummy * F.coeff().den()/dummy2; |
---|
| 139 | } |
---|
| 140 | else |
---|
| 141 | { |
---|
| 142 | dummy2 = MyDen(F.coeff()); |
---|
| 143 | dummy = dummy*dummy2/gcd(dummy,dummy2); |
---|
| 144 | } |
---|
| 145 | } |
---|
| 146 | if (bruch) |
---|
| 147 | On (SW_RATIONAL); |
---|
| 148 | else |
---|
| 149 | Off(SW_RATIONAL); |
---|
| 150 | return dummy; |
---|
| 151 | } |
---|
| 152 | |
---|
| 153 | /* Liefert die normierte Canonische Form a zurück, wenn LC(a) kein Nullteiler in Characteristic p ist */ |
---|
| 154 | /* sonst -1*/ |
---|
| 155 | CanonicalForm MyMonic(const CanonicalForm & a, const CanonicalForm & r, const int & l) |
---|
| 156 | { |
---|
| 157 | bool bruch = isOn(SW_RATIONAL); |
---|
| 158 | int zaehler; |
---|
| 159 | int Level = l; |
---|
| 160 | CanonicalForm dummy, dummy1, dummy2; |
---|
| 161 | CanonicalForm g = a; |
---|
| 162 | CanonicalForm p = r; |
---|
| 163 | |
---|
| 164 | On (SW_RATIONAL); |
---|
| 165 | |
---|
| 166 | if (Level == g.level()) |
---|
| 167 | { |
---|
| 168 | dummy = 1/g.LC(); |
---|
| 169 | } |
---|
| 170 | else |
---|
| 171 | { |
---|
| 172 | dummy = 1/g; |
---|
| 173 | } |
---|
| 174 | dummy1 = MyDen(dummy); |
---|
| 175 | dummy2 =dummy1; |
---|
| 176 | zaehler =1; |
---|
| 177 | |
---|
| 178 | Off (SW_RATIONAL); |
---|
| 179 | |
---|
| 180 | while ((mod(dummy2,p) != 1) && (mod(dummy2,p) !=0)) |
---|
| 181 | { |
---|
| 182 | dummy2 = dummy2*dummy1; |
---|
| 183 | dummy2 =mod(dummy2,p); |
---|
| 184 | zaehler++; |
---|
| 185 | } |
---|
| 186 | if (mod(dummy2,p).isZero()) |
---|
| 187 | { |
---|
| 188 | if (bruch) |
---|
| 189 | On (SW_RATIONAL); |
---|
| 190 | else |
---|
| 191 | Off(SW_RATIONAL); |
---|
| 192 | return -1; |
---|
| 193 | } |
---|
| 194 | else |
---|
| 195 | { |
---|
| 196 | zaehler--; |
---|
| 197 | dummy2 = power(dummy1,zaehler); |
---|
| 198 | dummy2 = mod(dummy2,p); |
---|
| 199 | dummy*= dummy1; |
---|
| 200 | dummy*= dummy2; |
---|
| 201 | g =mod(g*dummy,p); |
---|
| 202 | if (bruch) |
---|
| 203 | On (SW_RATIONAL); |
---|
| 204 | else |
---|
| 205 | Off(SW_RATIONAL); |
---|
| 206 | return g; |
---|
| 207 | } |
---|
| 208 | } |
---|
| 209 | |
---|
| 210 | |
---|
| 211 | |
---|
| 212 | /*Berechnet den ggT der Formen a und b in Characteristic p*/ |
---|
| 213 | |
---|
| 214 | CanonicalForm MyGCDlocal (const CanonicalForm & a, const CanonicalForm & b, const CanonicalForm & r, const int & l) |
---|
| 215 | { |
---|
| 216 | bool bruch = isOn(SW_RATIONAL); |
---|
| 217 | Off(SW_RATIONAL); |
---|
| 218 | CanonicalForm Rest, Result; |
---|
| 219 | |
---|
| 220 | CanonicalForm f=a; |
---|
| 221 | CanonicalForm g=b; |
---|
| 222 | CanonicalForm p=r; |
---|
| 223 | int Level =l; |
---|
| 224 | f = mod(f,p); |
---|
| 225 | g = mod(g,p); |
---|
| 226 | |
---|
| 227 | Rest=g; |
---|
| 228 | |
---|
| 229 | while (!Rest.isZero()) |
---|
| 230 | { |
---|
| 231 | g = MyMonic(g,p,Level); |
---|
| 232 | |
---|
| 233 | if (g == -1) |
---|
| 234 | { |
---|
| 235 | if (bruch) |
---|
| 236 | On(SW_RATIONAL); |
---|
| 237 | else |
---|
| 238 | Off(SW_RATIONAL); |
---|
| 239 | return -1; |
---|
| 240 | } |
---|
| 241 | else |
---|
| 242 | { |
---|
| 243 | Result =g; |
---|
| 244 | Rest = f%g; |
---|
| 245 | f = g; |
---|
| 246 | g = Rest; |
---|
| 247 | f =mod(f,p); |
---|
| 248 | g =mod(g,p); |
---|
| 249 | Rest =mod(Rest,p); |
---|
| 250 | } |
---|
| 251 | } |
---|
| 252 | if (bruch) |
---|
| 253 | On(SW_RATIONAL); |
---|
| 254 | else |
---|
| 255 | Off(SW_RATIONAL); |
---|
| 256 | return Result; |
---|
| 257 | } |
---|
| 258 | |
---|
| 259 | /* Chinese Remaindering für a mod m und b mod l */ |
---|
| 260 | |
---|
