cf_algorithm.cc
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1 /* emacs edit mode for this file is -*- C++ -*- */
2 
3 /**
4  *
5  *
6  * cf_algorithm.cc - simple mathematical algorithms.
7  *
8  * Hierarchy: mathematical algorithms on canonical forms
9  *
10  * Developers note:
11  * ----------------
12  * A "mathematical" algorithm is an algorithm which calculates
13  * some mathematical function in contrast to a "structural"
14  * algorithm which gives structural information on polynomials.
15  *
16  * Compare these functions to the functions in `cf_ops.cc', which
17  * are structural algorithms.
18  *
19 **/
20 
21 
22 #include "config.h"
23 
24 
25 #include "cf_assert.h"
26 
27 #include "cf_factory.h"
28 #include "cf_defs.h"
29 #include "canonicalform.h"
30 #include "cf_algorithm.h"
31 #include "variable.h"
32 #include "cf_iter.h"
34 #include "cfGcdAlgExt.h"
35 
36 void out_cf(const char *s1,const CanonicalForm &f,const char *s2);
37 
38 /** CanonicalForm psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
39  *
40  *
41  * psr() - return pseudo remainder of `f' and `g' with respect
42  * to `x'.
43  *
44  * `g' must not equal zero.
45  *
46  * For f and g in R[x], R an arbitrary ring, g != 0, there is a
47  * representation
48  *
49  * LC(g)^s*f = g*q + r
50  *
51  * with r = 0 or deg(r) < deg(g) and s = 0 if f = 0 or
52  * s = max( 0, deg(f)-deg(g)+1 ) otherwise.
53  * r = psr(f, g) and q = psq(f, g) are called "pseudo remainder"
54  * and "pseudo quotient", resp. They are uniquely determined if
55  * LC(g) is not a zero divisor in R.
56  *
57  * See H.-J. Reiffen/G. Scheja/U. Vetter - "Algebra", 2nd ed.,
58  * par. 15, for a reference.
59  *
60  * Type info:
61  * ----------
62  * f, g: Current
63  * x: Polynomial
64  *
65  * Polynomials over prime power domains are admissible if
66  * lc(LC(`g',`x')) is not a zero divisor. This is a slightly
67  * stronger precondition than mathematically necessary since
68  * pseudo remainder and quotient are well-defined if LC(`g',`x')
69  * is not a zero divisor.
70  *
71  * For example, psr(y^2, (13*x+1)*y) is well-defined in
72  * (Z/13^2[x])[y] since (13*x+1) is not a zero divisor. But
73  * calculating it with Factory would fail since 13 is a zero
74  * divisor in Z/13^2.
75  *
76  * Due to this inconsistency with mathematical notion, we decided
77  * not to declare type `CurrentPP' for `f' and `g'.
78  *
79  * Developers note:
80  * ----------------
81  * This is not an optimal implementation. Better would have been
82  * an implementation in `InternalPoly' avoiding the
83  * exponentiation of the leading coefficient of `g'. In contrast
84  * to `psq()' and `psqr()' it definitely seems worth to implement
85  * the pseudo remainder on the internal level.
86  *
87  * @sa psq(), psqr()
88 **/
90 #if 0
91 psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
92 {
93 
94  ASSERT( x.level() > 0, "type error: polynomial variable expected" );
95  ASSERT( ! g.isZero(), "math error: division by zero" );
96 
97  // swap variables such that x's level is larger or equal
98  // than both f's and g's levels.
