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mpr_numeric.cc
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1 /****************************************
2 * Computer Algebra System SINGULAR *
3 ****************************************/
4 
5 
6 /*
7 * ABSTRACT - multipolynomial resultants - numeric stuff
8 * ( root finder, vandermonde system solver, simplex )
9 */
10 
11 #include "kernel/mod2.h"
12 
13 //#define mprDEBUG_ALL
14 
15 #include "misc/options.h"
16 
17 #include "coeffs/numbers.h"
18 #include "coeffs/mpr_global.h"
19 
20 #include "polys/matpol.h"
21 
22 #include "kernel/polys.h"
23 
24 #include "mpr_base.h"
25 #include "mpr_numeric.h"
26 
27 #include <cmath>
28 //<-
29 
30 //-----------------------------------------------------------------------------
31 //-------------- vandermonde system solver ------------------------------------
32 //-----------------------------------------------------------------------------
33 
34 //-> vandermonde::*
35 vandermonde::vandermonde( const long _cn, const long _n, const long _maxdeg,
36  number *_p, const bool _homog )
37  : n(_n), cn(_cn), maxdeg(_maxdeg), p(_p), homog(_homog)
38 {
39  long j;
40  l= (long)pow((double)maxdeg+1,(int)n);
41  x= (number *)omAlloc( cn * sizeof(number) );
42  for ( j= 0; j < cn; j++ ) x[j]= nInit(1);
43  init();
44 }
45 
47 {
48  int j;
49  for ( j= 0; j < cn; j++ ) nDelete( x+j );
50  omFreeSize( (void *)x, cn * sizeof( number ) );
51 }
52 
54 {
55  int j;
56  long i,c,sum;
57  number tmp,tmp1;
58 
59  c=0;
60  sum=0;
61 
62  intvec exp( n );
63  for ( j= 0; j < n; j++ ) exp[j]=0;
64 
65  for ( i= 0; i < l; i++ )
66  {
67  if ( !homog || (sum == maxdeg) )
68  {
69  for ( j= 0; j < n; j++ )
70  {
71  nPower( p[j], exp[j], &tmp );
72  tmp1 = nMult( tmp, x[c] );
73  x[c]= tmp1;
74  nDelete( &tmp );
75  }
76  c++;
77  }
78  exp[0]++;
79  sum=0;
80  for ( j= 0; j < n - 1; j++ )
81  {
82  if ( exp[j] > maxdeg )
83  {
84  exp[j]= 0;
85  exp[j + 1]++;
86  }
87  sum+= exp[j];
88  }
89  sum+= exp[n - 1];
90  }
91 }
92 
93 poly vandermonde::numvec2poly( const number * q )
94 {
95  int j;
96  long i,/*c,*/sum;
97 
98  poly pnew,pit=NULL;
99 
100  // c=0;
101  sum=0;
102 
103  int *exp= (int *) omAlloc( (n+1) * sizeof(int) );
104 
105  for ( j= 0; j < n+1; j++ ) exp[j]=0;
106 
107  for ( i= 0; i < l; i++ )
108  {
109  if ( (!homog || (sum == maxdeg)) && q[i] && !nIsZero(q[i]) )
110  {
111  pnew= pOne();
112  pSetCoeff(pnew,q[i]);
113  pSetExpV(pnew,exp);
114  if ( pit )
115  {
116  pNext(pnew)= pit;
117  pit= pnew;
118  }
119  else
120  {
121  pit= pnew;
122  pNext(pnew)= NULL;
123  }
124  pSetm(pit);
125  }
126  exp[1]++;
127  sum=0;
128  for ( j= 1; j < n; j++ )
129  {
130  if ( exp[j] > maxdeg )
131  {
132  exp[j]= 0;
133  exp[j + 1]++;
134  }
135  sum+= exp[j];
136  }
137  sum+= exp[n];
138  }
139 
140  omFreeSize( (void *) exp, (n+1) * sizeof(int) );
141 
142  pSortAdd(pit);
143  return pit;
144 }
145 
146 number * vandermonde::interpolateDense( const number * q )
147 {
148  int i,j,k;
149  number newnum,tmp1;
150  number b,t,xx,s;
151  number *c;
152  number *w;
153 
154  b=t=xx=s=tmp1=NULL;
155 
156  w= (number *)omAlloc( cn * sizeof(number) );
157  c= (number *)omAlloc( cn * sizeof(number) );
158  for ( j= 0; j < cn; j++ )
159  {
160  w[j]= nInit(0);
161  c[j]= nInit(0);
162  }
163 
164  if ( cn == 1 )
165  {
166  nDelete( &w[0] );
167  w[0]= nCopy(q[0]);
168  }
169  else
170  {
171  nDelete( &c[cn-1] );
172  c[cn-1]= nCopy(x[0]);
173  c[cn-1]= nInpNeg(c[cn-1]); // c[cn]= -x[1]
174 
175  for ( i= 1; i < cn; i++ ) { // i=2; i <= cn
176  nDelete( &xx );
177  xx= nCopy(x[i]);
178  xx= nInpNeg(xx); // xx= -x[i]
179 
180  for ( j= (cn-i-1); j <= (cn-2); j++) { // j=(cn+1-i); j <= (cn-1)
181  nDelete( &tmp1 );
182  tmp1= nMult( xx, c[j+1] ); // c[j]= c[j] + (xx * c[j+1])
183  newnum= nAdd( c[j], tmp1 );
184  nDelete( &c[j] );
185  c[j]= newnum;
186  }
187 
188  newnum= nAdd( xx, c[cn-1] ); // c[cn-1]= c[cn-1] + xx
189  nDelete( &c[cn-1] );
190  c[cn-1]= newnum;
191  }
192 
193  for ( i= 0; i < cn; i++ ) { // i=1; i <= cn
194  nDelete( &xx );
195  xx= nCopy(x[i]); // xx= x[i]
196 
197  nDelete( &t );
198  t= nInit( 1 ); // t= b= 1
199  nDelete( &b );
200  b= nInit( 1 );
201  nDelete( &s ); // s= q[cn-1]
202  s= nCopy( q[cn-1] );
203 
204  for ( k= cn-1; k >= 1; k-- ) { // k=cn; k >= 2
205  nDelete( &tmp1 );
206  tmp1= nMult( xx, b ); // b= c[k] + (xx * b)
207  nDelete( &b );
208  b= nAdd( c[k], tmp1 );
209 
210  nDelete( &tmp1 );
211  tmp1= nMult( q[k-1], b ); // s= s + (q[k-1] * b)
212  newnum= nAdd( s, tmp1 );
213  nDelete( &s );
214  s= newnum;
215 
216  nDelete( &tmp1 );
217  tmp1= nMult( xx, t ); // t= (t * xx) + b
218  newnum= nAdd( tmp1, b );
219  nDelete( &t );
220  t= newnum;
221  }
222 
223  if (!nIsZero(t))
224  {
225  nDelete( &w[i] ); // w[i]= s/t
226  w[i]= nDiv( s, t );
227  nNormalize( w[i] );
228  }
229 
231  }
232  }
233  mprSTICKYPROT("\n");