| 261 | CanonicalForm MyChiRem(const CanonicalForm & a,const CanonicalForm & m,const CanonicalForm & b,const CanonicalForm & l) |
---|
| 262 | { |
---|
| 263 | bool bruch = isOn(SW_RATIONAL); |
---|
| 264 | |
---|
| 265 | CanonicalForm u,v,Runner; |
---|
| 266 | CanonicalForm Result(0); |
---|
| 267 | CanonicalForm LeadTerm; |
---|
| 268 | CanonicalForm x1=a; |
---|
| 269 | CanonicalForm m1 = m; |
---|
| 270 | CanonicalForm x2=b; |
---|
| 271 | CanonicalForm m2 = l; |
---|
| 272 | |
---|
| 273 | while (!x1.isZero() || !x2.isZero()) |
---|
| 274 | { |
---|
| 275 | if (x1.degree() > x2.degree()) |
---|
| 276 | { |
---|
| 277 | LeadTerm = power(x1.mvar(),x1.degree()); |
---|
| 278 | u = x1.LC()*LeadTerm; |
---|
| 279 | v = 0; |
---|
| 280 | x1 = x1-u; |
---|
| 281 | } |
---|
| 282 | else |
---|
| 283 | { |
---|
| 284 | if (x1.degree() < x2.degree()) |
---|
| 285 | { |
---|
| 286 | u = 0; |
---|
| 287 | LeadTerm = power(x2.mvar(),x2.degree()); |
---|
| 288 | v = x2.LC()*LeadTerm; |
---|
| 289 | x2 = x2-v; |
---|
| 290 | } |
---|
| 291 | else |
---|
| 292 | { |
---|
| 293 | if (x1.degree() == x2.degree()) |
---|
| 294 | { |
---|
| 295 | LeadTerm = power(x2.mvar(),x2.degree()); |
---|
| 296 | u = x1.LC()*LeadTerm; |
---|
| 297 | v = x2.LC()*LeadTerm; |
---|
| 298 | x1 = x1-u; |
---|
| 299 | x2 = x2-v; |
---|
| 300 | } |
---|
| 301 | } |
---|
| 302 | } |
---|
| 303 | |
---|
| 304 | if (u.LC().inBaseDomain() && v.LC().inBaseDomain()) |
---|
| 305 | { |
---|
| 306 | Runner = u.LC(); |
---|
| 307 | Off(SW_RATIONAL); |
---|
| 308 | while(mod(Runner,m2) !=v.LC()) |
---|
| 309 | { |
---|
| 310 | Runner = Runner+m1; |
---|
| 311 | } |
---|
| 312 | |
---|
| 313 | Result = Result+Runner*LeadTerm; |
---|
| 314 | |
---|
| 315 | } |
---|
| 316 | else |
---|
| 317 | { |
---|
| 318 | Result = Result + MyChiRem(u.LC(),m1, v.LC(), m2)*LeadTerm; |
---|
| 319 | } |
---|
| 320 | } |
---|
| 321 | if (bruch) |
---|
| 322 | On(SW_RATIONAL); |
---|
| 323 | else |
---|
| 324 | Off(SW_RATIONAL); |
---|
| 325 | return Result; |
---|
| 326 | } |
---|
| 327 | |
---|
| 328 | /*Rational Rekonstruction für a mod b*/ |
---|
| 329 | |
---|
| 330 | CanonicalForm MyRatRed(const CanonicalForm & a,const CanonicalForm & b) |
---|
| 331 | { |
---|
| 332 | bool bruch = isOn(SW_RATIONAL); |
---|
| 333 | |
---|
| 334 | CanonicalForm f,dummy,dummy1,dummy2, Wurzel; |
---|
| 335 | CanonicalForm q,u0,u1,v0,v1; |
---|
| 336 | CanonicalForm Result(0); |
---|
| 337 | |
---|
| 338 | CFIterator F =a; |
---|
| 339 | |
---|
| 340 | for ( ; F.hasTerms(); F++) |
---|
| 341 | { |
---|
| 342 | if (F.coeff().inBaseDomain()) |
---|
| 343 | { |
---|
| 344 | Wurzel = sqrt(b); |
---|
| 345 | u0 =b; |
---|
| 346 | u1= F.coeff(); |
---|
| 347 | v1 = 1; |
---|
| 348 | v0 = 0; |
---|
| 349 | |
---|
| 350 | int i=0; |
---|
| 351 | |
---|
| 352 | while(!(u1<Wurzel)) |
---|
| 353 | { |
---|
| 354 | Off(SW_RATIONAL); |
---|
| 355 | q=u0/u1; |
---|
| 356 | dummy = u0-q*u1; |
---|
| 357 | u0=u1; |
---|
| 358 | u1=dummy; |
---|
| 359 | dummy = v0+q*v1; |
---|
| 360 | v0=v1; |
---|
| 361 | v1=dummy; |
---|
| 362 | i++; |
---|
| 363 | } |
---|
| 364 | f = -1; |
---|
| 365 | |
---|
| 366 | On(SW_RATIONAL); |
---|
| 367 | dummy2 = u1/v1; |
---|
| 368 | |
---|
| 369 | f = power(f,i)*dummy2; |
---|
| 370 | |
---|
| 371 | if (f.isZero()) |
---|
| 372 | { |
---|
| 373 | Off(SW_RATIONAL); |
---|
| 374 | |
---|
| 375 | if (!mod(F.coeff(),b).isZero()) |
---|
| 376 | { |
---|
| 377 | if (bruch) |
---|
| 378 | On(SW_RATIONAL); |
---|
| 379 | else |
---|
| 380 | Off(SW_RATIONAL); |
---|
| 381 | return -1; |
---|
| 382 | } |
---|
| 383 | |
---|
| 384 | } |
---|