99  Variable X = tmax( tmax( f.mvar(), g.mvar() ), x );
100  CanonicalForm F = swapvar( f, x, X );
101  CanonicalForm G = swapvar( g, x, X );
102 
103  // now, we have to calculate the pseudo remainder of F and G
104  // w.r.t. X
105  int fDegree = degree( F, X );
106  int gDegree = degree( G, X );
107  if ( (fDegree < 0) || (fDegree < gDegree) )
108  return f;
109  else
110  {
111  CanonicalForm xresult = (power( LC( G, X ), fDegree-gDegree+1 ) * F) ;
112  CanonicalForm result = xresult -(xresult/G)*G;
113  return swapvar( result, x, X );
114  }
115 }
116 #else
117 psr ( const CanonicalForm &rr, const CanonicalForm &vv, const Variable & x )
118 {
119  CanonicalForm r=rr, v=vv, l, test, lu, lv, t, retvalue;
120  int dr, dv, d,n=0;
121 
122 
123  dr = degree( r, x );
124  if (dr>0)
125  {
126  dv = degree( v, x );
127  if (dv <= dr) {l=LC(v,x); v = v -l*power(x,dv);}
128  else { l = 1; }
129  d= dr-dv+1;
130  //out_cf("psr(",rr," ");
131  //out_cf("",vv," ");
132  //printf(" var=%d\n",x.level());
133  while ( ( dv <= dr ) && ( !r.isZero()) )
134  {
135  test = power(x,dr-dv)*v*LC(r,x);
136  if ( dr == 0 ) { r= CanonicalForm(0); }
137  else { r= r - LC(r,x)*power(x,dr); }
138  r= l*r -test;
139  dr= degree(r,x);
140  n+=1;
141  }
142  r= power(l, d-n)*r;
143  }
144  return r;
145 }
146 #endif
147 
148 /** CanonicalForm psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
149  *
150  *
151  * psq() - return pseudo quotient of `f' and `g' with respect
152  * to `x'.
153  *
154  * `g' must not equal zero.
155  *
156  * Type info:
157  * ----------
158  * f, g: Current
159  * x: Polynomial
160  *
161  * Developers note:
162  * ----------------
163  * This is not an optimal implementation. Better would have been
164  * an implementation in `InternalPoly' avoiding the
165  * exponentiation of the leading coefficient of `g'. It seemed
166  * not worth to do so.
167  *
168  * @sa psr(), psqr()
169  *
170 **/
172 psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x )
173 {
174  ASSERT( x.level() > 0, "type error: polynomial variable expected" );
175  ASSERT( ! g.isZero(), "math error: division by zero" );
176 
177  // swap variables such that x's level is larger or equal
178  // than both f's and g's levels.
179  Variable X = tmax( tmax( f.mvar(), g.mvar() ), x );
180  CanonicalForm F = swapvar( f, x, X );
181  CanonicalForm G = swapvar( g, x, X );
182 
183  // now, we have to calculate the pseudo remainder of F and G
184  // w.r.t. X
185  int fDegree = degree( F, X );
186  int gDegree = degree( G, X );
187  if ( fDegree < 0 || fDegree < gDegree )
188  return 0;
189  else {
190  CanonicalForm result = (power( LC( G, X ), fDegree-gDegree+1 ) * F) / G;
191  return swapvar( result, x, X );
192  }
193 }
194 
195 /** void psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r, const Variable & x )
196  *
197  *
198  * psqr() - calculate pseudo quotient and remainder of `f' and
199  * `g' with respect to `x'.
200  *
201  * Returns the pseudo quotient of `f' and `g' in `q', the pseudo
202  * remainder in `r'. `g' must not equal zero.
203  *
204  * See `psr()' for more detailed information.
205  *
206  * Type info:
207  * ----------
208  * f, g: Current
209  * q, r: Anything
210  * x: Polynomial
211  *
212  * Developers note:
213  * ----------------
214  * This is not an optimal implementation. Better would have been
215  * an implementation in `InternalPoly' avoiding the
216  * exponentiation of the leading coefficient of `g'. It seemed
217  * not worth to do so.
218  *
219  * @sa psr(), psq()
220  *
221 **/
222 void
223 psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r, const Variable& x )
224 {
225  ASSERT( x.level() > 0, "type error: polynomial variable expected" );
226  ASSERT( ! g.isZero(), "math error: division by zero" );
227 
228  // swap variables such that x's level is larger or equal
229  // than both f's and g's levels.
