3 * This file implements several functions that work on univariate and
4 * multivariate polynomials and rational functions.
5 * These functions include polynomial quotient and remainder, GCD and LCM
6 * computation, square-free factorization and rational function normalization. */
9 * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
11 * This program is free software; you can redistribute it and/or modify
12 * it under the terms of the GNU General Public License as published by
13 * the Free Software Foundation; either version 2 of the License, or
14 * (at your option) any later version.
16 * This program is distributed in the hope that it will be useful,
17 * but WITHOUT ANY WARRANTY; without even the implied warranty of
18 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19 * GNU General Public License for more details.
21 * You should have received a copy of the GNU General Public License
22 * along with this program; if not, write to the Free Software
23 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
34 #include "expairseq.h"
41 #include "relational.h"
42 #include "operators.h"
50 // If comparing expressions (ex::compare()) is fast, you can set this to 1.
51 // Some routines like quo(), rem() and gcd() will then return a quick answer
52 // when they are called with two identical arguments.
53 #define FAST_COMPARE 1
55 // Set this if you want divide_in_z() to use remembering
56 #define USE_REMEMBER 0
58 // Set this if you want divide_in_z() to use trial division followed by
59 // polynomial interpolation (always slower except for completely dense
61 #define USE_TRIAL_DIVISION 0
63 // Set this to enable some statistical output for the GCD routines
68 // Statistics variables
69 static int gcd_called = 0;
70 static int sr_gcd_called = 0;
71 static int heur_gcd_called = 0;
72 static int heur_gcd_failed = 0;
74 // Print statistics at end of program
75 static struct _stat_print {
78 std::cout << "gcd() called " << gcd_called << " times\n";
79 std::cout << "sr_gcd() called " << sr_gcd_called << " times\n";
80 std::cout << "heur_gcd() called " << heur_gcd_called << " times\n";
81 std::cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
87 /** Return pointer to first symbol found in expression. Due to GiNaCĀ“s
88 * internal ordering of terms, it may not be obvious which symbol this
89 * function returns for a given expression.
91 * @param e expression to search
92 * @param x first symbol found (returned)
93 * @return "false" if no symbol was found, "true" otherwise */
94 static bool get_first_symbol(const ex &e, ex &x)
96 if (is_a<symbol>(e)) {
99 } else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
100 for (size_t i=0; i<e.nops(); i++)
101 if (get_first_symbol(e.op(i), x))
103 } else if (is_exactly_a<power>(e)) {
104 if (get_first_symbol(e.op(0), x))
112 * Statistical information about symbols in polynomials
115 /** This structure holds information about the highest and lowest degrees
116 * in which a symbol appears in two multivariate polynomials "a" and "b".
117 * A vector of these structures with information about all symbols in
118 * two polynomials can be created with the function get_symbol_stats().
120 * @see get_symbol_stats */
122 /** Reference to symbol */
125 /** Highest degree of symbol in polynomial "a" */
128 /** Highest degree of symbol in polynomial "b" */
131 /** Lowest degree of symbol in polynomial "a" */
134 /** Lowest degree of symbol in polynomial "b" */
137 /** Maximum of deg_a and deg_b (Used for sorting) */
140 /** Maximum number of terms of leading coefficient of symbol in both polynomials */
143 /** Commparison operator for sorting */
144 bool operator<(const sym_desc &x) const
146 if (max_deg == x.max_deg)
147 return max_lcnops < x.max_lcnops;
149 return max_deg < x.max_deg;
153 // Vector of sym_desc structures
154 typedef std::vector<sym_desc> sym_desc_vec;
156 // Add symbol the sym_desc_vec (used internally by get_symbol_stats())
157 static void add_symbol(const ex &s, sym_desc_vec &v)
159 sym_desc_vec::const_iterator it = v.begin(), itend = v.end();
160 while (it != itend) {
161 if (it->sym.is_equal(s)) // If it's already in there, don't add it a second time
170 // Collect all symbols of an expression (used internally by get_symbol_stats())
171 static void collect_symbols(const ex &e, sym_desc_vec &v)
173 if (is_a<symbol>(e)) {
175 } else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
176 for (size_t i=0; i<e.nops(); i++)
177 collect_symbols(e.op(i), v);
178 } else if (is_exactly_a<power>(e)) {
179 collect_symbols(e.op(0), v);
183 /** Collect statistical information about symbols in polynomials.
184 * This function fills in a vector of "sym_desc" structs which contain
185 * information about the highest and lowest degrees of all symbols that
186 * appear in two polynomials. The vector is then sorted by minimum
187 * degree (lowest to highest). The information gathered by this
188 * function is used by the GCD routines to identify trivial factors
189 * and to determine which variable to choose as the main variable
190 * for GCD computation.
192 * @param a first multivariate polynomial
193 * @param b second multivariate polynomial
194 * @param v vector of sym_desc structs (filled in) */
195 static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
197 collect_symbols(a.eval(), v); // eval() to expand assigned symbols
198 collect_symbols(b.eval(), v);
199 sym_desc_vec::iterator it = v.begin(), itend = v.end();
200 while (it != itend) {
201 int deg_a = a.degree(it->sym);
202 int deg_b = b.degree(it->sym);
205 it->max_deg = std::max(deg_a, deg_b);
206 it->max_lcnops = std::max(a.lcoeff(it->sym).nops(), b.lcoeff(it->sym).nops());
207 it->ldeg_a = a.ldegree(it->sym);
208 it->ldeg_b = b.ldegree(it->sym);
211 std::sort(v.begin(), v.end());
214 std::clog << "Symbols:\n";
215 it = v.begin(); itend = v.end();
216 while (it != itend) {
217 std::clog << " " << it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl;
218 std::clog << " lcoeff_a=" << a.lcoeff(it->sym) << ", lcoeff_b=" << b.lcoeff(it->sym) << endl;
226 * Computation of LCM of denominators of coefficients of a polynomial
229 // Compute LCM of denominators of coefficients by going through the
230 // expression recursively (used internally by lcm_of_coefficients_denominators())
231 static numeric lcmcoeff(const ex &e, const numeric &l)
233 if (e.info(info_flags::rational))
234 return lcm(ex_to<numeric>(e).denom(), l);
235 else if (is_exactly_a<add>(e)) {
237 for (size_t i=0; i<e.nops(); i++)
238 c = lcmcoeff(e.op(i), c);
240 } else if (is_exactly_a<mul>(e)) {
242 for (size_t i=0; i<e.nops(); i++)
243 c *= lcmcoeff(e.op(i), _num1);
245 } else if (is_exactly_a<power>(e)) {
246 if (is_a<symbol>(e.op(0)))
249 return pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
254 /** Compute LCM of denominators of coefficients of a polynomial.
255 * Given a polynomial with rational coefficients, this function computes
256 * the LCM of the denominators of all coefficients. This can be used
257 * to bring a polynomial from Q[X] to Z[X].
259 * @param e multivariate polynomial (need not be expanded)
260 * @return LCM of denominators of coefficients */
261 static numeric lcm_of_coefficients_denominators(const ex &e)
263 return lcmcoeff(e, _num1);
266 /** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
267 * determined LCM of the coefficient's denominators.