| 385 | Result = Result+f*power(a.mvar(),F.exp()); |
---|
| 386 | |
---|
| 387 | } |
---|
| 388 | else |
---|
| 389 | { |
---|
| 390 | dummy1 = MyRatRed(F.coeff(),b); |
---|
| 391 | if (dummy1 == -1) |
---|
| 392 | { |
---|
| 393 | if (bruch) |
---|
| 394 | On(SW_RATIONAL); |
---|
| 395 | else |
---|
| 396 | Off(SW_RATIONAL); |
---|
| 397 | return -1; |
---|
| 398 | } |
---|
| 399 | else |
---|
| 400 | Result = Result + dummy1*power(a.mvar(),F.exp()); |
---|
| 401 | } |
---|
| 402 | } |
---|
| 403 | if (bruch) |
---|
| 404 | On(SW_RATIONAL); |
---|
| 405 | else |
---|
| 406 | Off(SW_RATIONAL); |
---|
| 407 | return Result; |
---|
| 408 | } |
---|
| 409 | |
---|
| 410 | /*Berechnet lokale ggT's der Formen a und b und liftet sie wieder hoch*/ |
---|
| 411 | |
---|
| 412 | CanonicalForm MyGCDmod( const CanonicalForm & a,const CanonicalForm & b) |
---|
| 413 | { |
---|
| 414 | bool bruch = isOn(SW_RATIONAL); |
---|
| 415 | // cout << "enter MyGCD mit a= "<<a<<endl; |
---|
| 416 | // cout << "und b= "<<b<<endl; |
---|
| 417 | CanonicalForm LeadA, LeadB; |
---|
| 418 | CanonicalForm Kandidat,Kandidat1; |
---|
| 419 | CanonicalForm f,g, Result; |
---|
| 420 | CanonicalForm NennerA =1; |
---|
| 421 | CanonicalForm NennerB=1; |
---|
| 422 | CanonicalForm ZahlerA=1; |
---|
| 423 | CanonicalForm ZahlerB=1; |
---|
| 424 | int treffer = 0; |
---|
| 425 | int Level; |
---|
| 426 | CanonicalForm Modulo; |
---|
| 427 | int i = 0; |
---|
| 428 | bool TryAgain = 1; |
---|
| 429 | int Primes[1228]; |
---|
| 430 | |
---|
| 431 | |
---|
| 432 | |
---|
| 433 | for (int i = 0;i <1228;i++) |
---|
| 434 | { |
---|
| 435 | Primes[i]=cf_getPrime(i+1); |
---|
| 436 | |
---|
| 437 | } |
---|
| 438 | Level=a.level(); |
---|
| 439 | |
---|
| 440 | if (a.degree() > b.degree()) |
---|
| 441 | { |
---|
| 442 | f = a; |
---|
| 443 | g = b; |
---|
| 444 | } |
---|
| 445 | else |
---|
| 446 | { |
---|
| 447 | g = a; |
---|
| 448 | f = b; |
---|
| 449 | } |
---|
| 450 | |
---|
| 451 | if (g.isZero()) |
---|
| 452 | { |
---|
| 453 | if (f.isZero()) return CanonicalForm(1); |
---|
| 454 | return f/f.LC(); |
---|
| 455 | } |
---|
| 456 | |
---|
| 457 | NennerA = MyDen(f); |
---|
| 458 | NennerB = MyDen(g); |
---|
| 459 | |
---|
| 460 | f = f*NennerA; |
---|
| 461 | g = g*NennerB; |
---|
| 462 | |
---|
| 463 | ZahlerA = MyNum(f); |
---|
| 464 | ZahlerB = MyNum(g); |
---|
| 465 | |
---|
| 466 | f=f/ZahlerA; |
---|
| 467 | g=g/ZahlerB; |
---|
| 468 | |
---|
| 469 | LeadA = f.LC(); |
---|
| 470 | while (!LeadA.inBaseDomain()) |
---|
| 471 | { |
---|
| 472 | LeadA =LeadA.LC(); |
---|
| 473 | } |
---|
| 474 | LeadB = g.LC(); |
---|
| 475 | while (!LeadB.inBaseDomain()) |
---|
| 476 | { |
---|
| 477 | LeadB =LeadB.LC(); |
---|
| 478 | } |
---|
| 479 | |
---|
| 480 | Off (SW_RATIONAL); |
---|
| 481 | |
---|
| 482 | |
---|
| 483 | while (TryAgain && i < 1228) |
---|
| 484 | { |
---|
| 485 | |
---|
| 486 | CanonicalForm p(Primes[i]); |
---|
| 487 | // cout << "p: "<<p<<endl; |
---|
| 488 | i++; |
---|
| 489 | if ( (mod(LeadA,p) != 0) && (mod(LeadB,p) != 0)) |
---|
| 490 | { |
---|
| 491 | Result = MyGCDlocal(f,g,p,Level); |
---|
| 492 | |
---|
| 493 | if (Result !=-1) |
---|
| 494 | { |
---|
| 495 | |
---|
| 496 | if (Result == 1) |
---|
| 497 | { |
---|
| 498 | if (bruch) |
---|
| 499 | On(SW_RATIONAL); |
---|
| 500 | else |
---|
| 501 | Off(SW_RATIONAL); |
---|
| 502 | return Result; |
---|
| 503 | } |
---|
| 504 | else |
---|
| 505 | { |
---|
| 506 | if (treffer == 0 || Kandidat.degree() > Result.degree()) |
---|
| 507 | { |
---|
| 508 | treffer = 1; |
---|
| 509 | Kandidat = Result; |
---|
| 510 | Modulo = p; |
---|
| 511 | } |
---|