234 
235  // free mem
236  for ( j= 0; j < cn; j++ ) nDelete( c+j );
237  omFreeSize( (void *)c, cn * sizeof( number ) );
238 
239  nDelete( &tmp1 );
240  nDelete( &s );
241  nDelete( &t );
242  nDelete( &b );
243  nDelete( &xx );
244 
245  // makes quotiens smaller
246  for ( j= 0; j < cn; j++ ) nNormalize( w[j] );
247 
248  return w;
249 }
250 //<-
251 
252 //-----------------------------------------------------------------------------
253 //-------------- rootContainer ------------------------------------------------
254 //-----------------------------------------------------------------------------
255 
256 //-> definitions
257 #define MR 8 // never change this value
258 #define MT 5
259 #define MMOD (MT*MR)
260 #define MAXIT (5*MMOD) // max number of iterations in laguer root finder
261 
262 
263 //-> rootContainer::rootContainer()
265 {
266  rt=none;
267 
268  coeffs= NULL;
269  ievpoint= NULL;
270  theroots= NULL;
271 
272  found_roots= false;
273 }
274 //<-
275 
276 //-> rootContainer::~rootContainer()
278 {
279  int i;
280  // free coeffs, ievpoint
281  if ( ievpoint != NULL )
282  {
283  for ( i=0; i < anz+2; i++ ) nDelete( ievpoint + i );
284  omFreeSize( (void *)ievpoint, (anz+2) * sizeof( number ) );
285  }
286 
287  for ( i=0; i <= tdg; i++ ) nDelete( coeffs + i );
288  omFreeSize( (void *)coeffs, (tdg+1) * sizeof( number ) );
289 
290  // theroots löschen
291  for ( i=0; i < tdg; i++ ) delete theroots[i];
292  omFreeSize( (void *) theroots, (tdg)*sizeof(gmp_complex*) );
293 
294  //mprPROTnl("~rootContainer()");
295 }
296 //<-
297 
298 //-> void rootContainer::fillContainer( ... )
299 void rootContainer::fillContainer( number *_coeffs, number *_ievpoint,
300  const int _var, const int _tdg,
301  const rootType _rt, const int _anz )
302 {
303  int i;
304  number nn= nInit(0);
305  var=_var;
306  tdg=_tdg;
307  coeffs=_coeffs;
308  rt=_rt;
309  anz=_anz;
310 
311  for ( i=0; i <= tdg; i++ )
312  {
313  if ( nEqual(coeffs[i],nn) )
314  {
315  nDelete( &coeffs[i] );
316  coeffs[i]=NULL;
317  }
318  }
319  nDelete( &nn );
320 
321  if ( rt == cspecialmu && _ievpoint ) // copy ievpoint
322  {
323  ievpoint= (number *)omAlloc( (anz+2) * sizeof( number ) );
324  for (i=0; i < anz+2; i++) ievpoint[i]= nCopy( _ievpoint[i] );
325  }
326 
327  theroots= NULL;
328  found_roots= false;
329 }
330 //<-
331 
332 //-> poly rootContainer::getPoly()
334 {
335  int i;
336 
337  poly result= NULL;
338  poly ppos;
339 
340  if ( (rt == cspecial) || ( rt == cspecialmu ) )
341  {
342  for ( i= tdg; i >= 0; i-- )
343  {
344  if ( coeffs[i] )
345  {
346  poly p= pOne();
347  //pSetExp( p, var+1, i);
348  pSetExp( p, 1, i);
349  pSetCoeff( p, nCopy( coeffs[i] ) );
350  pSetm( p );
351  if (result)
352  {
353  ppos->next=p;
354  ppos=ppos->next;
355  }
356  else
357  {
358  result=p;
359  ppos=p;
360  }
361 
362  }
363  }
364  if (result!=NULL) pSetm( result );
365  }
366 
367  return result;
368 }
369 //<-
370 
371 //-> const gmp_complex & rootContainer::opterator[] ( const int i )
372 // this is now inline!
373 // gmp_complex & rootContainer::operator[] ( const int i )
374 // {
375 // if ( found_roots && ( i >= 0) && ( i < tdg ) )
376 // {
377 // return *theroots[i];
378 // }
379 // // warning
380 // Warn("rootContainer::getRoot: Wrong index %d, found_roots %s",i,found_roots?"true":"false");
381 // gmp_complex *tmp= new gmp_complex();
382 // return *tmp;
383 // }
384 //<-
385 
386 //-> gmp_complex & rootContainer::evPointCoord( int i )
388 {
389  if (! ((i >= 0) && (i < anz+2) ) )
390  WarnS("rootContainer::evPointCoord: index out of range");
391  if (ievpoint == NULL)
392  WarnS("rootContainer::evPointCoord: ievpoint == NULL");
393 
394  if ( (rt == cspecialmu) && found_roots ) // FIX ME
395  {
396  if ( ievpoint[i] != NULL )
397  {
398  gmp_complex *tmp= new gmp_complex();
399  *tmp= numberToComplex(ievpoint[i], currRing->cf);
400  return *tmp;
401  }
402  else
403  {
404  Warn("rootContainer::evPointCoord: NULL index %d",i);
405  }
406  }
407 
408  // warning
409  Warn("rootContainer::evPointCoord: Wrong index %d, found_roots %s",i,found_roots?"true":"false");
410  gmp_complex *tmp= new gmp_complex();
411  return *tmp;
412 }
413 //<-
414 
415 //-> bool rootContainer::changeRoots( int from, int to )
416 bool rootContainer::swapRoots( const int from, const int to )
417 {
418  if ( found_roots && ( from >= 0) && ( from < tdg ) && ( to >= 0) && ( to < tdg ) )
419  {
420  if ( to != from )
421  {
422  gmp_complex tmp( *theroots[from] );
423  *theroots[from]= *theroots[to];
424  *theroots[to]= tmp;
425  }
426  return true;
427  }
428 
429  // warning
430  Warn(" rootContainer::changeRoots: Wrong index %d, %d",from,to);
431  return false;
432 }
433 //<-
434 
435 //-> void rootContainer::solver()
436 bool rootContainer::solver( const int polishmode )
437 {
438  int i;
439 
440  // there are maximal tdg roots, so *roots ranges form 0 to tdg-1.