269 * @param e multivariate polynomial (need not be expanded)
270 * @param lcm LCM to multiply in */
271 static ex multiply_lcm(const ex &e, const numeric &lcm)
273 if (is_exactly_a<mul>(e)) {
274 size_t num = e.nops();
275 exvector v; v.reserve(num + 1);
276 numeric lcm_accum = _num1;
277 for (size_t i=0; i<num; i++) {
278 numeric op_lcm = lcmcoeff(e.op(i), _num1);
279 v.push_back(multiply_lcm(e.op(i), op_lcm));
282 v.push_back(lcm / lcm_accum);
283 return (new mul(v))->setflag(status_flags::dynallocated);
284 } else if (is_exactly_a<add>(e)) {
285 size_t num = e.nops();
286 exvector v; v.reserve(num);
287 for (size_t i=0; i<num; i++)
288 v.push_back(multiply_lcm(e.op(i), lcm));
289 return (new add(v))->setflag(status_flags::dynallocated);
290 } else if (is_exactly_a<power>(e)) {
291 if (is_a<symbol>(e.op(0)))
294 return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
300 /** Compute the integer content (= GCD of all numeric coefficients) of an
301 * expanded polynomial. For a polynomial with rational coefficients, this
302 * returns g/l where g is the GCD of the coefficients' numerators and l
303 * is the LCM of the coefficients' denominators.
305 * @return integer content */
306 numeric ex::integer_content() const
308 return bp->integer_content();
311 numeric basic::integer_content() const
316 numeric numeric::integer_content() const
321 numeric add::integer_content() const
323 epvector::const_iterator it = seq.begin();
324 epvector::const_iterator itend = seq.end();
325 numeric c = _num0, l = _num1;
326 while (it != itend) {
327 GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
328 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
329 c = gcd(ex_to<numeric>(it->coeff).numer(), c);
330 l = lcm(ex_to<numeric>(it->coeff).denom(), l);
333 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
334 c = gcd(ex_to<numeric>(overall_coeff).numer(), c);
335 l = lcm(ex_to<numeric>(overall_coeff).denom(), l);
339 numeric mul::integer_content() const
341 #ifdef DO_GINAC_ASSERT
342 epvector::const_iterator it = seq.begin();
343 epvector::const_iterator itend = seq.end();
344 while (it != itend) {
345 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
348 #endif // def DO_GINAC_ASSERT
349 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
350 return abs(ex_to<numeric>(overall_coeff));
355 * Polynomial quotients and remainders
358 /** Quotient q(x) of polynomials a(x) and b(x) in Q[x].
359 * It satisfies a(x)=b(x)*q(x)+r(x).
361 * @param a first polynomial in x (dividend)
362 * @param b second polynomial in x (divisor)
363 * @param x a and b are polynomials in x
364 * @param check_args check whether a and b are polynomials with rational
365 * coefficients (defaults to "true")
366 * @return quotient of a and b in Q[x] */
367 ex quo(const ex &a, const ex &b, const ex &x, bool check_args)
370 throw(std::overflow_error("quo: division by zero"));
371 if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b))
377 if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
378 throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));
380 // Polynomial long division
384 int bdeg = b.degree(x);
385 int rdeg = r.degree(x);
386 ex blcoeff = b.expand().coeff(x, bdeg);
387 bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
388 exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
389 while (rdeg >= bdeg) {
390 ex term, rcoeff = r.coeff(x, rdeg);
391 if (blcoeff_is_numeric)
392 term = rcoeff / blcoeff;
394 if (!divide(rcoeff, blcoeff, term, false))
395 return (new fail())->setflag(status_flags::dynallocated);
397 term *= power(x, rdeg - bdeg);
399 r -= (term * b).expand();
404 return (new add(v))->setflag(status_flags::dynallocated);
408 /** Remainder r(x) of polynomials a(x) and b(x) in Q[x].
409 * It satisfies a(x)=b(x)*q(x)+r(x).
411 * @param a first polynomial in x (dividend)
412 * @param b second polynomial in x (divisor)
413 * @param x a and b are polynomials in x
414 * @param check_args check whether a and b are polynomials with rational
415 * coefficients (defaults to "true")
416 * @return remainder of a(x) and b(x) in Q[x] */
417 ex rem(const ex &a, const ex &b, const ex &x, bool check_args)
420 throw(std::overflow_error("rem: division by zero"));
421 if (is_exactly_a<numeric>(a)) {
422 if (is_exactly_a<numeric>(b))
431 if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
432 throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));
434 // Polynomial long division
438 int bdeg = b.degree(x);
439 int rdeg = r.degree(x);
440 ex blcoeff = b.expand().coeff(x, bdeg);
441 bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
442 while (rdeg >= bdeg) {
443 ex term, rcoeff = r.coeff(x, rdeg);
444 if (blcoeff_is_numeric)
445 term = rcoeff / blcoeff;
447 if (!divide(rcoeff, blcoeff, term, false))
448 return (new fail())->setflag(status_flags::dynallocated);
450 term *= power(x, rdeg - bdeg);
451 r -= (term * b).expand();
460 /** Decompose rational function a(x)=N(x)/D(x) into P(x)+n(x)/D(x)
461 * with degree(n, x) < degree(D, x).
463 * @param a rational function in x
464 * @param x a is a function of x
465 * @return decomposed function. */
466 ex decomp_rational(const ex &a, const ex &x)
468 ex nd = numer_denom(a);
469 ex numer = nd.op(0), denom = nd.op(1);
470 ex q = quo(numer, denom, x);
471 if (is_exactly_a<fail>(q))
474 return q + rem(numer, denom, x) / denom;
478 /** Pseudo-remainder of polynomials a(x) and b(x) in Q[x].
480 * @param a first polynomial in x (dividend)
481 * @param b second polynomial in x (divisor)
482 * @param x a and b are polynomials in x
483 * @param check_args check whether a and b are polynomials with rational
484 * coefficients (defaults to "true")
485 * @return pseudo-remainder of a(x) and b(x) in Q[x] */
486 ex prem(const ex &a, const ex &b, const ex &x, bool check_args)
489 throw(std::overflow_error("prem: division by zero"));
490 if (is_exactly_a<numeric>(a)) {
491 if (is_exactly_a<numeric>(b))
496 if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
497 throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));
499 // Polynomial long division
502 int rdeg = r.degree(x);
503 int bdeg = eb.degree(x);
506 blcoeff = eb.coeff(x, bdeg);
510 eb -= blcoeff * power(x, bdeg);
514 int delta = rdeg - bdeg + 1, i = 0;
515 while (rdeg >= bdeg && !r.is_zero()) {
516 ex rlcoeff = r.coeff(x, rdeg);
517 ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
521 r -= rlcoeff * power(x, rdeg);
522 r = (blcoeff * r).expand() - term;
526 return power(blcoeff, delta - i) * r;
530 /** Sparse pseudo-remainder of polynomials a(x) and b(x) in Q[x].
532 * @param a first polynomial in x (dividend)
533 * @param b second polynomial in x (divisor)
534 * @param x a and b are polynomials in x
535 * @param check_args check whether a and b are polynomials with rational
536 * coefficients (defaults to "true")
537 * @return sparse pseudo-remainder of a(x) and b(x) in Q[x] */
538 ex sprem(const ex &a, const ex &b, const ex &x, bool check_args)
541 throw(std::overflow_error("prem: division by zero"));
542 if (is_exactly_a<numeric>(a)) {
543 if (is_exactly_a<numeric>(b))
548 if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
549 throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));
551 // Polynomial long division
554 int rdeg = r.degree(x);
555 int bdeg = eb.degree(x);
558 blcoeff = eb.coeff(x, bdeg);
562 eb -= blcoeff * power(x, bdeg);
566 while (rdeg >= bdeg && !r.is_zero()) {
567 ex rlcoeff = r.coeff(x, rdeg);
568 ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
572 r -= rlcoeff * power(x, rdeg);
573 r = (blcoeff * r).expand() - term;
580 /** Exact polynomial division of a(X) by b(X) in Q[X].