| 512 | else |
---|
| 513 | { |
---|
| 514 | if (Kandidat.degree() == Result.degree()) |
---|
| 515 | { |
---|
| 516 | Kandidat = MyChiRem(Kandidat,Modulo,Result,p); |
---|
| 517 | Modulo = Modulo*p; |
---|
| 518 | treffer++; |
---|
| 519 | } |
---|
| 520 | } |
---|
| 521 | if (mod(treffer,4) ==1) |
---|
| 522 | { |
---|
| 523 | Kandidat1=MyRatRed(Kandidat, Modulo); |
---|
| 524 | |
---|
| 525 | |
---|
| 526 | if (Kandidat1 !=-1) |
---|
| 527 | { |
---|
| 528 | Off(SW_RATIONAL); |
---|
| 529 | if (mod(f,Kandidat1) == 0 && mod(g,Kandidat1) == 0) |
---|
| 530 | { |
---|
| 531 | break; |
---|
| 532 | } |
---|
| 533 | |
---|
| 534 | } |
---|
| 535 | } |
---|
| 536 | |
---|
| 537 | |
---|
| 538 | } |
---|
| 539 | } |
---|
| 540 | } |
---|
| 541 | else |
---|
| 542 | { |
---|
| 543 | |
---|
| 544 | } |
---|
| 545 | |
---|
| 546 | } |
---|
| 547 | |
---|
| 548 | if (bruch) |
---|
| 549 | On(SW_RATIONAL); |
---|
| 550 | else |
---|
| 551 | Off(SW_RATIONAL); |
---|
| 552 | return Kandidat1; |
---|
| 553 | } |
---|
| 554 | |
---|
| 555 | /* Berechnet die Norm eines Form h über zwei Körpererweiterungen und faktorisiert sie*/ |
---|
| 556 | |
---|
| 557 | CFFList FactorizeNorm (const CanonicalForm & h, const int & i ) |
---|
| 558 | { |
---|
| 559 | bool bruch = isOn(SW_RATIONAL); |
---|
| 560 | if (i ==0) |
---|
| 561 | { return factorize(h); |
---|
| 562 | } |
---|
| 563 | |
---|
| 564 | |
---|
| 565 | CanonicalForm g =h; |
---|
| 566 | |
---|
| 567 | Variable x =g.mvar(); |
---|
| 568 | |
---|
| 569 | int AnzExt = i; // Über welcher Erweiterung arbeite ich gerade ... |
---|
| 570 | Variable l(-AnzExt); //... und welche algebr. Variable gehört dazu ? |
---|
| 571 | Variable y('_'); |
---|
| 572 | |
---|
| 573 | |
---|
| 574 | CanonicalForm MiPo, Norm, NormAbl, Factor_Norm,dummy1, dummy2, Nenner,LeaC; |
---|
| 575 | |
---|
| 576 | CFFList Result; |
---|
| 577 | CFFList dummy; |
---|
| 578 | |
---|
| 579 | bool is = true; |
---|
| 580 | |
---|
| 581 | int k = 0; |
---|
| 582 | g = g(y,l); //die algeb. Variable wird durch eine Polynomvariable ersetzt |
---|
| 583 | MiPo = getMipo(l,y); |
---|
| 584 | |
---|
| 585 | Norm = resultant(MiPo,g,y); //norm von g als Funk. in x und y (l->y) bzgl y |
---|
| 586 | NormAbl = Norm.deriv(); |
---|
| 587 | // Off(SW_RATIONAL); |
---|
| 588 | is = !MyGCDmod(Norm,NormAbl).inBaseDomain(); //ist die Norm quadratfrei ? |
---|
| 589 | while (is) |
---|
| 590 | { |
---|
| 591 | k++; |
---|
| 592 | CanonicalForm t = g; |
---|
| 593 | t = t(x-k*y,x); //wenn nicht, wird g gestört und die neue Norm berechnet |
---|
| 594 | |
---|
| 595 | On(SW_RATIONAL); |
---|
| 596 | Norm = resultant(MiPo,t,y); //Problem tritt hier auf, bei AnzExt = 1 |
---|
| 597 | Off(SW_RATIONAL); |
---|
| 598 | NormAbl = Norm.deriv(); |
---|
| 599 | is = ! MyGCDmod(Norm,NormAbl).inBaseDomain(); |
---|
| 600 | //cout << "ggt der Norm: "<< MyGCDmod(Norm,NormAbl)<<endl; |
---|
| 601 | } |
---|
| 602 | AnzExt--; |
---|
| 603 | if (bruch) |
---|
| 604 | On(SW_RATIONAL); |
---|
| 605 | else |
---|
| 606 | Off(SW_RATIONAL); |
---|
| 607 | if (AnzExt == 0) //sind alle Erweiterungen abgearbeitet... |
---|
| 608 | { |
---|
| 609 | Result = factorize(Norm); //... wird die Norm Faktorisiert |
---|
| 610 | } |
---|
| 611 | else |
---|
| 612 | { |
---|
| 613 | Result = FactorizeNorm(Norm, AnzExt); //wenn nicht, kommt die nächste erweiterung dran |
---|
| 614 | } |
---|
| 615 | CFFListIterator J=Result; |
---|
| 616 | for ( ; J.hasItem(); J++) |
---|
[806c18] | 617 | { |
---|
[6ead9d] | 618 | Factor_Norm = J.getItem().factor(); |
---|
| 619 | Factor_Norm = Factor_Norm(x+k*l,x); // die Störungen werden rückgänig gemacht |