230  Variable X = tmax( tmax( f.mvar(), g.mvar() ), x );
231  CanonicalForm F = swapvar( f, x, X );
232  CanonicalForm G = swapvar( g, x, X );
233 
234  // now, we have to calculate the pseudo remainder of F and G
235  // w.r.t. X
236  int fDegree = degree( F, X );
237  int gDegree = degree( G, X );
238  if ( fDegree < 0 || fDegree < gDegree ) {
239  q = 0; r = f;
240  } else {
241  divrem( power( LC( G, X ), fDegree-gDegree+1 ) * F, G, q, r );
242  q = swapvar( q, x, X );
243  r = swapvar( r, x, X );
244  }
245 }
246 
247 /** static CanonicalForm internalBCommonDen ( const CanonicalForm & f )
248  *
249  *
250  * internalBCommonDen() - recursively calculate multivariate
251  * common denominator of coefficients of `f'.
252  *
253  * Used by: bCommonDen()
254  *
255  * Type info:
256  * ----------
257  * f: Poly( Q )
258  * Switches: isOff( SW_RATIONAL )
259  *
260 **/
261 static CanonicalForm
263 {
264  if ( f.inBaseDomain() )
265  return f.den();
266  else {
267  CanonicalForm result = 1;
268  for ( CFIterator i = f; i.hasTerms(); i++ )
269  result = blcm( result, internalBCommonDen( i.coeff() ) );
270  return result;
271  }
272 }
273 
274 /** CanonicalForm bCommonDen ( const CanonicalForm & f )
275  *
276  *
277  * bCommonDen() - calculate multivariate common denominator of
278  * coefficients of `f'.
279  *
280  * The common denominator is calculated with respect to all
281  * coefficients of `f' which are in a base domain. In other
282  * words, common_den( `f' ) * `f' is guaranteed to have integer
283  * coefficients only. The common denominator of zero is one.
284  *
285  * Returns something non-trivial iff the current domain is Q.
286  *
287  * Type info:
288  * ----------
289  * f: CurrentPP
290  *
291 **/
294 {
295  if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) {
296  // otherwise `bgcd()' returns one
297  Off( SW_RATIONAL );
299  On( SW_RATIONAL );
300  return result;
301  } else
302  return CanonicalForm( 1 );
303 }
304 
305 /** bool fdivides ( const CanonicalForm & f, const CanonicalForm & g )
306  *
307  *
308  * fdivides() - check whether `f' divides `g'.
309  *
310  * Returns true iff `f' divides `g'. Uses some extra heuristic
311  * to avoid polynomial division. Without the heuristic, the test
312  * essentialy looks like `divremt(g, f, q, r) && r.isZero()'.
313  *
314  * Type info:
315  * ----------
316  * f, g: Current
317  *
318  * Elements from prime power domains (or polynomials over such
319  * domains) are admissible if `f' (or lc(`f'), resp.) is not a
320  * zero divisor. This is a slightly stronger precondition than
321  * mathematically necessary since divisibility is a well-defined
322  * notion in arbitrary rings. Hence, we decided not to declare
323  * the weaker type `CurrentPP'.
324  *
325  * Developers note:
326  * ----------------
327  * One may consider the the test `fdivides( f.LC(), g.LC() )' in
328  * the main `if'-test superfluous since `divremt()' in the
329  * `if'-body repeats the test. However, `divremt()' does not use
330  * any heuristic to do so.
331  *
332  * It seems not reasonable to call `fdivides()' from `divremt()'
333  * to check divisibility of leading coefficients. `fdivides()' is
334  * on a relatively high level compared to `divremt()'.