582 * @param a first multivariate polynomial (dividend)
583 * @param b second multivariate polynomial (divisor)
584 * @param q quotient (returned)
585 * @param check_args check whether a and b are polynomials with rational
586 * coefficients (defaults to "true")
587 * @return "true" when exact division succeeds (quotient returned in q),
588 * "false" otherwise (q left untouched) */
589 bool divide(const ex &a, const ex &b, ex &q, bool check_args)
592 throw(std::overflow_error("divide: division by zero"));
597 if (is_exactly_a<numeric>(b)) {
600 } else if (is_exactly_a<numeric>(a))
608 if (check_args && (!a.info(info_flags::rational_polynomial) ||
609 !b.info(info_flags::rational_polynomial)))
610 throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));
614 if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
615 throw(std::invalid_argument("invalid expression in divide()"));
617 // Polynomial long division (recursive)
623 int bdeg = b.degree(x);
624 int rdeg = r.degree(x);
625 ex blcoeff = b.expand().coeff(x, bdeg);
626 bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
627 exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
628 while (rdeg >= bdeg) {
629 ex term, rcoeff = r.coeff(x, rdeg);
630 if (blcoeff_is_numeric)
631 term = rcoeff / blcoeff;
633 if (!divide(rcoeff, blcoeff, term, false))
635 term *= power(x, rdeg - bdeg);
637 r -= (term * b).expand();
639 q = (new add(v))->setflag(status_flags::dynallocated);
653 typedef std::pair<ex, ex> ex2;
654 typedef std::pair<ex, bool> exbool;
657 bool operator() (const ex2 &p, const ex2 &q) const
659 int cmp = p.first.compare(q.first);
660 return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0));
664 typedef std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
668 /** Exact polynomial division of a(X) by b(X) in Z[X].
669 * This functions works like divide() but the input and output polynomials are
670 * in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike
671 * divide(), it doesn't check whether the input polynomials really are integer
672 * polynomials, so be careful of what you pass in. Also, you have to run
673 * get_symbol_stats() over the input polynomials before calling this function
674 * and pass an iterator to the first element of the sym_desc vector. This
675 * function is used internally by the heur_gcd().
677 * @param a first multivariate polynomial (dividend)
678 * @param b second multivariate polynomial (divisor)
679 * @param q quotient (returned)
680 * @param var iterator to first element of vector of sym_desc structs
681 * @return "true" when exact division succeeds (the quotient is returned in
682 * q), "false" otherwise.
683 * @see get_symbol_stats, heur_gcd */
684 static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var)
688 throw(std::overflow_error("divide_in_z: division by zero"));
689 if (b.is_equal(_ex1)) {
693 if (is_exactly_a<numeric>(a)) {
694 if (is_exactly_a<numeric>(b)) {
696 return q.info(info_flags::integer);
709 static ex2_exbool_remember dr_remember;
710 ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b));
711 if (remembered != dr_remember.end()) {
712 q = remembered->second.first;
713 return remembered->second.second;
718 const ex &x = var->sym;
721 int adeg = a.degree(x), bdeg = b.degree(x);
725 #if USE_TRIAL_DIVISION
727 // Trial division with polynomial interpolation
730 // Compute values at evaluation points 0..adeg
731 vector<numeric> alpha; alpha.reserve(adeg + 1);
732 exvector u; u.reserve(adeg + 1);
733 numeric point = _num0;
735 for (i=0; i<=adeg; i++) {
736 ex bs = b.subs(x == point, subs_options::no_pattern);
737 while (bs.is_zero()) {
739 bs = b.subs(x == point, subs_options::no_pattern);
741 if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1))
743 alpha.push_back(point);
749 vector<numeric> rcp; rcp.reserve(adeg + 1);
750 rcp.push_back(_num0);
751 for (k=1; k<=adeg; k++) {
752 numeric product = alpha[k] - alpha[0];
754 product *= alpha[k] - alpha[i];
755 rcp.push_back(product.inverse());
758 // Compute Newton coefficients
759 exvector v; v.reserve(adeg + 1);
761 for (k=1; k<=adeg; k++) {
763 for (i=k-2; i>=0; i--)
764 temp = temp * (alpha[k] - alpha[i]) + v[i];
765 v.push_back((u[k] - temp) * rcp[k]);
768 // Convert from Newton form to standard form
770 for (k=adeg-1; k>=0; k--)
771 c = c * (x - alpha[k]) + v[k];
773 if (c.degree(x) == (adeg - bdeg)) {
781 // Polynomial long division (recursive)
787 ex blcoeff = eb.coeff(x, bdeg);
788 exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
789 while (rdeg >= bdeg) {
790 ex term, rcoeff = r.coeff(x, rdeg);
791 if (!divide_in_z(rcoeff, blcoeff, term, var+1))
793 term = (term * power(x, rdeg - bdeg)).expand();
795 r -= (term * eb).expand();
797 q = (new add(v))->setflag(status_flags::dynallocated);
799 dr_remember[ex2(a, b)] = exbool(q, true);
806 dr_remember[ex2(a, b)] = exbool(q, false);
815 * Separation of unit part, content part and primitive part of polynomials
818 /** Compute unit part (= sign of leading coefficient) of a multivariate
819 * polynomial in Q[x]. The product of unit part, content part, and primitive
820 * part is the polynomial itself.
822 * @param x variable in which to compute the unit part
824 * @see ex::content, ex::primpart */
825 ex ex::unit(const ex &x) const
827 ex c = expand().lcoeff(x);
828 if (is_exactly_a<numeric>(c))
829 return c < _ex0 ? _ex_1 : _ex1;
832 if (get_first_symbol(c, y))
835 throw(std::invalid_argument("invalid expression in unit()"));
840 /** Compute content part (= unit normal GCD of all coefficients) of a
841 * multivariate polynomial in Q[x]. The product of unit part, content part,
842 * and primitive part is the polynomial itself.
844 * @param x variable in which to compute the content part
845 * @return content part
846 * @see ex::unit, ex::primpart */
847 ex ex::content(const ex &x) const
851 if (is_exactly_a<numeric>(*this))
852 return info(info_flags::negative) ? -*this : *this;
857 // First, divide out the integer content (which we can calculate very efficiently).
858 // If the leading coefficient of the quotient is an integer, we are done.
859 ex c = e.integer_content();
861 ex lcoeff = r.lcoeff(x);
862 if (lcoeff.info(info_flags::integer))
865 // GCD of all coefficients
866 int deg = r.degree(x);
867 int ldeg = r.ldegree(x);
871 for (int i=ldeg; i<=deg; i++)
872 cont = gcd(r.coeff(x, i), cont, NULL, NULL, false);
877 /** Compute primitive part of a multivariate polynomial in Q[x].
878 * The product of unit part, content part, and primitive part is the
881 * @param x variable in which to compute the primitive part
882 * @return primitive part
883 * @see ex::unit, ex::content */
884 ex ex::primpart(const ex &x) const
888 if (is_exactly_a<numeric>(*this))
895 if (is_exactly_a<numeric>(c))
896 return *this / (c * u);
898 return quo(*this, c * u, x, false);
902 /** Compute primitive part of a multivariate polynomial in Q[x] when the
903 * content part is already known. This function is faster in computing the
904 * primitive part than the previous function.
906 * @param x variable in which to compute the primitive part
907 * @param c previously computed content part
908 * @return primitive part */
909 ex ex::primpart(const ex &x, const ex &c) const
915 if (is_exactly_a<numeric>(*this))
919 if (is_exactly_a<numeric>(c))
920 return *this / (c * u);
922 return quo(*this, c * u, x, false);
927 * GCD of multivariate polynomials
930 /** Compute GCD of multivariate polynomials using the subresultant PRS
931 * algorithm. This function is used internally by gcd().