---|
| 620 | dummy.append(CFFactor(Factor_Norm)); |
---|
| 621 | } |
---|
| 622 | return dummy; |
---|
| 623 | } |
---|
| 624 | |
---|
| 625 | |
---|
| 626 | /* Bereitet die Form h vor, ruft FactorizeNorm(h) auf und rekonstruiert daraus die |
---|
| 627 | Faktoren von h */ |
---|
| 628 | |
---|
| 629 | CFFList MyFactorize(const CanonicalForm & h) |
---|
| 630 | { |
---|
| 631 | bool bruch = isOn(SW_RATIONAL); |
---|
| 632 | |
---|
| 633 | CanonicalForm g = h; |
---|
| 634 | CFFList FacNorm, Result; // Faktorisierung der Norm und das Ergebniss |
---|
| 635 | CanonicalForm Abl_g, LeaCoeff_g, normiert_g, g_quadrat ; //Ableitung, führender Koeff. und Normierung von g |
---|
| 636 | CanonicalForm Norm,NormAbl; |
---|
| 637 | CanonicalForm Factor_Norm, Fac; |
---|
| 638 | CanonicalForm dummy, g_origin; |
---|
| 639 | CanonicalForm Nenner,warte; |
---|
| 640 | |
---|
| 641 | Variable x = g.mvar(); |
---|
| 642 | |
---|
| 643 | int exp =0; |
---|
| 644 | int DegAlt, DegNeu; |
---|
| 645 | On(SW_RATIONAL); |
---|
| 646 | |
---|
| 647 | /* Initzialisierung, faktorisiert wird CF g */ |
---|
| 648 | g_origin = g; |
---|
| 649 | LeaCoeff_g = g.LC(); |
---|
| 650 | //g /=LeaCoeff_g; |
---|
| 651 | Nenner = MyDen(g); |
---|
| 652 | g *= power(Nenner, g.degree()); |
---|
| 653 | g *= power(LeaCoeff_g,g.degree()-1); |
---|
| 654 | g = g(x/(Nenner*LeaCoeff_g),x); |
---|
| 655 | Abl_g = g.deriv(); |
---|
| 656 | DegAlt = g.degree(); |
---|
| 657 | g_quadrat=g; |
---|
| 658 | g /= MyGCDmod(g,Abl_g); // g wird quadratfrei gemacht |
---|
| 659 | DegNeu = g.degree(); |
---|
| 660 | |
---|
| 661 | //g = g/LeaCoeff_g; // und normiert |
---|
| 662 | //CurrentExtension(); |
---|
| 663 | FacNorm = FactorizeNorm(g,ExtensionLevel()); |
---|
| 664 | CFFListIterator J=FacNorm; |
---|
| 665 | J.lastItem(); |
---|
| 666 | // g = g*MyDen(g); |
---|
| 667 | // |
---|
| 668 | g = h ; |
---|
| 669 | |
---|
| 670 | for ( ; J.hasItem(); J--) // Iteration über die Faktoren der Norm |
---|
| 671 | { |
---|
| 672 | Factor_Norm = J.getItem().factor(); |
---|
| 673 | |
---|
| 674 | Fac = MyGCDmod(g,Factor_Norm); //Ergebniss wird hochgeliftet |
---|
| 675 | |
---|
| 676 | Fac = Fac/Fac.LC(); // und normiert |
---|
| 677 | |
---|
| 678 | /* Ermittlung der Exponenten der einzelnen Faktoren */ |
---|
| 679 | |
---|
| 680 | exp = 1; // für den FaKtor mit Grad 0 |
---|
| 681 | dummy = g_quadrat; |
---|
| 682 | |
---|
| 683 | if (!Fac.inBaseDomain()) // echter Faktor ? |
---|
| 684 | { |
---|
| 685 | exp = 0; |
---|
| 686 | while ( 0==dummy%Fac && !dummy.inBaseDomain()) // Wie oft Teilt der Faktor das Polynom ? |
---|
| 687 | { |
---|
| 688 | dummy =dummy/Fac; |
---|
| 689 | exp++; |
---|
| 690 | } |
---|
| 691 | Fac = Fac(x*(Nenner*LeaCoeff_g),x); |
---|
| 692 | |
---|
| 693 | Fac /= Fac.LC(); |
---|
| 694 | } |
---|
| 695 | |
---|
| 696 | else |
---|
| 697 | { |
---|
| 698 | Fac *= LeaCoeff_g; |
---|
| 699 | g *= LeaCoeff_g; |
---|
| 700 | } |
---|
| 701 | |
---|
| 702 | g /=power(Fac,exp); |
---|
| 703 | |
---|
| 704 | Result.append(CFFactor( Fac, exp )); // Faktor wird an Result gehängt |
---|
| 705 | } |
---|
| 706 | if (bruch) |
---|
| 707 | On(SW_RATIONAL); |
---|
| 708 | else |
---|
| 709 | Off(SW_RATIONAL); |
---|
| 710 | return Result; // und zurückgegeben |
---|
| 711 | } |
---|
| 712 | |
---|
| 713 | CFFList AbsFactorize(const CanonicalForm & a) |
---|
| 714 | { |
---|
| 715 | CanonicalForm f = a; |
---|
| 716 | CanonicalForm Factor, NewFactor,dummy3, Nenner,LeadC; |
---|
| 717 | CanonicalForm Coeff=f.LC(); |
---|
| 718 | |
---|
| 719 | Variable x =a.mvar(); |
---|
| 720 | CFFList dummy, dummy2; |