441  theroots= (gmp_complex**)omAlloc( tdg*sizeof(gmp_complex*) );
442  for ( i=0; i < tdg; i++ ) theroots[i]= new gmp_complex();
443 
444  // copy the coefficients of type number to type gmp_complex
445  gmp_complex **ad= (gmp_complex**)omAlloc( (tdg+1)*sizeof(gmp_complex*) );
446  for ( i=0; i <= tdg; i++ )
447  {
448  ad[i]= new gmp_complex();
449  if ( coeffs[i] ) *ad[i] = numberToComplex( coeffs[i], currRing->cf );
450  }
451 
452  // now solve
453  found_roots= laguer_driver( ad, theroots, polishmode != 0 );
454  if (!found_roots)
455  WarnS("rootContainer::solver: No roots found!");
456 
457  // free memory
458  for ( i=0; i <= tdg; i++ ) delete ad[i];
459  omFreeSize( (void *) ad, (tdg+1)*sizeof(gmp_complex*) );
460 
461  return found_roots;
462 }
463 //<-
464 
465 //-> gmp_complex* rootContainer::laguer_driver( bool polish )
466 bool rootContainer::laguer_driver(gmp_complex ** a, gmp_complex ** roots, bool polish )
467 {
468  int i,j,k,its;
469  gmp_float zero(0.0);
470  gmp_complex x(0.0),o(1.0);
471  bool ret= true, isf=isfloat(a), type=true;
472 
473  gmp_complex ** ad= (gmp_complex**)omAlloc( (tdg+1)*sizeof(gmp_complex*) );
474  for ( i=0; i <= tdg; i++ ) ad[i]= new gmp_complex( *a[i] );
475 
476  k = 0;
477  i = tdg;
478  j = i-1;
479  while (i>2)
480  {
481  // run laguer alg
482  x = zero;
483  laguer(ad, i, &x, &its, type);
484  if ( its > MAXIT )
485  {
486  type = !type;
487  x = zero;
488  laguer(ad, i, &x, &its, type);
489  }
490 
492  if ( its > MAXIT )
493  { // error
494  WarnS("Laguerre solver: Too many iterations!");
495  ret= false;
496  goto theend;
497  }
498  if ( polish )
499  {
500  laguer( a, tdg, &x, &its , type);
501  if ( its > MAXIT )
502  { // error
503  WarnS("Laguerre solver: Too many iterations in polish!");
504  ret= false;
505  goto theend;
506  }
507  }
508  if ((!type)&&(!((x.real()==zero)&&(x.imag()==zero)))) x = o/x;
509  if (x.imag() == zero)
510  {
511  *roots[k] = x;
512  k++;
513  divlin(ad,x,i);
514  i--;
515  }
516  else
517  {
518  if(isf)
519  {
520  *roots[j] = x;
521  *roots[j-1]= gmp_complex(x.real(),-x.imag());
522  j -= 2;
523  divquad(ad,x,i);
524  i -= 2;
525  }
526  else
527  {
528  *roots[j] = x;
529  j--;
530  divlin(ad,x,i);
531  i--;
532  }
533  }
534  type = !type;
535  }
536  solvequad(ad,roots,k,j);
537  sortroots(roots,k,j,isf);
538 
539 theend:
540  mprSTICKYPROT("\n");
541  for ( i=0; i <= tdg; i++ ) delete ad[i];
542  omFreeSize( (void *) ad, (tdg+1)*sizeof( gmp_complex* ));
543 
544  return ret;
545 }
546 //<-
547 
548 //-> void rootContainer::laguer(...)
549 void rootContainer::laguer(gmp_complex ** a, int m, gmp_complex *x, int *its, bool type)
550 {
551  int iter,j;
552  gmp_float zero(0.0),one(1.0),deg(m);
553  gmp_float abx_g, err_g, fabs;
554  gmp_complex dx, x1, b, d, f, g, h, sq, gp, gm, g2;
555  gmp_float frac_g[MR+1] = { 0.0, 0.5, 0.25, 0.75, 0.125, 0.375, 0.625, 0.875, 1.0 };
556 
557  gmp_float epss(0.1);
558  mpf_pow_ui(*epss._mpfp(),*epss.mpfp(),gmp_output_digits);
559 
560  for ( iter= 1; iter <= MAXIT; iter++ )
561  {
563  *its=iter;
564  if (type)
565  computefx(a,*x,m,b,d,f,abx_g,err_g);
566  else
567  computegx(a,*x,m,b,d,f,abx_g,err_g);
568  err_g *= epss; // EPSS;
569 
570  fabs = abs(b);
571  if (fabs <= err_g)
572  {
573  if ((fabs==zero) || (abs(d)==zero)) return;
574  *x -= (b/d); // a last newton-step
575  goto ende;
576  }
577 
578  g= d / b;
579  g2 = g * g;
580  h= g2 - (((f+f) / b ));
581  sq= sqrt(( ( h * deg ) - g2 ) * (deg - one));
582  gp= g + sq;
583  gm= g - sq;
584  if (abs(gp)<abs(gm))
585  {
586  dx = deg/gm;
587  }
588  else
589  {
590  if((gp.real()==zero)&&(gp.imag()==zero))
591  {
592  dx.real(cos((mprfloat)iter));
593  dx.imag(sin((mprfloat)iter));
594  dx = dx*(one+abx_g);
595  }
596  else
597  {
598  dx = deg/gp;
599  }
600  }
601  x1= *x - dx;
602 
603  if (*x == x1) goto ende;
604 
605  j = iter%MMOD;
606  if (j==0) j=MT;
607  if ( j % MT ) *x= x1;
608  else *x -= ( dx * frac_g[ j / MT ] );
609  }
610 
611  *its= MAXIT+1;
612 ende:
613  checkimag(x,epss);
614 }
615 
617 {
618  if(abs(x->imag())<abs(x->real())*e)
619  {
620  x->imag(0.0);
621  }
622 }
623 
625 {
626  gmp_float z(0.0);
627  gmp_complex *b;
628  for (int i=tdg; i >= 0; i-- )
629  {
630  b = &(*a[i]);
631  if (!(b->imag()==z))
632  return false;
633  }
634  return true;
635 }
636 
638 {
639  int i;
640  gmp_float o(1.0);
641 
642  if (abs(x)<o)
643  {
644  for (i= j-1; i > 0; i-- )
645  *a[i] += (*a[i+1]*x);
646  for (i= 0; i < j; i++ )
647  *a[i] = *a[i+1];
648  }
649  else
650  {
651  gmp_complex y(o/x);
652  for (i= 1; i < j; i++)
653  *a[i] += (*a[i-1]*y);
654  }
655 }
656 
658 {
659  int i;