335  *
336 **/
337 bool
338 fdivides ( const CanonicalForm & f, const CanonicalForm & g )
339 {
340  // trivial cases
341  if ( g.isZero() )
342  return true;
343  else if ( f.isZero() )
344  return false;
345 
346  if ( (f.inCoeffDomain() || g.inCoeffDomain())
347  && ((getCharacteristic() == 0 && isOn( SW_RATIONAL ))
348  || (getCharacteristic() > 0) ))
349  {
350  // if we are in a field all elements not equal to zero are units
351  if ( f.inCoeffDomain() )
352  return true;
353  else
354  // g.inCoeffDomain()
355  return false;
356  }
357 
358  // we may assume now that both levels either equal LEVELBASE
359  // or are greater zero
360  int fLevel = f.level();
361  int gLevel = g.level();
362  if ( (gLevel > 0) && (fLevel == gLevel) )
363  // f and g are polynomials in the same main variable
364  if ( degree( f ) <= degree( g )
365  && fdivides( f.tailcoeff(), g.tailcoeff() )
366  && fdivides( f.LC(), g.LC() ) )
367  {
368  CanonicalForm q, r;
369  return divremt( g, f, q, r ) && r.isZero();
370  }
371  else
372  return false;
373  else if ( gLevel < fLevel )
374  // g is a coefficient w.r.t. f
375  return false;
376  else
377  {
378  // either f is a coefficient w.r.t. polynomial g or both
379  // f and g are from a base domain (should be Z or Z/p^n,
380  // then)
381  CanonicalForm q, r;
382  return divremt( g, f, q, r ) && r.isZero();
383  }
384 }
385 
386 /// same as fdivides if true returns quotient quot of g by f otherwise quot == 0
387 bool
388 fdivides ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm& quot )
389 {
390  quot= 0;
391  // trivial cases
392  if ( g.isZero() )
393  return true;
394  else if ( f.isZero() )
395  return false;
396 
397  if ( (f.inCoeffDomain() || g.inCoeffDomain())
398  && ((getCharacteristic() == 0 && isOn( SW_RATIONAL ))
399  || (getCharacteristic() > 0) ))
400  {
401  // if we are in a field all elements not equal to zero are units
402  if ( f.inCoeffDomain() )
403  {
404  quot= g/f;
405  return true;
406  }
407  else
408  // g.inCoeffDomain()
409  return false;
410  }
411 
412  // we may assume now that both levels either equal LEVELBASE
413  // or are greater zero
414  int fLevel = f.level();
415  int gLevel = g.level();
416  if ( (gLevel > 0) && (fLevel == gLevel) )
417  // f and g are polynomials in the same main variable
418  if ( degree( f ) <= degree( g )
419  && fdivides( f.tailcoeff(), g.tailcoeff() )
420  && fdivides( f.LC(), g.LC() ) )
421  {
422  CanonicalForm q, r;
423  if (divremt( g, f, q, r ) && r.isZero())
424  {
425  quot= q;
426  return true;
427  }
428  else
429  return false;
430  }
431  else
432  return false;
433  else if ( gLevel < fLevel )
434  // g is a coefficient w.r.t. f
435  return false;
436  else
437  {
438  // either f is a coefficient w.r.t. polynomial g or both
439  // f and g are from a base domain (should be Z or Z/p^n,
440  // then)
441  CanonicalForm q, r;
442  if (divremt( g, f, q, r ) && r.isZero())
443  {
444  quot= q;
445  return true;
446  }
447  else
448  return false;
449  }
450 }
451 
452 /// same as fdivides but handles zero divisors in Z_p[t]/(f)[x1,...,xn] for reducible f
453 bool
454 tryFdivides ( const CanonicalForm & f, const CanonicalForm & g, const CanonicalForm& M, bool& fail )
455 {
456  fail= false;
457  // trivial cases
458  if ( g.isZero() )
459  return true;
460  else if ( f.isZero() )
461  return false;
462 
463  if (f.inCoeffDomain() || g.inCoeffDomain())
464  {
465  // if we are in a field all elements not equal to zero are units