933 * @param a first multivariate polynomial
934 * @param b second multivariate polynomial
935 * @param var iterator to first element of vector of sym_desc structs
936 * @return the GCD as a new expression
939 static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
945 // The first symbol is our main variable
946 const ex &x = var->sym;
948 // Sort c and d so that c has higher degree
950 int adeg = a.degree(x), bdeg = b.degree(x);
964 // Remove content from c and d, to be attached to GCD later
965 ex cont_c = c.content(x);
966 ex cont_d = d.content(x);
967 ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
970 c = c.primpart(x, cont_c);
971 d = d.primpart(x, cont_d);
973 // First element of subresultant sequence
974 ex r = _ex0, ri = _ex1, psi = _ex1;
975 int delta = cdeg - ddeg;
979 // Calculate polynomial pseudo-remainder
980 r = prem(c, d, x, false);
982 return gamma * d.primpart(x);
986 if (!divide_in_z(r, ri * pow(psi, delta), d, var))
987 throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
990 if (is_exactly_a<numeric>(r))
993 return gamma * r.primpart(x);
996 // Next element of subresultant sequence
997 ri = c.expand().lcoeff(x);
1001 divide_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
1002 delta = cdeg - ddeg;
1007 /** Return maximum (absolute value) coefficient of a polynomial.
1008 * This function is used internally by heur_gcd().
1010 * @return maximum coefficient
1012 numeric ex::max_coefficient() const
1014 return bp->max_coefficient();
1017 /** Implementation ex::max_coefficient().
1019 numeric basic::max_coefficient() const
1024 numeric numeric::max_coefficient() const
1029 numeric add::max_coefficient() const
1031 epvector::const_iterator it = seq.begin();
1032 epvector::const_iterator itend = seq.end();
1033 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1034 numeric cur_max = abs(ex_to<numeric>(overall_coeff));
1035 while (it != itend) {
1037 GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
1038 a = abs(ex_to<numeric>(it->coeff));
1046 numeric mul::max_coefficient() const
1048 #ifdef DO_GINAC_ASSERT
1049 epvector::const_iterator it = seq.begin();
1050 epvector::const_iterator itend = seq.end();
1051 while (it != itend) {
1052 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
1055 #endif // def DO_GINAC_ASSERT
1056 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1057 return abs(ex_to<numeric>(overall_coeff));
1061 /** Apply symmetric modular homomorphism to an expanded multivariate
1062 * polynomial. This function is usually used internally by heur_gcd().
1065 * @return mapped polynomial
1067 ex basic::smod(const numeric &xi) const
1072 ex numeric::smod(const numeric &xi) const
1074 return GiNaC::smod(*this, xi);
1077 ex add::smod(const numeric &xi) const
1080 newseq.reserve(seq.size()+1);
1081 epvector::const_iterator it = seq.begin();
1082 epvector::const_iterator itend = seq.end();
1083 while (it != itend) {
1084 GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
1085 numeric coeff = GiNaC::smod(ex_to<numeric>(it->coeff), xi);
1086 if (!coeff.is_zero())
1087 newseq.push_back(expair(it->rest, coeff));
1090 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1091 numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
1092 return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
1095 ex mul::smod(const numeric &xi) const
1097 #ifdef DO_GINAC_ASSERT
1098 epvector::const_iterator it = seq.begin();
1099 epvector::const_iterator itend = seq.end();
1100 while (it != itend) {
1101 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
1104 #endif // def DO_GINAC_ASSERT
1105 mul * mulcopyp = new mul(*this);
1106 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1107 mulcopyp->overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
1108 mulcopyp->clearflag(status_flags::evaluated);
1109 mulcopyp->clearflag(status_flags::hash_calculated);
1110 return mulcopyp->setflag(status_flags::dynallocated);
1114 /** xi-adic polynomial interpolation */
1115 static ex interpolate(const ex &gamma, const numeric &xi, const ex &x, int degree_hint = 1)
1117 exvector g; g.reserve(degree_hint);
1119 numeric rxi = xi.inverse();
1120 for (int i=0; !e.is_zero(); i++) {
1122 g.push_back(gi * power(x, i));
1125 return (new add(g))->setflag(status_flags::dynallocated);
1128 /** Exception thrown by heur_gcd() to signal failure. */
1129 class gcdheu_failed {};
1131 /** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
1132 * get_symbol_stats() must have been called previously with the input
1133 * polynomials and an iterator to the first element of the sym_desc vector
1134 * passed in. This function is used internally by gcd().
1136 * @param a first multivariate polynomial (expanded)
1137 * @param b second multivariate polynomial (expanded)
1138 * @param ca cofactor of polynomial a (returned), NULL to suppress
1139 * calculation of cofactor
1140 * @param cb cofactor of polynomial b (returned), NULL to suppress
1141 * calculation of cofactor
1142 * @param var iterator to first element of vector of sym_desc structs
1143 * @return the GCD as a new expression
1145 * @exception gcdheu_failed() */
1146 static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
1152 // Algorithm only works for non-vanishing input polynomials
1153 if (a.is_zero() || b.is_zero())
1154 return (new fail())->setflag(status_flags::dynallocated);
1156 // GCD of two numeric values -> CLN
1157 if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
1158 numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
1160 *ca = ex_to<numeric>(a) / g;
1162 *cb = ex_to<numeric>(b) / g;
1166 // The first symbol is our main variable
1167 const ex &x = var->sym;
1169 // Remove integer content
1170 numeric gc = gcd(a.integer_content(), b.integer_content());
1171 numeric rgc = gc.inverse();
1174 int maxdeg = std::max(p.degree(x), q.degree(x));
1176 // Find evaluation point
1177 numeric mp = p.max_coefficient();
1178 numeric mq = q.max_coefficient();
1181 xi = mq * _num2 + _num2;
1183 xi = mp * _num2 + _num2;
1186 for (int t=0; t<6; t++) {
1187 if (xi.int_length() * maxdeg > 100000) {
1188 throw gcdheu_failed();
1191 // Apply evaluation homomorphism and calculate GCD
1193 ex gamma = heur_gcd(p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1).expand();
1194 if (!is_exactly_a<fail>(gamma)) {
1196 // Reconstruct polynomial from GCD of mapped polynomials
1197 ex g = interpolate(gamma, xi, x, maxdeg);
1199 // Remove integer content
1200 g /= g.integer_content();
1202 // If the calculated polynomial divides both p and q, this is the GCD
1204 if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
1206 ex lc = g.lcoeff(x);
1207 if (is_exactly_a<numeric>(lc) && ex_to<numeric>(lc).is_negative())
1214 // Next evaluation point
1215 xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
1217 return (new fail())->setflag(status_flags::dynallocated);
1221 /** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
1224 * @param a first multivariate polynomial
1225 * @param b second multivariate polynomial
1226 * @param ca pointer to expression that will receive the cofactor of a, or NULL
1227 * @param cb pointer to expression that will receive the cofactor of b, or NULL
1228 * @param check_args check whether a and b are polynomials with rational
1229 * coefficients (defaults to "true")
1230 * @return the GCD as a new expression */
1231 ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
1237 // GCD of numerics -> CLN
1238 if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