660  gmp_float o(1.0),p(x.real()+x.real()),
661  q((x.real()*x.real())+(x.imag()*x.imag()));
662 
663  if (abs(x)<o)
664  {
665  *a[j-1] += (*a[j]*p);
666  for (i= j-2; i > 1; i-- )
667  *a[i] += ((*a[i+1]*p)-(*a[i+2]*q));
668  for (i= 0; i < j-1; i++ )
669  *a[i] = *a[i+2];
670  }
671  else
672  {
673  p = p/q;
674  q = o/q;
675  *a[1] += (*a[0]*p);
676  for (i= 2; i < j-1; i++)
677  *a[i] += ((*a[i-1]*p)-(*a[i-2]*q));
678  }
679 }
680 
682 {
683  gmp_float zero(0.0);
684 
685  if ((j>k)
686  &&((!(*a[2]).real().isZero())||(!(*a[2]).imag().isZero())))
687  {
688  gmp_complex sq(zero);
689  gmp_complex h1(*a[1]/(*a[2] + *a[2])), h2(*a[0] / *a[2]);
690  gmp_complex disk((h1 * h1) - h2);
691  if (disk.imag().isZero())
692  {
693  if (disk.real()<zero)
694  {
695  sq.real(zero);
696  sq.imag(sqrt(-disk.real()));
697  }
698  else
699  sq = (gmp_complex)sqrt(disk.real());
700  }
701  else
702  sq = sqrt(disk);
703  *r[k+1] = sq - h1;
704  sq += h1;
705  *r[k] = (gmp_complex)0.0-sq;
706  if(sq.imag().isZero())
707  {
708  k = j;
709  j++;
710  }
711  else
712  {
713  j = k;
714  k--;
715  }
716  }
717  else
718  {
719  if (((*a[1]).real().isZero()) && ((*a[1]).imag().isZero()))
720  {
721  WerrorS("precision lost, try again with higher precision");
722  }
723  else
724  {
725  *r[k]= (gmp_complex)0.0-(*a[0] / *a[1]);
726  if(r[k]->imag().isZero())
727  j++;
728  else
729  k--;
730  }
731  }
732 }
733 
734 void rootContainer::sortroots(gmp_complex **ro, int r, int c, bool isf)
735 {
736  int j;
737 
738  for (j=0; j<r; j++) // the real roots
739  sortre(ro, j, r, 1);
740  if (c>=tdg) return;
741  if (isf)
742  {
743  for (j=c; j+2<tdg; j+=2) // the complex roots for a real poly
744  sortre(ro, j, tdg-1, 2);
745  }
746  else
747  {
748  for (j=c; j+1<tdg; j++) // the complex roots for a general poly
749  sortre(ro, j, tdg-1, 1);
750  }
751 }
752 
753 void rootContainer::sortre(gmp_complex **r, int l, int u, int inc)
754 {
755  int pos,i;
756  gmp_complex *x,*y;
757 
758  pos = l;
759  x = r[pos];
760  for (i=l+inc; i<=u; i+=inc)
761  {
762  if (r[i]->real()<x->real())
763  {
764  pos = i;
765  x = r[pos];
766  }
767  }
768  if (pos>l)
769  {
770  if (inc==1)
771  {
772  for (i=pos; i>l; i--)
773  r[i] = r[i-1];
774  r[l] = x;
775  }
776  else
777  {
778  y = r[pos+1];
779  for (i=pos+1; i+1>l; i--)
780  r[i] = r[i-2];
781  if (x->imag()>y->imag())
782  {
783  r[l] = x;
784  r[l+1] = y;
785  }
786  else
787  {
788  r[l] = y;
789  r[l+1] = x;
790  }
791  }
792  }
793  else if ((inc==2)&&(x->imag()<r[l+1]->imag()))
794  {
795  r[l] = r[l+1];
796  r[l+1] = x;
797  }
798 }
799 
801  gmp_complex &f0, gmp_complex &f1, gmp_complex &f2,
802  gmp_float &ex, gmp_float &ef)
803 {
804  int k;
805 
806  f0= *a[m];
807  ef= abs(f0);
808  f1= gmp_complex(0.0);
809  f2= f1;
810  ex= abs(x);
811 
812  for ( k= m-1; k >= 0; k-- )
813  {
814  f2 = ( x * f2 ) + f1;
815  f1 = ( x * f1 ) + f0;
816  f0 = ( x * f0 ) + *a[k];
817  ef = abs( f0 ) + ( ex * ef );
818  }
819 }
820 
822  gmp_complex &f0, gmp_complex &f1, gmp_complex &f2,
823  gmp_float &ex, gmp_float &ef)
824 {
825  int k;
826 
827  f0= *a[0];
828  ef= abs(f0);
829  f1= gmp_complex(0.0);
830  f2= f1;
831  ex= abs(x);
832 
833  for ( k= 1; k <= m; k++ )
834  {
835  f2 = ( x * f2 ) + f1;
836  f1 = ( x * f1 ) + f0;
837  f0 = ( x * f0 ) + *a[k];
838  ef = abs( f0 ) + ( ex * ef );
839  }
840 }
841 
842 //-----------------------------------------------------------------------------
843 //-------------- rootArranger -------------------------------------------------
844 //-----------------------------------------------------------------------------
845 
846 //-> rootArranger::rootArranger(...)
848  rootContainer ** _mu,
849  const int _howclean )
850  : roots(_roots), mu(_mu), howclean(_howclean)
851 {
852  found_roots=false;
853 }
854 //<-
855 
856 //-> void rootArranger::solve_all()
858 {
859  int i;
860  found_roots= true;
861 
862  // find roots of polys given by coeffs in roots
863  rc= roots[0]->getAnzElems();
864  for ( i= 0; i < rc; i++ )
865  if ( !roots[i]->solver( howclean ) )
866  {
867  found_roots= false;
868  return;
869  }
870  // find roots of polys given by coeffs in mu
871  mc= mu[0]->getAnzElems();
872  for ( i= 0; i < mc; i++ )
873  if ( ! mu[i]->solver( howclean ) )
874  {
875  found_roots= false;
876  return;
877  }
878 }
879 //<-
880 
881 //-> void rootArranger::arrange()
883 {
884  gmp_complex tmp,zwerg;
885  int anzm= mu[0]->getAnzElems();
886  int anzr= roots[0]->getAnzRoots();
887  int xkoord, r, rtest, xk, mtest;
888  bool found;
889  //gmp_complex mprec(1.0/pow(10,gmp_output_digits-5),1.0/pow(10,gmp_output_digits-5));
890 
891  for ( xkoord= 0; xkoord < anzm; xkoord++ ) { // für x1,x2, x1,x2,x3, x1,x2,...,xn