---|
| 721 | CFFList result, empty; |
---|
| 722 | empty.append(CFFactor(1,1)); |
---|
| 723 | bool NewRoot = false; |
---|
| 724 | bool fertig; |
---|
| 725 | |
---|
| 726 | LeadC = f.LC(); |
---|
| 727 | f *= power(LeadC, f.degree()-1); |
---|
| 728 | Nenner = MyDen(f); |
---|
| 729 | f *= power(Nenner, f.degree()); |
---|
| 730 | f = f(x/(Nenner*LeadC), x); |
---|
| 731 | result = MyFactorize(f); |
---|
| 732 | |
---|
| 733 | CFFListIterator L = result; |
---|
| 734 | fertig = true; |
---|
| 735 | for(; L.hasItem();L++) |
---|
| 736 | { |
---|
| 737 | if (L.getItem().factor().degree() >1) |
---|
| 738 | { |
---|
| 739 | fertig = false; |
---|
| 740 | } |
---|
| 741 | } |
---|
| 742 | |
---|
| 743 | while(!fertig) |
---|
| 744 | { |
---|
| 745 | dummy = result; |
---|
| 746 | CFFListIterator J = dummy; |
---|
| 747 | result = empty; |
---|
| 748 | for ( ; J.hasItem(); J++) // Iteration über die Faktoren der Norm |
---|
| 749 | { |
---|
| 750 | Factor = J.getItem().factor(); |
---|
| 751 | |
---|
| 752 | if (Factor.degree() != 0 && Factor.degree() != 1 && !NewRoot) |
---|
| 753 | { |
---|
| 754 | Reduce(false); |
---|
| 755 | Variable u = rootOf(Factor); |
---|
| 756 | Reduce(true); |
---|
| 757 | NewRoot = true; |
---|
| 758 | result.append(CFFactor((x-u),1)); |
---|
| 759 | Factor /= (x-u); |
---|
| 760 | } |
---|
| 761 | |
---|
| 762 | |
---|
| 763 | if (Factor.degree() != 0 && Factor.degree() != 1 && NewRoot) |
---|
| 764 | { |
---|
| 765 | dummy2 = MyFactorize(Factor); |
---|
| 766 | |
---|
| 767 | CFFListIterator H = dummy2; |
---|
| 768 | for ( ; H.hasItem(); H++) // Iteration über die Faktoren der Norm |
---|
| 769 | { |
---|
| 770 | NewFactor = H.getItem().factor(); |
---|
| 771 | if (!NewFactor.inBaseDomain()) |
---|
| 772 | { |
---|
| 773 | result.append(CFFactor(NewFactor, H.getItem().exp()*J.getItem().exp())); |
---|
| 774 | } |
---|
| 775 | else |
---|
| 776 | { |
---|
| 777 | Coeff *=H.getItem().factor(); |
---|
| 778 | } |
---|
| 779 | } |
---|
| 780 | } |
---|
| 781 | if ( Factor.degree() == 0) |
---|
| 782 | { |
---|
| 783 | Coeff *=Factor; |
---|
| 784 | } |
---|
| 785 | if( Factor.degree() == 1) |
---|
| 786 | { |
---|
| 787 | result.append(CFFactor(Factor,J.getItem().exp())); |
---|
| 788 | } |
---|
| 789 | } |
---|
| 790 | NewRoot = false; |
---|
| 791 | CFFListIterator K = result; |
---|
| 792 | fertig = true; |
---|
| 793 | |
---|
| 794 | for(; K.hasItem();K++) |
---|
| 795 | { |
---|
| 796 | |
---|
| 797 | if (K.getItem().factor().degree() >1) |
---|
| 798 | { |
---|
| 799 | fertig = false; |
---|
| 800 | } |
---|
| 801 | } |
---|
| 802 | } |
---|
| 803 | CFFList result2; |
---|
| 804 | //result2.append(CFFactor(Coeff)); |
---|
| 805 | CFFListIterator K = result; |
---|
| 806 | for(; K.hasItem();K++) |
---|
| 807 | { |
---|
| 808 | dummy3 = K.getItem().factor(); |
---|
| 809 | if (dummy3.degree() == 0) |
---|
| 810 | { dummy3 *= Coeff; |
---|
| 811 | } |
---|
| 812 | else |
---|
| 813 | { |
---|
| 814 | dummy3 = dummy3(x*Nenner*LeadC,x); |
---|
| 815 | dummy3 /= dummy3.LC(); |
---|
| 816 | } |
---|
| 817 | |
---|
| 818 | result2.append(CFFactor(dummy3,K.getItem().exp())); |
---|
| 819 | } |
---|
| 820 | return result2; |
---|
| 821 | } |
---|
| 822 | |
---|
| 823 | |
---|
| 824 | // |
---|
| 825 | // |
---|
| 826 | CanonicalForm Bigcd( const CanonicalForm& f, const CanonicalForm& g) |
---|
| 827 | { |
---|
| 828 | |
---|
| 829 | |
---|
| 830 | if( f.level() < 0 ) return 1; |
---|
| 831 | if( g.level() < 0 ) return 1; |
---|
| 832 | |
---|
| 833 | CFArray A; |
---|
| 834 | |
---|
| 835 | int i=0; |
---|
| 836 | int r; |
---|
| 837 | |
---|