1239 numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
1248 *ca = ex_to<numeric>(a) / g;
1250 *cb = ex_to<numeric>(b) / g;
1257 if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) {
1258 throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
1261 // Partially factored cases (to avoid expanding large expressions)
1262 if (is_exactly_a<mul>(a)) {
1263 if (is_exactly_a<mul>(b) && b.nops() > a.nops())
1266 size_t num = a.nops();
1267 exvector g; g.reserve(num);
1268 exvector acc_ca; acc_ca.reserve(num);
1270 for (size_t i=0; i<num; i++) {
1271 ex part_ca, part_cb;
1272 g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
1273 acc_ca.push_back(part_ca);
1277 *ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
1280 return (new mul(g))->setflag(status_flags::dynallocated);
1281 } else if (is_exactly_a<mul>(b)) {
1282 if (is_exactly_a<mul>(a) && a.nops() > b.nops())
1285 size_t num = b.nops();
1286 exvector g; g.reserve(num);
1287 exvector acc_cb; acc_cb.reserve(num);
1289 for (size_t i=0; i<num; i++) {
1290 ex part_ca, part_cb;
1291 g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
1292 acc_cb.push_back(part_cb);
1298 *cb = (new mul(acc_cb))->setflag(status_flags::dynallocated);
1299 return (new mul(g))->setflag(status_flags::dynallocated);
1303 // Input polynomials of the form poly^n are sometimes also trivial
1304 if (is_exactly_a<power>(a)) {
1306 if (is_exactly_a<power>(b)) {
1307 if (p.is_equal(b.op(0))) {
1308 // a = p^n, b = p^m, gcd = p^min(n, m)
1309 ex exp_a = a.op(1), exp_b = b.op(1);
1310 if (exp_a < exp_b) {
1314 *cb = power(p, exp_b - exp_a);
1315 return power(p, exp_a);
1318 *ca = power(p, exp_a - exp_b);
1321 return power(p, exp_b);
1325 if (p.is_equal(b)) {
1326 // a = p^n, b = p, gcd = p
1328 *ca = power(p, a.op(1) - 1);
1334 } else if (is_exactly_a<power>(b)) {
1336 if (p.is_equal(a)) {
1337 // a = p, b = p^n, gcd = p
1341 *cb = power(p, b.op(1) - 1);
1347 // Some trivial cases
1348 ex aex = a.expand(), bex = b.expand();
1349 if (aex.is_zero()) {
1356 if (bex.is_zero()) {
1363 if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) {
1371 if (a.is_equal(b)) {
1380 // Gather symbol statistics
1381 sym_desc_vec sym_stats;
1382 get_symbol_stats(a, b, sym_stats);
1384 // The symbol with least degree is our main variable
1385 sym_desc_vec::const_iterator var = sym_stats.begin();
1386 const ex &x = var->sym;
1388 // Cancel trivial common factor
1389 int ldeg_a = var->ldeg_a;
1390 int ldeg_b = var->ldeg_b;
1391 int min_ldeg = std::min(ldeg_a,ldeg_b);
1393 ex common = power(x, min_ldeg);
1394 return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
1397 // Try to eliminate variables
1398 if (var->deg_a == 0) {
1399 ex c = bex.content(x);
1400 ex g = gcd(aex, c, ca, cb, false);
1402 *cb *= bex.unit(x) * bex.primpart(x, c);
1404 } else if (var->deg_b == 0) {
1405 ex c = aex.content(x);
1406 ex g = gcd(c, bex, ca, cb, false);
1408 *ca *= aex.unit(x) * aex.primpart(x, c);
1412 // Try heuristic algorithm first, fall back to PRS if that failed
1415 g = heur_gcd(aex, bex, ca, cb, var);
1416 } catch (gcdheu_failed) {
1419 if (is_exactly_a<fail>(g)) {
1423 g = sr_gcd(aex, bex, var);
1424 if (g.is_equal(_ex1)) {
1425 // Keep cofactors factored if possible
1432 divide(aex, g, *ca, false);
1434 divide(bex, g, *cb, false);
1437 if (g.is_equal(_ex1)) {
1438 // Keep cofactors factored if possible
1450 /** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
1452 * @param a first multivariate polynomial
1453 * @param b second multivariate polynomial
1454 * @param check_args check whether a and b are polynomials with rational
1455 * coefficients (defaults to "true")
1456 * @return the LCM as a new expression */
1457 ex lcm(const ex &a, const ex &b, bool check_args)
1459 if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b))
1460 return lcm(ex_to<numeric>(a), ex_to<numeric>(b));
1461 if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
1462 throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
1465 ex g = gcd(a, b, &ca, &cb, false);
1471 * Square-free factorization
1474 /** Compute square-free factorization of multivariate polynomial a(x) using
1475 * YunĀ“s algorithm. Used internally by sqrfree().
1477 * @param a multivariate polynomial over Z[X], treated here as univariate
1479 * @param x variable to factor in
1480 * @return vector of factors sorted in ascending degree */
1481 static exvector sqrfree_yun(const ex &a, const symbol &x)
1487 if (g.is_equal(_ex1)) {
1498 } while (!z.is_zero());
1503 /** Compute a square-free factorization of a multivariate polynomial in Q[X].
1505 * @param a multivariate polynomial over Q[X]
1506 * @param l lst of variables to factor in, may be left empty for autodetection
1507 * @return a square-free factorization of \p a.
1510 * A polynomial \f$p(X) \in C[X]\f$ is said <EM>square-free</EM>
1511 * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$
1514 * p(X) = q(X)^2 r(X),
1516 * we have \f$q(X) \in C\f$.
1517 * This means that \f$p(X)\f$ has no repeated factors, apart
1518 * eventually from constants.
1519 * Given a polynomial \f$p(X) \in C[X]\f$, we say that the
1522 * p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r}
1524 * is a <EM>square-free factorization</EM> of \f$p(X)\f$ if the
1525 * following conditions hold:
1526 * -# \f$b \in C\f$ and \f$b \neq 0\f$;
1527 * -# \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$;
1528 * -# the degree of the polynomial \f$p_i\f$ is strictly positive
1529 * for \f$i = 1, \ldots, r\f$;
1530 * -# the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free.
1532 * Square-free factorizations need not be unique. For example, if
1533 * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$
1534 * into \f$-p_i(X)\f$.
1535 * Observe also that the factors \f$p_i(X)\f$ need not be irreducible
1538 ex sqrfree(const ex &a, const lst &l)
1540 if (is_exactly_a<numeric>(a) || // algorithm does not trap a==0
1541 is_a<symbol>(a)) // shortcut
1544 // If no lst of variables to factorize in was specified we have to
1545 // invent one now. Maybe one can optimize here by reversing the order
1546 // or so, I don't know.
1550 get_symbol_stats(a, _ex0, sdv);
1551 sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end();
1552 while (it != itend) {
1553 args.append(it->sym);
1560 // Find the symbol to factor in at this stage
1561 if (!is_a<symbol>(args.op(0)))
1562 throw (std::runtime_error("sqrfree(): invalid factorization variable"));
1563 const symbol &x = ex_to<symbol>(args.op(0));
1565 // convert the argument from something in Q[X] to something in Z[X]
1566 const numeric lcm = lcm_of_coefficients_denominators(a);
1567 const ex tmp = multiply_lcm(a,lcm);
1570 exvector factors = sqrfree_yun(tmp, x);
1572 // construct the next list of symbols with the first element popped
1574 newargs.remove_first();
1576 // recurse down the factors in remaining variables
1577 if (newargs.nops()>0) {
1578 exvector::iterator i = factors.begin();
1579 while (i != factors.end()) {
1580 *i = sqrfree(*i, newargs);
1585 // Done with recursion, now construct the final result
1587 exvector::const_iterator it = factors.begin(), itend = factors.end();
1588 for (int p = 1; it!=itend; ++it, ++p)
1589 result *= power(*it, p);
1591 // Yun's algorithm does not account for constant factors. (For univariate
1592 // polynomials it works only in the monic case.) We can correct this by
1593 // inserting what has been lost back into the result. For completeness
1594 // we'll also have to recurse down that factor in the remaining variables.