892  gmp_float mprec(1.0/pow(10.0,(int)(gmp_output_digits/3)));
893  for ( r= 0; r < anzr; r++ ) { // für jede Nullstelle
894  // (x1-koordinate) * evp[1] + (x2-koordinate) * evp[2] +
895  // ... + (xkoord-koordinate) * evp[xkoord]
896  tmp= gmp_complex();
897  for ( xk =0; xk <= xkoord; xk++ )
898  {
899  tmp -= (*roots[xk])[r] * mu[xkoord]->evPointCoord(xk+1); //xk+1
900  }
901  found= false;
902  do { // while not found
903  for ( rtest= r; rtest < anzr; rtest++ ) { // für jede Nullstelle
904  zwerg = tmp - (*roots[xk])[rtest] * mu[xkoord]->evPointCoord(xk+1); // xk+1, xkoord+2
905  for ( mtest= 0; mtest < anzr; mtest++ )
906  {
907  // if ( tmp == (*mu[xkoord])[mtest] )
908  // {
909  if ( ((zwerg.real() <= (*mu[xkoord])[mtest].real() + mprec) &&
910  (zwerg.real() >= (*mu[xkoord])[mtest].real() - mprec)) &&
911  ((zwerg.imag() <= (*mu[xkoord])[mtest].imag() + mprec) &&
912  (zwerg.imag() >= (*mu[xkoord])[mtest].imag() - mprec)) )
913  {
914  roots[xk]->swapRoots( r, rtest );
915  found= true;
916  break;
917  }
918  }
919  } // rtest
920  if (!found)
921  {
922  WarnS("rootArranger::arrange: precision lost");
923  mprec*=10;
924  }
925  } while(!found);
926 #if 0
927  if ( !found )
928  {
929  Warn("rootArranger::arrange: No match? coord %d, root %d.",xkoord,r);
930 //#ifdef mprDEBUG_PROT
931  WarnS("One of these ...");
932  for ( rtest= r; rtest < anzr; rtest++ )
933  {
934  tmp= gmp_complex();
935  for ( xk =0; xk <= xkoord; xk++ )
936  {
937  tmp-= (*roots[xk])[r] * mu[xkoord]->evPointCoord(xk+1);
938  }
939  tmp-= (*roots[xk])[rtest] * mu[xkoord]->evPointCoord(xk+1); // xkoord+2
940  Warn(" %s",complexToStr(tmp,gmp_output_digits+1),rtest);
941  }
942  WarnS(" ... must match to one of these:");
943  for ( mtest= 0; mtest < anzr; mtest++ )
944  {
945  Warn(" %s",complexToStr((*mu[xkoord])[mtest],gmp_output_digits+1));
946  }
947 //#endif
948  }
949 #endif
950  } // r
951  } // xkoord
952 }
953 //<-
954 
955 //-----------------------------------------------------------------------------
956 //-------------- simplex ----- ------------------------------------------------
957 //-----------------------------------------------------------------------------
958 
959 // #ifdef mprDEBUG_PROT
960 // #define error(a) a
961 // #else
962 // #define error(a)
963 // #endif
964 
965 #define error(a) a
966 
967 #define MAXPOINTS 1000
968 
969 //-> simplex::*
970 //
971 simplex::simplex( int rows, int cols )
972  : LiPM_cols(cols), LiPM_rows(rows)
973 {
974  int i;
975 
978 
979  LiPM = (mprfloat **)omAlloc( LiPM_rows * sizeof(mprfloat *) ); // LP matrix
980  for( i= 0; i < LiPM_rows; i++ )
981  {
982  // Mem must be allocated aligned, also for type double!
983  LiPM[i] = (mprfloat *)omAlloc0Aligned( LiPM_cols * sizeof(mprfloat) );
984  }
985 
986  iposv = (int *)omAlloc0( 2*LiPM_rows*sizeof(int) );
987  izrov = (int *)omAlloc0( 2*LiPM_rows*sizeof(int) );
988 
989  m=n=m1=m2=m3=icase=0;
990 
991 #ifdef mprDEBUG_ALL
992  Print("LiPM size: %d, %d\n",LiPM_rows,LiPM_cols);
993 #endif
994 }
995 
997 {
998  // clean up
999  int i;
1000  for( i= 0; i < LiPM_rows; i++ )
1001  {
1002  omFreeSize( (void *) LiPM[i], LiPM_cols * sizeof(mprfloat) );
1003  }
1004  omFreeSize( (void *) LiPM, LiPM_rows * sizeof(mprfloat *) );
1005 
1006  omFreeSize( (void *) iposv, 2*LiPM_rows*sizeof(int) );
1007  omFreeSize( (void *) izrov, 2*LiPM_rows*sizeof(int) );
1008 }
1009 
1011 {
1012  int i,j;
1013 // if ( MATROWS( m ) > LiPM_rows || MATCOLS( m ) > LiPM_cols ) {
1014 // WarnS("");
1015 // return FALSE;
1016 // }
1017 
1018  number coef;
1019  for ( i= 1; i <= MATROWS( mm ); i++ )
1020  {
1021  for ( j= 1; j <= MATCOLS( mm ); j++ )
1022  {
1023  if ( MATELEM(mm,i,j) != NULL )
1024  {
1025  coef= pGetCoeff( MATELEM(mm,i,j) );
1026  if ( coef != NULL && !nIsZero(coef) )
1027  LiPM[i][j]= (double)(*(gmp_float*)coef);
1028  //#ifdef mpr_DEBUG_PROT
1029  //Print("%f ",LiPM[i][j]);
1030  //#endif
1031  }
1032  }
1033  // PrintLn();
1034  }
1035 
1036  return TRUE;
1037 }
1038 
1040 {
1041  int i,j;
1042 // if ( MATROWS( mm ) < LiPM_rows-3 || MATCOLS( m ) < LiPM_cols-2 ) {
1043 // WarnS("");
1044 // return NULL;
1045 // }
1046 
1047 //Print(" %d x %d\n",MATROWS( mm ),MATCOLS( mm ));
1048 
1049  number coef;
1050  gmp_float * bla;
1051  for ( i= 1; i <= MATROWS( mm ); i++ )
1052  {
1053  for ( j= 1; j <= MATCOLS( mm ); j++ )
1054  {
1055  pDelete( &(MATELEM(mm,i,j)) );
1056  MATELEM(mm,i,j)= NULL;
1057 //Print(" %3.0f ",LiPM[i][j]);
1058  if ( LiPM[i][j] != 0.0 )
1059  {
1060  bla= new gmp_float(LiPM[i][j]);
1061  coef= (number)bla;
1062  MATELEM(mm,i,j)= pOne();
1063  pSetCoeff( MATELEM(mm,i,j), coef );