466  if ( f.inCoeffDomain() )
467  {
468  CanonicalForm inv;
469  tryInvert (f, M, inv, fail);
470  return !fail;
471  }
472  else
473  {
474  return false;
475  }
476  }
477 
478  // we may assume now that both levels either equal LEVELBASE
479  // or are greater zero
480  int fLevel = f.level();
481  int gLevel = g.level();
482  if ( (gLevel > 0) && (fLevel == gLevel) )
483  {
484  if (degree( f ) > degree( g ))
485  return false;
486  bool dividestail= tryFdivides (f.tailcoeff(), g.tailcoeff(), M, fail);
487 
488  if (fail || !dividestail)
489  return false;
490  bool dividesLC= tryFdivides (f.LC(),g.LC(), M, fail);
491  if (fail || !dividesLC)
492  return false;
493  CanonicalForm q,r;
494  bool divides= tryDivremt (g, f, q, r, M, fail);
495  if (fail || !divides)
496  return false;
497  return r.isZero();
498  }
499  else if ( gLevel < fLevel )
500  {
501  // g is a coefficient w.r.t. f
502  return false;
503  }
504  else
505  {
506  // either f is a coefficient w.r.t. polynomial g or both
507  // f and g are from a base domain (should be Z or Z/p^n,
508  // then)
509  CanonicalForm q, r;
510  bool divides= tryDivremt (g, f, q, r, M, fail);
511  if (fail || !divides)
512  return false;
513  return r.isZero();
514  }
515 }
516 
517 /** CanonicalForm maxNorm ( const CanonicalForm & f )
518  *
519  *
520  * maxNorm() - return maximum norm of `f'.
521  *
522  * That is, the base coefficient of `f' with the largest absolute
523  * value.
524  *
525  * Valid for arbitrary polynomials over arbitrary domains, but
526  * most useful for multivariate polynomials over Z.
527  *
528  * Type info:
529  * ----------
530  * f: CurrentPP
531  *
532 **/
534 maxNorm ( const CanonicalForm & f )
535 {
536  if ( f.inBaseDomain() )
537  return abs( f );
538  else {
539  CanonicalForm result = 0;
540  for ( CFIterator i = f; i.hasTerms(); i++ ) {
541  CanonicalForm coeffMaxNorm = maxNorm( i.coeff() );
542  if ( coeffMaxNorm > result )
543  result = coeffMaxNorm;
544  }
545  return result;
546  }
547 }
548 
549 /** CanonicalForm euclideanNorm ( const CanonicalForm & f )
550  *
551  *
552  * euclideanNorm() - return Euclidean norm of `f'.
553  *
554  * Returns the largest integer smaller or equal norm(`f') =
555  * sqrt(sum( `f'[i]^2 )).
556  *
557  * Type info:
558  * ----------
559  * f: UVPoly( Z )
560  *
561 **/
564 {
565  ASSERT( (f.inBaseDomain() || f.isUnivariate()) && f.LC().inZ(),
566  "type error: univariate poly over Z expected" );
567 
568  CanonicalForm result = 0;
569  for ( CFIterator i = f; i.hasTerms(); i++ ) {
570  CanonicalForm coeff = i.coeff();
571  result += coeff*coeff;
572  }
573  return sqrt( result );
574 }
CanonicalForm power(const CanonicalForm &f, int n)
exponentiation
void psqr(const CanonicalForm &f, const CanonicalForm &g, CanonicalForm &q, CanonicalForm &r, const Variable &x)
void psqr ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r...
GCD over Q(a)
void Off(int sw)
switches
some useful template functions.
template CanonicalForm tmax(const CanonicalForm &, const CanonicalForm &)
bool divremt(const CanonicalForm &f, const CanonicalForm &g, CanonicalForm &q, CanonicalForm &r)
f
Definition: cfModGcd.cc:4022
factory&#39;s class for variables
Definition: factory.h:115
CF_NO_INLINE CanonicalForm coeff() const
get the current coefficient
CF_NO_INLINE bool isZero() const
Definition: cf_inline.cc:372
void out_cf(const char *s1, const CanonicalForm &f, const char *s2)
cf_algorithm.cc - simple mathematical algorithms.