1595 if (newargs.nops()>0)
1596 result *= sqrfree(quo(tmp, result, x), newargs);
1598 result *= quo(tmp, result, x);
1600 // Put in the reational overall factor again and return
1601 return result * lcm.inverse();
1605 /** Compute square-free partial fraction decomposition of rational function
1608 * @param a rational function over Z[x], treated as univariate polynomial
1610 * @param x variable to factor in
1611 * @return decomposed rational function */
1612 ex sqrfree_parfrac(const ex & a, const symbol & x)
1614 // Find numerator and denominator
1615 ex nd = numer_denom(a);
1616 ex numer = nd.op(0), denom = nd.op(1);
1617 //clog << "numer = " << numer << ", denom = " << denom << endl;
1619 // Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D)
1620 ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand();
1621 //clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;
1623 // Factorize denominator and compute cofactors
1624 exvector yun = sqrfree_yun(denom, x);
1625 //clog << "yun factors: " << exprseq(yun) << endl;
1626 size_t num_yun = yun.size();
1627 exvector factor; factor.reserve(num_yun);
1628 exvector cofac; cofac.reserve(num_yun);
1629 for (size_t i=0; i<num_yun; i++) {
1630 if (!yun[i].is_equal(_ex1)) {
1631 for (size_t j=0; j<=i; j++) {
1632 factor.push_back(pow(yun[i], j+1));
1634 for (size_t k=0; k<num_yun; k++) {
1636 prod *= pow(yun[k], i-j);
1638 prod *= pow(yun[k], k+1);
1640 cofac.push_back(prod.expand());
1644 size_t num_factors = factor.size();
1645 //clog << "factors : " << exprseq(factor) << endl;
1646 //clog << "cofactors: " << exprseq(cofac) << endl;
1648 // Construct coefficient matrix for decomposition
1649 int max_denom_deg = denom.degree(x);
1650 matrix sys(max_denom_deg + 1, num_factors);
1651 matrix rhs(max_denom_deg + 1, 1);
1652 for (int i=0; i<=max_denom_deg; i++) {
1653 for (size_t j=0; j<num_factors; j++)
1654 sys(i, j) = cofac[j].coeff(x, i);
1655 rhs(i, 0) = red_numer.coeff(x, i);
1657 //clog << "coeffs: " << sys << endl;
1658 //clog << "rhs : " << rhs << endl;
1660 // Solve resulting linear system
1661 matrix vars(num_factors, 1);
1662 for (size_t i=0; i<num_factors; i++)
1663 vars(i, 0) = symbol();
1664 matrix sol = sys.solve(vars, rhs);
1666 // Sum up decomposed fractions
1668 for (size_t i=0; i<num_factors; i++)
1669 sum += sol(i, 0) / factor[i];
1671 return red_poly + sum;
1676 * Normal form of rational functions
1680 * Note: The internal normal() functions (= basic::normal() and overloaded
1681 * functions) all return lists of the form {numerator, denominator}. This
1682 * is to get around mul::eval()'s automatic expansion of numeric coefficients.
1683 * E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
1684 * the information that (a+b) is the numerator and 3 is the denominator.
1688 /** Create a symbol for replacing the expression "e" (or return a previously
1689 * assigned symbol). The symbol and expression are appended to repl, for
1690 * a later application of subs().
1691 * @see ex::normal */
1692 static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup)
1694 // Expression already replaced? Then return the assigned symbol
1695 exmap::const_iterator it = rev_lookup.find(e);
1696 if (it != rev_lookup.end())
1699 // Otherwise create new symbol and add to list, taking care that the
1700 // replacement expression doesn't itself contain symbols from repl,
1701 // because subs() is not recursive
1702 ex es = (new symbol)->setflag(status_flags::dynallocated);
1703 ex e_replaced = e.subs(repl, subs_options::no_pattern);
1704 repl.insert(std::make_pair(es, e_replaced));
1705 rev_lookup.insert(std::make_pair(e_replaced, es));
1709 /** Create a symbol for replacing the expression "e" (or return a previously
1710 * assigned symbol). The symbol and expression are appended to repl, and the
1711 * symbol is returned.
1712 * @see basic::to_rational
1713 * @see basic::to_polynomial */
1714 static ex replace_with_symbol(const ex & e, exmap & repl)
1716 // Expression already replaced? Then return the assigned symbol
1717 for (exmap::const_iterator it = repl.begin(); it != repl.end(); ++it)
1718 if (it->second.is_equal(e))
1721 // Otherwise create new symbol and add to list, taking care that the
1722 // replacement expression doesn't itself contain symbols from repl,
1723 // because subs() is not recursive
1724 ex es = (new symbol)->setflag(status_flags::dynallocated);
1725 ex e_replaced = e.subs(repl, subs_options::no_pattern);
1726 repl.insert(std::make_pair(es, e_replaced));
1731 /** Function object to be applied by basic::normal(). */
1732 struct normal_map_function : public map_function {
1734 normal_map_function(int l) : level(l) {}
1735 ex operator()(const ex & e) { return normal(e, level); }
1738 /** Default implementation of ex::normal(). It normalizes the children and
1739 * replaces the object with a temporary symbol.
1740 * @see ex::normal */
1741 ex basic::normal(exmap & repl, exmap & rev_lookup, int level) const
1744 return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1747 return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1748 else if (level == -max_recursion_level)
1749 throw(std::runtime_error("max recursion level reached"));
1751 normal_map_function map_normal(level - 1);
1752 return (new lst(replace_with_symbol(map(map_normal), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1758 /** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
1759 * @see ex::normal */
1760 ex symbol::normal(exmap & repl, exmap & rev_lookup, int level) const
1762 return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
1766 /** Implementation of ex::normal() for a numeric. It splits complex numbers
1767 * into re+I*im and replaces I and non-rational real numbers with a temporary
1769 * @see ex::normal */
1770 ex numeric::normal(exmap & repl, exmap & rev_lookup, int level) const
1772 numeric num = numer();
1775 if (num.is_real()) {
1776 if (!num.is_integer())
1777 numex = replace_with_symbol(numex, repl, rev_lookup);
1779 numeric re = num.real(), im = num.imag();
1780 ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl, rev_lookup);
1781 ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl, rev_lookup);
1782 numex = re_ex + im_ex * replace_with_symbol(I, repl, rev_lookup);
1785 // Denominator is always a real integer (see numeric::denom())
1786 return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
1790 /** Fraction cancellation.
1791 * @param n numerator
1792 * @param d denominator
1793 * @return cancelled fraction {n, d} as a list */
1794 static ex frac_cancel(const ex &n, const ex &d)
1798 numeric pre_factor = _num1;
1800 //std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
1802 // Handle trivial case where denominator is 1
1803 if (den.is_equal(_ex1))
1804 return (new lst(num, den))->setflag(status_flags::dynallocated);
1806 // Handle special cases where numerator or denominator is 0
1808 return (new lst(num, _ex1))->setflag(status_flags::dynallocated);
1809 if (den.expand().is_zero())
1810 throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
1812 // Bring numerator and denominator to Z[X] by multiplying with
1813 // LCM of all coefficients' denominators
1814 numeric num_lcm = lcm_of_coefficients_denominators(num);
1815 numeric den_lcm = lcm_of_coefficients_denominators(den);
1816 num = multiply_lcm(num, num_lcm);
1817 den = multiply_lcm(den, den_lcm);
1818 pre_factor = den_lcm / num_lcm;
1820 // Cancel GCD from numerator and denominator
1822 if (gcd(num, den, &cnum, &cden, false) != _ex1) {
1827 // Make denominator unit normal (i.e. coefficient of first symbol
1828 // as defined by get_first_symbol() is made positive)
1829 if (is_exactly_a<numeric>(den)) {
1830 if (ex_to<numeric>(den).is_negative()) {
1836 if (get_first_symbol(den, x)) {
1837 GINAC_ASSERT(is_exactly_a<numeric>(den.unit(x)));
1838 if (ex_to<numeric>(den.unit(x)).is_negative()) {
1845 // Return result as list
1846 //std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
1847 return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
1851 /** Implementation of ex::normal() for a sum. It expands terms and performs
1852 * fractional addition.