1064  }
1065  }
1066 //PrintLn();
1067  }
1068 
1069  return mm;
1070 }
1071 
1073 {
1074  int i;
1075  intvec * iv = new intvec( m );
1076  for ( i= 1; i <= m; i++ )
1077  {
1078  IMATELEM(*iv,i,1)= iposv[i];
1079  }
1080  return iv;
1081 }
1082 
1084 {
1085  int i;
1086  intvec * iv = new intvec( n );
1087  for ( i= 1; i <= n; i++ )
1088  {
1089  IMATELEM(*iv,i,1)= izrov[i];
1090  }
1091  return iv;
1092 }
1093 
1095 {
1096  int i,ip,ir,is,k,kh,kp,m12,nl1,nl2;
1097  int *l1,*l2,*l3;
1098  mprfloat q1, bmax;
1099 
1100  if ( m != (m1+m2+m3) )
1101  {
1102  // error: bad input
1103  error(WarnS("simplex::compute: Bad input constraint counts!");)
1104  icase=-2;
1105  return;
1106  }
1107 
1108  l1= (int *) omAlloc0( (n+1) * sizeof(int) );
1109  l2= (int *) omAlloc0( (m+1) * sizeof(int) );
1110  l3= (int *) omAlloc0( (m+1) * sizeof(int) );
1111 
1112  nl1= n;
1113  for ( k=1; k<=n; k++ ) l1[k]=izrov[k]=k;
1114  nl2=m;
1115  for ( i=1; i<=m; i++ )
1116  {
1117  if ( LiPM[i+1][1] < 0.0 )
1118  {
1119  // error: bad input
1120  error(WarnS("simplex::compute: Bad input tableau!");)
1121  error(Warn("simplex::compute: in input Matrix row %d, column 1, value %f",i+1,LiPM[i+1][1]);)
1122  icase=-2;
1123  // free mem l1,l2,l3;
1124  omFreeSize( (void *) l3, (m+1) * sizeof(int) );
1125  omFreeSize( (void *) l2, (m+1) * sizeof(int) );
1126  omFreeSize( (void *) l1, (n+1) * sizeof(int) );
1127  return;
1128  }
1129  l2[i]= i;
1130  iposv[i]= n+i;
1131  }
1132  for ( i=1; i<=m2; i++) l3[i]= 1;
1133  ir= 0;
1134  if (m2+m3)
1135  {
1136  ir=1;
1137  for ( k=1; k <= (n+1); k++ )
1138  {
1139  q1=0.0;
1140  for ( i=m1+1; i <= m; i++ ) q1+= LiPM[i+1][k];
1141  LiPM[m+2][k]= -q1;
1142  }
1143 
1144  do
1145  {
1146  simp1(LiPM,m+1,l1,nl1,0,&kp,&bmax);
1147  if ( bmax <= SIMPLEX_EPS && LiPM[m+2][1] < -SIMPLEX_EPS )
1148  {
1149  icase= -1; // no solution found
1150  // free mem l1,l2,l3;
1151  omFreeSize( (void *) l3, (m+1) * sizeof(int) );
1152  omFreeSize( (void *) l2, (m+1) * sizeof(int) );
1153  omFreeSize( (void *) l1, (n+1) * sizeof(int) );
1154  return;
1155  }
1156  else if ( bmax <= SIMPLEX_EPS && LiPM[m+2][1] <= SIMPLEX_EPS )
1157  {
1158  m12= m1+m2+1;
1159  if ( m12 <= m )
1160  {
1161  for ( ip= m12; ip <= m; ip++ )
1162  {
1163  if ( iposv[ip] == (ip+n) )
1164  {
1165  simp1(LiPM,ip,l1,nl1,1,&kp,&bmax);
1166  if ( fabs(bmax) >= SIMPLEX_EPS)
1167  goto one;
1168  }
1169  }
1170  }
1171  ir= 0;
1172  --m12;
1173  if ( m1+1 <= m12 )
1174  for ( i=m1+1; i <= m12; i++ )
1175  if ( l3[i-m1] == 1 )
1176  for ( k=1; k <= n+1; k++ )
1177  LiPM[i+1][k] = -(LiPM[i+1][k]);
1178  break;
1179  }
1180  //#if DEBUG
1181  //print_bmat( a, m+2, n+3);
1182  //#endif
1183  simp2(LiPM,n,l2,nl2,&ip,kp,&q1);
1184  if ( ip == 0 )
1185  {
1186  icase = -1; // no solution found
1187  // free mem l1,l2,l3;
1188  omFreeSize( (void *) l3, (m+1) * sizeof(int) );
1189  omFreeSize( (void *) l2, (m+1) * sizeof(int) );
1190  omFreeSize( (void *) l1, (n+1) * sizeof(int) );
1191  return;
1192  }
1193  one: simp3(LiPM,m+1,n,ip,kp);
1194  // #if DEBUG
1195  // print_bmat(a,m+2,n+3);
1196  // #endif
1197  if ( iposv[ip] >= (n+m1+m2+1))
1198  {
1199  for ( k= 1; k <= nl1; k++ )
1200  if ( l1[k] == kp ) break;
1201  --nl1;
1202  for ( is=k; is <= nl1; is++ ) l1[is]= l1[is+1];
1203  ++(LiPM[m+2][kp+1]);
1204  for ( i= 1; i <= m+2; i++ ) LiPM[i][kp+1] = -(LiPM[i][kp+1]);
1205  }
1206  else
1207  {
1208  if ( iposv[ip] >= (n+m1+1) )
1209  {
1210  kh= iposv[ip]-m1-n;
1211  if ( l3[kh] )
1212  {
1213  l3[kh]= 0;
1214  ++(LiPM[m+2][kp+1]);
1215  for ( i=1; i<= m+2; i++ )
1216  LiPM[i][kp+1] = -(LiPM[i][kp+1]);
1217  }
1218  }
1219  }
1220  is= izrov[kp];
1221  izrov[kp]= iposv[ip];
1222  iposv[ip]= is;
1223  } while (ir);
1224  }
1225  /* end of phase 1, have feasible sol, now optimize it */
1226  loop
1227  {
1228  // #if DEBUG
1229  // print_bmat( a, m+1, n+5);
1230  // #endif
1231  simp1(LiPM,0,l1,nl1,0,&kp,&bmax);
1232  if (bmax <= /*SIMPLEX_EPS*/0.0)
1233  {
1234  icase=0; // finite solution found
1235  // free mem l1,l2,l3
1236  omFreeSize( (void *) l3, (m+1) * sizeof(int) );
1237  omFreeSize( (void *) l2, (m+1) * sizeof(int) );
1238  omFreeSize( (void *) l1, (n+1) * sizeof(int) );
1239  return;
1240  }
1241  simp2(LiPM,n,l2,nl2,&ip,kp,&q1);
1242  if (ip == 0)
1243  {
1244  //printf("Unbounded:");
1245  // #if DEBUG
1246  // print_bmat( a, m+1, n+1);
1247  // #endif
1248  icase=1; /* unbounded */
1249  // free mem
1250  omFreeSize( (void *) l3, (m+1) * sizeof(int) );
1251  omFreeSize( (void *) l2, (m+1) * sizeof(int) );
1252  omFreeSize( (void *) l1, (n+1) * sizeof(int) );
1253  return;
1254  }
1255  simp3(LiPM,m,n,ip,kp);
1256  is= izrov[kp];