| 838 | Variable x = f.mvar(); |
---|
| 839 | Variable y = g.mvar(); |
---|
| 840 | |
---|
| 841 | // Wahl als Hauptvariable ? |
---|
| 842 | // |
---|
| 843 | if( x.level() >= y.level() ) x = y; |
---|
| 844 | |
---|
| 845 | CanonicalForm Cf, Cg, gamma, c,T; |
---|
| 846 | CanonicalForm F=f; |
---|
| 847 | CanonicalForm G=g; |
---|
| 848 | |
---|
| 849 | Cf = MyContent(f,x); //changed |
---|
| 850 | Cg = MyContent(g,x); //changed |
---|
| 851 | F = F/Cf; |
---|
| 852 | G = G/Cg; |
---|
| 853 | gamma = MyGCDmod( LC(F,x), LC(G,x) ); |
---|
| 854 | |
---|
| 855 | A = subResChain(F,G,x); |
---|
| 856 | |
---|
| 857 | c = MyGCDmod( Cf, Cg ); |
---|
| 858 | |
---|
| 859 | r = A.size(); |
---|
| 860 | |
---|
| 861 | while( A[i].isZero() ) i++; |
---|
| 862 | |
---|
| 863 | F = A[i]; |
---|
| 864 | |
---|
[806c18] | 865 | if( degree(F,x) == 0 ) |
---|
| 866 | if( c.level() < 0 ) return 1; else return c; |
---|
[6ead9d] | 867 | |
---|
| 868 | F = gamma*F/LC(F, x); |
---|
| 869 | F = F/content(F,x); |
---|
| 870 | |
---|
| 871 | F = c*F; |
---|
| 872 | |
---|
| 873 | c = F.LC(); |
---|
| 874 | |
---|
| 875 | while( c.level()>0 ) c = c.LC(); |
---|
| 876 | |
---|
| 877 | F=F/c; |
---|
| 878 | |
---|
| 879 | if( F.level() < 0 ) return 1; |
---|
| 880 | |
---|
| 881 | return F; |
---|
| 882 | } |
---|
| 883 | |
---|
| 884 | |
---|
| 885 | |
---|
| 886 | |
---|
| 887 | CanonicalForm MYGCD( const CanonicalForm& f, const CanonicalForm& g) |
---|
| 888 | { |
---|
| 889 | |
---|
| 890 | // FIX ME: CONSTANT FIELD |
---|
| 891 | // |
---|
| 892 | // |
---|
| 893 | // |
---|
| 894 | // |
---|
| 895 | // |
---|
| 896 | if( f.level() < 0 && g.level() < 0) return 1; |
---|
| 897 | if( (f.level() < 0 && g.level() > 0) || |
---|
| 898 | (f.level() > 0 && g.level() <0 ) ) return 1; |
---|
| 899 | |
---|
| 900 | int i; |
---|
| 901 | |
---|
| 902 | CFList L; |
---|
| 903 | for (i=1; i<= level(f); i++) |
---|
| 904 | if( f != f(0,i) ) |
---|
| 905 | L.append(i); |
---|
| 906 | |
---|
| 907 | int nvf = L.length(); |
---|
| 908 | |
---|
| 909 | for (i=1; i<= level(g); i++) |
---|
| 910 | if( g != g(0,i) ) |
---|
| 911 | L.append(i); |
---|
[806c18] | 912 | |
---|
[6ead9d] | 913 | int nvg = L.length(); |
---|
| 914 | |
---|
[806c18] | 915 | |
---|
[6ead9d] | 916 | |
---|
| 917 | if( f.level() < 0 && g.level() < 0 ) { ; |
---|
| 918 | return 1; } |
---|
[806c18] | 919 | |
---|
[6ead9d] | 920 | CFArray A; |
---|
[806c18] | 921 | |
---|
[6ead9d] | 922 | i=0; |
---|
| 923 | int r; |
---|
| 924 | |
---|
| 925 | Variable x = f.mvar(); |
---|
| 926 | Variable y = g.mvar(); |
---|
| 927 | |
---|
| 928 | // Wahl als Hauptvariable ? |
---|
| 929 | // |
---|
| 930 | if( x.level() >= y.level() ) x = y; |
---|
| 931 | |
---|
| 932 | CanonicalForm Cf, Cg, gamma, c,T; |
---|
| 933 | CanonicalForm F=f; |
---|
| 934 | CanonicalForm G=g; |
---|
| 935 | |
---|
| 936 | |
---|
| 937 | Cf = MyContent(f,x); |
---|
| 938 | Cg = MyContent(g,x); |
---|
| 939 | F = F/Cf; |
---|
| 940 | G = G/Cg; |
---|
| 941 | |
---|
| 942 | if( nvf <= 1 && nvg <=1 ) |
---|
| 943 | { |
---|
[806c18] | 944 | gamma = MyGCDmod( LC(F,x), LC(G,x) ); |
---|
[6ead9d] | 945 | c = MyGCDmod( Cf, Cg ); |
---|
| 946 | } |
---|
| 947 | else |
---|
| 948 | { |
---|
| 949 | gamma = MYGCD( LC(F,x), LC(G,x) ); |
---|
| 950 | c = MYGCD( Cf, Cg ); |
---|
| 951 | } |
---|
| 952 | A = subResChain(F,G,x); |
---|
| 953 | |
---|
| 954 | r = A.size(); |
---|
| 955 | |
---|
| 956 | while( A[i].isZero() ) i++; |
---|
| 957 | |
---|
| 958 | F = A[i]; |
---|
| 959 | |
---|
[806c18] | 960 | if( degree(F,x) == 0 ) |
---|
| 961 | if( c.level() < 0 ) return 1; else return c; |
---|