Definition: cf_factor.cc:90
static CanonicalForm * retvalue
Definition: readcf.cc:121
factory&#39;s main class
Definition: canonicalform.h:75
assertions for Factory
g
Definition: cfModGcd.cc:4031
static TreeM * G
Definition: janet.cc:38
CanonicalForm psq(const CanonicalForm &f, const CanonicalForm &g, const Variable &x)
CanonicalForm psq ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) ...
void tryInvert(const CanonicalForm &F, const CanonicalForm &M, CanonicalForm &inv, bool &fail)
Definition: cfGcdAlgExt.cc:219
int getCharacteristic()
Definition: cf_char.cc:51
Rational abs(const Rational &a)
Definition: GMPrat.cc:443
#define M
Definition: sirandom.c:24
CanonicalForm swapvar(const CanonicalForm &, const Variable &, const Variable &)
swapvar() - swap variables x1 and x2 in f.
Definition: cf_ops.cc:168
const ring r
Definition: syzextra.cc:208
bool inBaseDomain() const
CanonicalForm blcm(const CanonicalForm &f, const CanonicalForm &g)
int level() const
Definition: factory.h:132
bool isUnivariate() const
Interface to generate InternalCF&#39;s over various domains from intrinsic types or mpz_t&#39;s.
gmp_float sqrt(const gmp_float &a)
Definition: mpr_complex.cc:329
static const int SW_RATIONAL
set to 1 for computations over Q
Definition: cf_defs.h:28
bool isOn(int sw)
switches
void On(int sw)
switches
Iterators for CanonicalForm&#39;s.
int i
Definition: cfEzgcd.cc:123
factory switches.
Variable mvar() const
mvar() returns the main variable of CO or Variable() if CO is in a base domain.
declarations of higher level algorithms.
CanonicalForm tailcoeff() const
tailcoeff() - return least coefficient
CanonicalForm bCommonDen(const CanonicalForm &f)
CanonicalForm bCommonDen ( const CanonicalForm & f )
CanonicalForm maxNorm(const CanonicalForm &f)
CanonicalForm maxNorm ( const CanonicalForm & f )
class to iterate through CanonicalForm&#39;s
Definition: cf_iter.h:44
CanonicalForm test
Definition: cfModGcd.cc:4037
operations on variables
bool fdivides(const CanonicalForm &f, const CanonicalForm &g)
bool fdivides ( const CanonicalForm & f, const CanonicalForm & g )
const Variable & v
< [in] a sqrfree bivariate poly
Definition: facBivar.h:37
CanonicalForm den() const
den() returns the denominator of CO if CO is a rational number, 1 (from the current domain!) otherwis...
CanonicalForm psr(const CanonicalForm &rr, const CanonicalForm &vv, const Variable &x)
CanonicalForm psr ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) ...
bool tryDivremt(const CanonicalForm &f, const CanonicalForm &g, CanonicalForm &q, CanonicalForm &r, const CanonicalForm &M, bool &fail)
same as divremt but handles zero divisors in case we are in Z_p[x]/(f) where f is not irreducible ...
bool inZ() const
predicates
Variable x
Definition: cfModGcd.cc:4023
static CanonicalForm internalBCommonDen(const CanonicalForm &f)
static CanonicalForm internalBCommonDen ( const CanonicalForm & f )
CanonicalForm euclideanNorm(const CanonicalForm &f)
CanonicalForm euclideanNorm ( const CanonicalForm & f )
int level() const
level() returns the level of CO.
#define ASSERT(expression, message)
Definition: cf_assert.h:99
int degree(const CanonicalForm &f)
bool tryFdivides(const CanonicalForm &f, const CanonicalForm &g, const CanonicalForm &M, bool &fail)
same as fdivides but handles zero divisors in Z_p[t]/(f)[x1,...,xn] for reducible f ...
CanonicalForm LC(const CanonicalForm &f)
int l
Definition: cfEzgcd.cc:94
return result
Definition: facAbsBiFact.cc:76
Header for factory&#39;s main class CanonicalForm.
void divrem(const CanonicalForm &f, const CanonicalForm &g, CanonicalForm &q, CanonicalForm &r)
bool inCoeffDomain() const
CanonicalForm LC() const