1257  izrov[kp]= iposv[ip];
1258  iposv[ip]= is;
1259  }/*for ;;*/
1260 }
1261 
1262 void simplex::simp1( mprfloat **a, int mm, int ll[], int nll, int iabf, int *kp, mprfloat *bmax )
1263 {
1264  int k;
1265  mprfloat test;
1266 
1267  if( nll <= 0)
1268  { /* init'tion: fixed */
1269  *bmax = 0.0;
1270  return;
1271  }
1272  *kp=ll[1];
1273  *bmax=a[mm+1][*kp+1];
1274  for (k=2;k<=nll;k++)
1275  {
1276  if (iabf == 0)
1277  {
1278  test=a[mm+1][ll[k]+1]-(*bmax);
1279  if (test > 0.0)
1280  {
1281  *bmax=a[mm+1][ll[k]+1];
1282  *kp=ll[k];
1283  }
1284  }
1285  else
1286  { /* abs values: have fixed it */
1287  test=fabs(a[mm+1][ll[k]+1])-fabs(*bmax);
1288  if (test > 0.0)
1289  {
1290  *bmax=a[mm+1][ll[k]+1];
1291  *kp=ll[k];
1292  }
1293  }
1294  }
1295 }
1296 
1297 void simplex::simp2( mprfloat **a, int nn, int l2[], int nl2, int *ip, int kp, mprfloat *q1 )
1298 {
1299  int k,ii,i;
1300  mprfloat qp,q0,q;
1301 
1302  *ip= 0;
1303  for ( i=1; i <= nl2; i++ )
1304  {
1305  if ( a[l2[i]+1][kp+1] < -SIMPLEX_EPS )
1306  {
1307  *q1= -a[l2[i]+1][1] / a[l2[i]+1][kp+1];
1308  *ip= l2[i];
1309  for ( i= i+1; i <= nl2; i++ )
1310  {
1311  ii= l2[i];
1312  if (a[ii+1][kp+1] < -SIMPLEX_EPS)
1313  {
1314  q= -a[ii+1][1] / a[ii+1][kp+1];
1315  if (q - *q1 < -SIMPLEX_EPS)
1316  {
1317  *ip=ii;
1318  *q1=q;
1319  }
1320  else if (q - *q1 < SIMPLEX_EPS)
1321  {
1322  for ( k=1; k<= nn; k++ )
1323  {
1324  qp= -a[*ip+1][k+1]/a[*ip+1][kp+1];
1325  q0= -a[ii+1][k+1]/a[ii+1][kp+1];
1326  if ( q0 != qp ) break;
1327  }
1328  if ( q0 < qp ) *ip= ii;
1329  }
1330  }
1331  }
1332  }
1333  }
1334 }
1335 
1336 void simplex::simp3( mprfloat **a, int i1, int k1, int ip, int kp )
1337 {
1338  int kk,ii;
1339  mprfloat piv;
1340 
1341  piv= 1.0 / a[ip+1][kp+1];
1342  for ( ii=1; ii <= i1+1; ii++ )
1343  {
1344  if ( ii -1 != ip )
1345  {
1346  a[ii][kp+1] *= piv;
1347  for ( kk=1; kk <= k1+1; kk++ )
1348  if ( kk-1 != kp )
1349  a[ii][kk] -= a[ip+1][kk] * a[ii][kp+1];
1350  }
1351  }
1352  for ( kk=1; kk<= k1+1; kk++ )
1353  if ( kk-1 != kp ) a[ip+1][kk] *= -piv;
1354  a[ip+1][kp+1]= piv;
1355 }
1356 //<-
1357 
1358 //-----------------------------------------------------------------------------
1359 
1360 // local Variables: ***
1361 // folded-file: t ***
1362 // compile-command-1: "make installg" ***
1363 // compile-command-2: "make install" ***
1364 // End: ***
1365 
Rational pow(const Rational &a, int e)
Definition: GMPrat.cc:411
Rational abs(const Rational &a)
Definition: GMPrat.cc:436
int BOOLEAN
Definition: auxiliary.h:87
#define TRUE
Definition: auxiliary.h:100
int l
Definition: cfEzgcd.cc:93
int m
Definition: cfEzgcd.cc:121
int i
Definition: cfEzgcd.cc:125
int k
Definition: cfEzgcd.cc:92
Variable x
Definition: cfModGcd.cc:4023
int p
Definition: cfModGcd.cc:4019
g
Definition: cfModGcd.cc:4031
CanonicalForm gp
Definition: cfModGcd.cc:4043
CanonicalForm test
Definition: cfModGcd.cc:4037
CanonicalForm b
Definition: cfModGcd.cc:4044
void mu(int **points, int sizePoints)
FILE * f
Definition: checklibs.c:9
gmp_complex numbers based on
Definition: mpr_complex.h:179
gmp_float imag() const
Definition: mpr_complex.h:235
gmp_float real() const
Definition: mpr_complex.h:234
bool isZero()
Definition: mpr_complex.h:241
const mpf_t * mpfp() const
Definition: mpr_complex.h:133
mpf_t * _mpfp()
Definition: mpr_complex.h:134
bool isZero() const
Definition: mpr_complex.cc:252
Definition: intvec.h:23
void solve_all()
Definition: mpr_numeric.cc:857
rootContainer ** roots
Definition: mpr_numeric.h:167
rootArranger(rootContainer **_roots, rootContainer **_mu, const int _howclean=PM_CORRUPT)
Definition: mpr_numeric.cc:847
rootContainer ** mu
Definition: mpr_numeric.h:168
bool found_roots
Definition: mpr_numeric.h:172
void arrange()
Definition: mpr_numeric.cc:882
complex root finder for univariate polynomials based on laguers algorithm
Definition: mpr_numeric.h:66
void sortre(gmp_complex **r, int l, int u, int inc)
Definition: mpr_numeric.cc:753
bool laguer_driver(gmp_complex **a, gmp_complex **roots, bool polish=true)
Given the degree tdg and the tdg+1 complex coefficients ad0..tdg of the polynomial this routine succe...
Definition: mpr_numeric.cc:466
void computegx(gmp_complex **a, gmp_complex x, int m, gmp_complex &f0, gmp_complex &f1, gmp_complex &f2, gmp_float &ex, gmp_float &ef)
Definition: mpr_numeric.cc:821
void fillContainer(number *_coeffs, number *_ievpoint, const int _var, const int _tdg, const rootType _rt, const int _anz)
Definition: mpr_numeric.cc:299
void laguer(gmp_complex **a, int m, gmp_complex *x, int *its, bool type)
Given the degree m and the m+1 complex coefficients a[0..m] of the polynomial, and given the complex ...