1853 * @see ex::normal */
1854 ex add::normal(exmap & repl, exmap & rev_lookup, int level) const
1857 return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1858 else if (level == -max_recursion_level)
1859 throw(std::runtime_error("max recursion level reached"));
1861 // Normalize children and split each one into numerator and denominator
1862 exvector nums, dens;
1863 nums.reserve(seq.size()+1);
1864 dens.reserve(seq.size()+1);
1865 epvector::const_iterator it = seq.begin(), itend = seq.end();
1866 while (it != itend) {
1867 ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(repl, rev_lookup, level-1);
1868 nums.push_back(n.op(0));
1869 dens.push_back(n.op(1));
1872 ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, level-1);
1873 nums.push_back(n.op(0));
1874 dens.push_back(n.op(1));
1875 GINAC_ASSERT(nums.size() == dens.size());
1877 // Now, nums is a vector of all numerators and dens is a vector of
1879 //std::clog << "add::normal uses " << nums.size() << " summands:\n";
1881 // Add fractions sequentially
1882 exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
1883 exvector::const_iterator den_it = dens.begin(), den_itend = dens.end();
1884 //std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
1885 ex num = *num_it++, den = *den_it++;
1886 while (num_it != num_itend) {
1887 //std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
1888 ex next_num = *num_it++, next_den = *den_it++;
1890 // Trivially add sequences of fractions with identical denominators
1891 while ((den_it != den_itend) && next_den.is_equal(*den_it)) {
1892 next_num += *num_it;
1896 // Additiion of two fractions, taking advantage of the fact that
1897 // the heuristic GCD algorithm computes the cofactors at no extra cost
1898 ex co_den1, co_den2;
1899 ex g = gcd(den, next_den, &co_den1, &co_den2, false);
1900 num = ((num * co_den2) + (next_num * co_den1)).expand();
1901 den *= co_den2; // this is the lcm(den, next_den)
1903 //std::clog << " common denominator = " << den << std::endl;
1905 // Cancel common factors from num/den
1906 return frac_cancel(num, den);
1910 /** Implementation of ex::normal() for a product. It cancels common factors
1912 * @see ex::normal() */
1913 ex mul::normal(exmap & repl, exmap & rev_lookup, int level) const
1916 return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1917 else if (level == -max_recursion_level)
1918 throw(std::runtime_error("max recursion level reached"));
1920 // Normalize children, separate into numerator and denominator
1921 exvector num; num.reserve(seq.size());
1922 exvector den; den.reserve(seq.size());
1924 epvector::const_iterator it = seq.begin(), itend = seq.end();
1925 while (it != itend) {
1926 n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(repl, rev_lookup, level-1);
1927 num.push_back(n.op(0));
1928 den.push_back(n.op(1));
1931 n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, level-1);
1932 num.push_back(n.op(0));
1933 den.push_back(n.op(1));
1935 // Perform fraction cancellation
1936 return frac_cancel((new mul(num))->setflag(status_flags::dynallocated),
1937 (new mul(den))->setflag(status_flags::dynallocated));
1941 /** Implementation of ex::normal([B) for powers. It normalizes the basis,
1942 * distributes integer exponents to numerator and denominator, and replaces
1943 * non-integer powers by temporary symbols.
1944 * @see ex::normal */
1945 ex power::normal(exmap & repl, exmap & rev_lookup, int level) const
1948 return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1949 else if (level == -max_recursion_level)
1950 throw(std::runtime_error("max recursion level reached"));
1952 // Normalize basis and exponent (exponent gets reassembled)
1953 ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup, level-1);
1954 ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup, level-1);
1955 n_exponent = n_exponent.op(0) / n_exponent.op(1);
1957 if (n_exponent.info(info_flags::integer)) {
1959 if (n_exponent.info(info_flags::positive)) {
1961 // (a/b)^n -> {a^n, b^n}
1962 return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated);
1964 } else if (n_exponent.info(info_flags::negative)) {
1966 // (a/b)^-n -> {b^n, a^n}
1967 return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated);
1972 if (n_exponent.info(info_flags::positive)) {
1974 // (a/b)^x -> {sym((a/b)^x), 1}
1975 return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1977 } else if (n_exponent.info(info_flags::negative)) {
1979 if (n_basis.op(1).is_equal(_ex1)) {
1981 // a^-x -> {1, sym(a^x)}
1982 return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), repl, rev_lookup)))->setflag(status_flags::dynallocated);
1986 // (a/b)^-x -> {sym((b/a)^x), 1}
1987 return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1992 // (a/b)^x -> {sym((a/b)^x, 1}
1993 return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
1997 /** Implementation of ex::normal() for pseries. It normalizes each coefficient
1998 * and replaces the series by a temporary symbol.
1999 * @see ex::normal */
2000 ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const
2003 epvector::const_iterator i = seq.begin(), end = seq.end();
2005 ex restexp = i->rest.normal();
2006 if (!restexp.is_zero())
2007 newseq.push_back(expair(restexp, i->coeff));
2010 ex n = pseries(relational(var,point), newseq);
2011 return (new lst(replace_with_symbol(n, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
2015 /** Normalization of rational functions.
2016 * This function converts an expression to its normal form
2017 * "numerator/denominator", where numerator and denominator are (relatively
2018 * prime) polynomials. Any subexpressions which are not rational functions
2019 * (like non-rational numbers, non-integer powers or functions like sin(),
2020 * cos() etc.) are replaced by temporary symbols which are re-substituted by
2021 * the (normalized) subexpressions before normal() returns (this way, any
2022 * expression can be treated as a rational function). normal() is applied
2023 * recursively to arguments of functions etc.
2025 * @param level maximum depth of recursion
2026 * @return normalized expression */
2027 ex ex::normal(int level) const
2029 exmap repl, rev_lookup;
2031 ex e = bp->normal(repl, rev_lookup, level);
2032 GINAC_ASSERT(is_a<lst>(e));
2034 // Re-insert replaced symbols
2036 e = e.subs(repl, subs_options::no_pattern);
2038 // Convert {numerator, denominator} form back to fraction
2039 return e.op(0) / e.op(1);
2042 /** Get numerator of an expression. If the expression is not of the normal
2043 * form "numerator/denominator", it is first converted to this form and
2044 * then the numerator is returned.
2047 * @return numerator */
2048 ex ex::numer() const
2050 exmap repl, rev_lookup;
2052 ex e = bp->normal(repl, rev_lookup, 0);
2053 GINAC_ASSERT(is_a<lst>(e));
2055 // Re-insert replaced symbols
2059 return e.op(0).subs(repl, subs_options::no_pattern);
2062 /** Get denominator of an expression. If the expression is not of the normal
2063 * form "numerator/denominator", it is first converted to this form and
2064 * then the denominator is returned.
2067 * @return denominator */
2068 ex ex::denom() const
2070 exmap repl, rev_lookup;
2072 ex e = bp->normal(repl, rev_lookup, 0);
2073 GINAC_ASSERT(is_a<lst>(e));
2075 // Re-insert replaced symbols
2079 return e.op(1).subs(repl, subs_options::no_pattern);
2082 /** Get numerator and denominator of an expression. If the expresison is not
2083 * of the normal form "numerator/denominator", it is first converted to this
2084 * form and then a list [numerator, denominator] is returned.
2087 * @return a list [numerator, denominator] */
2088 ex ex::numer_denom() const
2090 exmap repl, rev_lookup;
2092 ex e = bp->normal(repl, rev_lookup, 0);
2093 GINAC_ASSERT(is_a<lst>(e));
2095 // Re-insert replaced symbols
2099 return e.subs(repl, subs_options::no_pattern);
2103 /** Rationalization of non-rational functions.
2104 * This function converts a general expression to a rational function
2105 * by replacing all non-rational subexpressions (like non-rational numbers,
2106 * non-integer powers or functions like sin(), cos() etc.) to temporary
2107 * symbols. This makes it possible to use functions like gcd() and divide()
2108 * on non-rational functions by applying to_rational() on the arguments,
2109 * calling the desired function and re-substituting the temporary symbols
2110 * in the result. To make the last step possible, all temporary symbols and
2111 * their associated expressions are collected in the map specified by the
2112 * repl parameter, ready to be passed as an argument to ex::subs().