[6ead9d] | 962 | |
---|
| 963 | F = gamma*F/LC(F, x); |
---|
| 964 | F = F/MyContent(F,x); |
---|
| 965 | |
---|
| 966 | F = c*F; |
---|
| 967 | |
---|
| 968 | c = F.LC(); |
---|
| 969 | |
---|
| 970 | while( c.level()>0 ) c = c.LC(); |
---|
| 971 | |
---|
| 972 | F=F/c; |
---|
| 973 | |
---|
| 974 | //if( F.level() < 0 ) return 1; |
---|
| 975 | |
---|
| 976 | return F; |
---|
| 977 | } |
---|
| 978 | |
---|
| 979 | |
---|
| 980 | |
---|
| 981 | CFFList Mysqrfree_local( const CanonicalForm& F, const Variable& v) |
---|
| 982 | { |
---|
| 983 | int i=1; |
---|
| 984 | CanonicalForm f=F; |
---|
| 985 | CanonicalForm g, qi, fp, wp, temp1, temp2, temp3, temp4, pA; |
---|
| 986 | CFFList L; |
---|
| 987 | |
---|
| 988 | |
---|
| 989 | g = MyContent(f,v); |
---|
| 990 | |
---|
| 991 | if( g != 1 ) |
---|
| 992 | L.append( CFFactor(g,1) ); |
---|
| 993 | |
---|
| 994 | pA = f/g; |
---|
| 995 | |
---|
| 996 | fp = deriv( pA, v); |
---|
| 997 | |
---|
| 998 | temp1 = MYGCD( pA, fp ); |
---|
| 999 | |
---|
| 1000 | if( temp1 == 1 ){ L.append( CFFactor(pA,1) ); return L; } |
---|
| 1001 | else |
---|
| 1002 | { |
---|
| 1003 | temp2 = pA/temp1; |
---|
| 1004 | temp3 = fp/temp1; |
---|
| 1005 | wp = deriv(temp2,v); |
---|
| 1006 | temp4 = temp3 - wp; |
---|
| 1007 | |
---|
| 1008 | while( !temp4.isZero() ) |
---|
| 1009 | { |
---|
| 1010 | CanonicalForm qi = MYGCD( temp2, temp4); |
---|
[806c18] | 1011 | if( qi != 1 ) L.append( CFFactor( qi, i ) ); |
---|
[6ead9d] | 1012 | i++; |
---|
| 1013 | temp2 = temp2/qi; |
---|
| 1014 | temp3 = temp4/qi; |
---|
| 1015 | temp4 = temp3-deriv(temp2, v); |
---|
| 1016 | } |
---|
| 1017 | |
---|
| 1018 | if( temp2 != 1 ) L.append( CFFactor( temp2, i ) ); |
---|
| 1019 | |
---|
| 1020 | } |
---|
| 1021 | |
---|
| 1022 | return L; |
---|
| 1023 | } |
---|
| 1024 | |
---|
| 1025 | CFFList Mysqrfree( const CanonicalForm& F ) |
---|
| 1026 | { |
---|
| 1027 | CFFList L, M, V; |
---|
| 1028 | CFFList N; |
---|
| 1029 | CanonicalForm vars=getVars(F); |
---|
| 1030 | Variable v; |
---|
| 1031 | CanonicalForm s; |
---|
| 1032 | bool b; |
---|
| 1033 | |
---|
| 1034 | L.append( CFFactor(F,1) ); |
---|
| 1035 | |
---|
| 1036 | int n = F.level(); |
---|
| 1037 | int *vrs = new int[n+1]; |
---|
| 1038 | for ( int i = 0; i <= n; i++ ) vars[i] = 0; |
---|
| 1039 | for ( CFIterator I = F; I.hasTerms(); ++I ) fillVarsRec( I.coeff(), vrs ); |
---|
| 1040 | |
---|
[806c18] | 1041 | N.append( CFFactor(F,1) ); |
---|
[6ead9d] | 1042 | |
---|
| 1043 | int i = n+1; |
---|
| 1044 | |
---|
| 1045 | while( i >= 0 ) |
---|
| 1046 | { |
---|
[806c18] | 1047 | b = 0; |
---|
[6ead9d] | 1048 | |
---|
| 1049 | if( i == 0 ){ v = mvar(F); b=1 ;} |
---|
| 1050 | else |
---|
[806c18] | 1051 | if( vrs[i] != 0 ){ b=1; v= Variable(i);} |
---|
| 1052 | if( vrs[i] == 0 ) i--; |
---|
[6ead9d] | 1053 | |
---|
| 1054 | if( b ) |
---|
[806c18] | 1055 | { |
---|
[6ead9d] | 1056 | for( CFFListIterator J = L; J.hasItem(); J++ ) |
---|
| 1057 | { |
---|
| 1058 | M = Mysqrfree_local( J.getItem().factor() , v ); |
---|
| 1059 | |
---|
| 1060 | for( CFFListIterator K = M; K.hasItem(); K++ ) |
---|
| 1061 | { |
---|
| 1062 | if( K.getItem().factor().level() > 0 ) |
---|
| 1063 | { |
---|
| 1064 | N.append( CFFactor( K.getItem().factor(), K.getItem().exp()+J.getItem().exp()-1 )); } |
---|
| 1065 | } |
---|
| 1066 | N.removeFirst(); |
---|
| 1067 | } |
---|
| 1068 | if( N.length() == L.length() ) i -= 1; |
---|
[806c18] | 1069 | L=N; |
---|
[6ead9d] | 1070 | } |
---|
| 1071 | } |
---|
| 1072 | |
---|
| 1073 | return L; |
---|
| 1074 | |
---|
| 1075 | } |
---|
[e4fe2b] | 1076 | #endif /* HAVE_BIFAC */ |
---|