Definition: mpr_numeric.cc:549
void divlin(gmp_complex **a, gmp_complex x, int j)
Definition: mpr_numeric.cc:637
void sortroots(gmp_complex **roots, int r, int c, bool isf)
Definition: mpr_numeric.cc:734
rootType rt
Definition: mpr_numeric.h:137
void divquad(gmp_complex **a, gmp_complex x, int j)
Definition: mpr_numeric.cc:657
bool swapRoots(const int from, const int to)
Definition: mpr_numeric.cc:416
gmp_complex ** theroots
Definition: mpr_numeric.h:139
void computefx(gmp_complex **a, gmp_complex x, int m, gmp_complex &f0, gmp_complex &f1, gmp_complex &f2, gmp_float &ex, gmp_float &ef)
Definition: mpr_numeric.cc:800
void solvequad(gmp_complex **a, gmp_complex **r, int &k, int &j)
Definition: mpr_numeric.cc:681
gmp_complex & evPointCoord(const int i)
Definition: mpr_numeric.cc:387
bool isfloat(gmp_complex **a)
Definition: mpr_numeric.cc:624
void checkimag(gmp_complex *x, gmp_float &e)
Definition: mpr_numeric.cc:616
int getAnzRoots()
Definition: mpr_numeric.h:97
bool solver(const int polishmode=PM_NONE)
Definition: mpr_numeric.cc:436
number * ievpoint
Definition: mpr_numeric.h:136
int getAnzElems()
Definition: mpr_numeric.h:95
intvec * zrovToIV()
mprfloat ** LiPM
Definition: mpr_numeric.h:205
int * iposv
Definition: mpr_numeric.h:203
int LiPM_rows
Definition: mpr_numeric.h:225
BOOLEAN mapFromMatrix(matrix m)
int * izrov
Definition: mpr_numeric.h:203
int icase
Definition: mpr_numeric.h:201
void compute()
int LiPM_cols
Definition: mpr_numeric.h:225
simplex(int rows, int cols)
#rows should be >= m+2, #cols >= n+1
Definition: mpr_numeric.cc:971
void simp2(mprfloat **a, int n, int l2[], int nl2, int *ip, int kp, mprfloat *q1)
void simp3(mprfloat **a, int i1, int k1, int ip, int kp)
void simp1(mprfloat **a, int mm, int ll[], int nll, int iabf, int *kp, mprfloat *bmax)
matrix mapToMatrix(matrix m)
intvec * posvToIV()
void init()
Definition: mpr_numeric.cc:53
poly numvec2poly(const number *q)
Definition: mpr_numeric.cc:93
number * x
Definition: mpr_numeric.h:55
vandermonde(const long _cn, const long _n, const long _maxdeg, number *_p, const bool _homog=true)
Definition: mpr_numeric.cc:35
number * p
Definition: mpr_numeric.h:54
number * interpolateDense(const number *q)
Solves the Vandermode linear system \sum_{i=1}^{n} x_i^k-1 w_i = q_k, k=1,..,n.
Definition: mpr_numeric.cc:146
#define Print
Definition: emacs.cc:80
#define Warn
Definition: emacs.cc:77
#define WarnS
Definition: emacs.cc:78
CFFListIterator iter
Definition: facAbsBiFact.cc:54
return result
Definition: facAbsBiFact.cc:76
const CanonicalForm int s
Definition: facAbsFact.cc:55
const CanonicalForm int const CFList const Variable & y
Definition: facAbsFact.cc:57
const CanonicalForm & w
Definition: facAbsFact.cc:55
bool found
Definition: facFactorize.cc:56
CFList tmp1
Definition: facFqBivar.cc:70
int j
Definition: facHensel.cc:105
bool isZero(const CFArray &A)
checks if entries of A are zero
void WerrorS(const char *s)
Definition: feFopen.cc:24
#define IMATELEM(M, I, J)
Definition: intvec.h:85
STATIC_VAR Poly * h
Definition: janet.cc:971
#define MATELEM(mat, i, j)
1-based access to matrix
Definition: matpol.h:29
#define MATROWS(i)
Definition: matpol.h:26
#define MATCOLS(i)
Definition: matpol.h:27
#define pNext(p)
Definition: monomials.h:36
static number & pGetCoeff(poly p)
return an alias to the leading coefficient of p assumes that p != NULL NOTE: not copy
Definition: monomials.h:44
EXTERN_VAR size_t gmp_output_digits
Definition: mpr_base.h:115
gmp_float sin(const gmp_float &a)
Definition: mpr_complex.cc:333
gmp_float sqrt(const gmp_float &a)
Definition: mpr_complex.cc:327
gmp_float exp(const gmp_float &a)
Definition: mpr_complex.cc:357
gmp_float cos(const gmp_float &a)
Definition: mpr_complex.cc:338
char * complexToStr(gmp_complex &c, const unsigned int oprec, const coeffs src)
Definition: mpr_complex.cc:704
gmp_complex numberToComplex(number num, const coeffs r)
Definition: mpr_complex.h:312
#define mprSTICKYPROT(msg)
Definition: mpr_global.h:54
#define ST_VANDER_STEP
Definition: mpr_global.h:84
#define ST_ROOTS_LG
Definition: mpr_global.h:82
double mprfloat
Definition: mpr_global.h:17
#define ST_ROOTS_LGSTEP
Definition: mpr_global.h:80
#define MT
Definition: mpr_numeric.cc:258
#define MMOD
Definition: mpr_numeric.cc:259
#define MR
Definition: mpr_numeric.cc:257
#define error(a)
Definition: mpr_numeric.cc:965
#define MAXIT
Definition: mpr_numeric.cc:260
#define SIMPLEX_EPS
Definition: mpr_numeric.h:181
The main handler for Singular numbers which are suitable for Singular polynomials.
#define nDiv(a, b)
Definition: numbers.h:32
#define nDelete(n)
Definition: numbers.h:16
#define nInpNeg(n)
Definition: numbers.h:21
#define nIsZero(n)
Definition: numbers.h:19
#define nEqual(n1, n2)
Definition: numbers.h:20
#define nCopy(n)
Definition: numbers.h:15
#define nAdd(n1, n2)
Definition: numbers.h:18
#define nNormalize(n)
Definition: numbers.h:30
#define nInit(i)
Definition: numbers.h:24
#define nMult(n1, n2)
Definition: numbers.h:17
#define nPower(a, b, res)
Definition: numbers.h:38
#define omFreeSize(addr, size)
Definition: omAllocDecl.h:260
#define omAlloc0Aligned
Definition: omAllocDecl.h:274
#define omAlloc(size)
Definition: omAllocDecl.h:210
#define omAlloc0(size)
Definition: omAllocDecl.h:211
#define NULL
Definition: omList.c:12
VAR ring currRing
Widely used global variable which specifies the current polynomial ring for Singular interpreter and ...
Definition: polys.cc:13
Compatiblity layer for legacy polynomial operations (over currRing)
#define pDelete(p_ptr)
Definition: polys.h:182
#define pSetm(p)
Definition: polys.h:267
#define pSortAdd(p)
sorts p, p may have equal monomials
Definition: polys.h:217
#define pSetCoeff(p, n)
deletes old coeff before setting the new one
Definition: polys.h:31
#define pSetExpV(p, e)
Definition: polys.h:97
#define pSetExp(p, i, v)
Definition: polys.h:42
#define pOne()
Definition: polys.h:311
#define loop
Definition: structs.h:80