2114 * @param repl collects all temporary symbols and their replacements
2115 * @return rationalized expression */
2116 ex ex::to_rational(exmap & repl) const
2118 return bp->to_rational(repl);
2121 // GiNaC 1.1 compatibility function
2122 ex ex::to_rational(lst & repl_lst) const
2124 // Convert lst to exmap
2126 for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it)
2127 m.insert(std::make_pair(it->op(0), it->op(1)));
2129 ex ret = bp->to_rational(m);
2131 // Convert exmap back to lst
2132 repl_lst.remove_all();
2133 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it)
2134 repl_lst.append(it->first == it->second);
2139 ex ex::to_polynomial(exmap & repl) const
2141 return bp->to_polynomial(repl);
2144 // GiNaC 1.1 compatibility function
2145 ex ex::to_polynomial(lst & repl_lst) const
2147 // Convert lst to exmap
2149 for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it)
2150 m.insert(std::make_pair(it->op(0), it->op(1)));
2152 ex ret = bp->to_polynomial(m);
2154 // Convert exmap back to lst
2155 repl_lst.remove_all();
2156 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it)
2157 repl_lst.append(it->first == it->second);
2162 /** Default implementation of ex::to_rational(). This replaces the object with
2163 * a temporary symbol. */
2164 ex basic::to_rational(exmap & repl) const
2166 return replace_with_symbol(*this, repl);
2169 ex basic::to_polynomial(exmap & repl) const
2171 return replace_with_symbol(*this, repl);
2175 /** Implementation of ex::to_rational() for symbols. This returns the
2176 * unmodified symbol. */
2177 ex symbol::to_rational(exmap & repl) const
2182 /** Implementation of ex::to_polynomial() for symbols. This returns the
2183 * unmodified symbol. */
2184 ex symbol::to_polynomial(exmap & repl) const
2190 /** Implementation of ex::to_rational() for a numeric. It splits complex
2191 * numbers into re+I*im and replaces I and non-rational real numbers with a
2192 * temporary symbol. */
2193 ex numeric::to_rational(exmap & repl) const
2197 return replace_with_symbol(*this, repl);
2199 numeric re = real();
2200 numeric im = imag();
2201 ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl);
2202 ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl);
2203 return re_ex + im_ex * replace_with_symbol(I, repl);
2208 /** Implementation of ex::to_polynomial() for a numeric. It splits complex
2209 * numbers into re+I*im and replaces I and non-integer real numbers with a
2210 * temporary symbol. */
2211 ex numeric::to_polynomial(exmap & repl) const
2215 return replace_with_symbol(*this, repl);
2217 numeric re = real();
2218 numeric im = imag();
2219 ex re_ex = re.is_integer() ? re : replace_with_symbol(re, repl);
2220 ex im_ex = im.is_integer() ? im : replace_with_symbol(im, repl);
2221 return re_ex + im_ex * replace_with_symbol(I, repl);
2227 /** Implementation of ex::to_rational() for powers. It replaces non-integer
2228 * powers by temporary symbols. */
2229 ex power::to_rational(exmap & repl) const
2231 if (exponent.info(info_flags::integer))
2232 return power(basis.to_rational(repl), exponent);
2234 return replace_with_symbol(*this, repl);
2237 /** Implementation of ex::to_polynomial() for powers. It replaces non-posint
2238 * powers by temporary symbols. */
2239 ex power::to_polynomial(exmap & repl) const
2241 if (exponent.info(info_flags::posint))
2242 return power(basis.to_rational(repl), exponent);
2244 return replace_with_symbol(*this, repl);
2248 /** Implementation of ex::to_rational() for expairseqs. */
2249 ex expairseq::to_rational(exmap & repl) const
2252 s.reserve(seq.size());
2253 epvector::const_iterator i = seq.begin(), end = seq.end();
2255 s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl)));
2258 ex oc = overall_coeff.to_rational(repl);
2259 if (oc.info(info_flags::numeric))
2260 return thisexpairseq(s, overall_coeff);
2262 s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1));
2263 return thisexpairseq(s, default_overall_coeff());
2266 /** Implementation of ex::to_polynomial() for expairseqs. */
2267 ex expairseq::to_polynomial(exmap & repl) const
2270 s.reserve(seq.size());
2271 epvector::const_iterator i = seq.begin(), end = seq.end();
2273 s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_polynomial(repl)));
2276 ex oc = overall_coeff.to_polynomial(repl);
2277 if (oc.info(info_flags::numeric))
2278 return thisexpairseq(s, overall_coeff);
2280 s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1));
2281 return thisexpairseq(s, default_overall_coeff());
2285 /** Remove the common factor in the terms of a sum 'e' by calculating the GCD,
2286 * and multiply it into the expression 'factor' (which needs to be initialized
2287 * to 1, unless you're accumulating factors). */
2288 static ex find_common_factor(const ex & e, ex & factor, exmap & repl)
2290 if (is_exactly_a<add>(e)) {
2292 size_t num = e.nops();
2293 exvector terms; terms.reserve(num);
2296 // Find the common GCD
2297 for (size_t i=0; i<num; i++) {
2298 ex x = e.op(i).to_polynomial(repl);
2300 if (is_exactly_a<add>(x) || is_exactly_a<mul>(x)) {
2302 x = find_common_factor(x, f, repl);
2314 if (gc.is_equal(_ex1))
2317 // The GCD is the factor we pull out
2320 // Now divide all terms by the GCD
2321 for (size_t i=0; i<num; i++) {
2324 // Try to avoid divide() because it expands the polynomial
2326 if (is_exactly_a<mul>(t)) {
2327 for (size_t j=0; j<t.nops(); j++) {
2328 if (t.op(j).is_equal(gc)) {
2329 exvector v; v.reserve(t.nops());
2330 for (size_t k=0; k<t.nops(); k++) {
2334 v.push_back(t.op(k));
2336 t = (new mul(v))->setflag(status_flags::dynallocated);
2346 return (new add(terms))->setflag(status_flags::dynallocated);
2348 } else if (is_exactly_a<mul>(e)) {
2350 size_t num = e.nops();
2351 exvector v; v.reserve(num);
2353 for (size_t i=0; i<num; i++)
2354 v.push_back(find_common_factor(e.op(i), factor, repl));
2356 return (new mul(v))->setflag(status_flags::dynallocated);
2358 } else if (is_exactly_a<power>(e)) {
2360 return e.to_polynomial(repl);
2367 /** Collect common factors in sums. This converts expressions like
2368 * 'a*(b*x+b*y)' to 'a*b*(x+y)'. */
2369 ex collect_common_factors(const ex & e)
2371 if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
2375 ex r = find_common_factor(e, factor, repl);
2376 return factor.subs(repl, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern);
2383 /** Resultant of two expressions e1,e2 with respect to symbol s.
2384 * Method: Compute determinant of Sylvester matrix of e1,e2,s. */
2385 ex resultant(const ex & e1, const ex & e2, const ex & s)
2387 const ex ee1 = e1.expand();
2388 const ex ee2 = e2.expand();
2389 if (!ee1.info(info_flags::polynomial) ||
2390 !ee2.info(info_flags::polynomial))
2391 throw(std::runtime_error("resultant(): arguments must be polynomials"));
2393 const int h1 = ee1.degree(s);
2394 const int l1 = ee1.ldegree(s);
2395 const int h2 = ee2.degree(s);
2396 const int l2 = ee2.ldegree(s);
2398 const int msize = h1 + h2;
2399 matrix m(msize, msize);
2401 for (int l = h1; l >= l1; --l) {
2402 const ex e = ee1.coeff(s, l);
2403 for (int k = 0; k < h2; ++k)
2406 for (int l = h2; l >= l2; --l) {
2407 const ex e = ee2.coeff(s, l);
2408 for (int k = 0; k < h1; ++k)
2409 m(k+h2, k+h2-l) = e;
2412 return m.determinant();
2416 } // namespace GiNaC