3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
33 #include "operators.h"
34 #include "relational.h"
46 static ex conjugate_evalf(const ex & arg)
48 if (is_exactly_a<numeric>(arg)) {
49 return ex_to<numeric>(arg).conjugate();
51 return conjugate_function(arg).hold();
54 static ex conjugate_eval(const ex & arg)
56 return arg.conjugate();
59 static void conjugate_print_latex(const ex & arg, const print_context & c)
61 c.s << "\\bar{"; arg.print(c); c.s << "}";
64 static ex conjugate_conjugate(const ex & arg)
69 static ex conjugate_real_part(const ex & arg)
71 return arg.real_part();
74 static ex conjugate_imag_part(const ex & arg)
76 return -arg.imag_part();
79 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
80 evalf_func(conjugate_evalf).
81 print_func<print_latex>(conjugate_print_latex).
82 conjugate_func(conjugate_conjugate).
83 real_part_func(conjugate_real_part).
84 imag_part_func(conjugate_imag_part).
85 set_name("conjugate","conjugate"));
91 static ex real_part_evalf(const ex & arg)
93 if (is_exactly_a<numeric>(arg)) {
94 return ex_to<numeric>(arg).real();
96 return real_part_function(arg).hold();
99 static ex real_part_eval(const ex & arg)
101 return arg.real_part();
104 static void real_part_print_latex(const ex & arg, const print_context & c)
106 c.s << "\\Re"; arg.print(c); c.s << "";
109 static ex real_part_conjugate(const ex & arg)
111 return real_part_function(arg).hold();
114 static ex real_part_real_part(const ex & arg)
116 return real_part_function(arg).hold();
119 static ex real_part_imag_part(const ex & arg)
124 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
125 evalf_func(real_part_evalf).
126 print_func<print_latex>(real_part_print_latex).
127 conjugate_func(real_part_conjugate).
128 real_part_func(real_part_real_part).
129 imag_part_func(real_part_imag_part).
130 set_name("real_part","real_part"));
136 static ex imag_part_evalf(const ex & arg)
138 if (is_exactly_a<numeric>(arg)) {
139 return ex_to<numeric>(arg).imag();
141 return imag_part_function(arg).hold();
144 static ex imag_part_eval(const ex & arg)
146 return arg.imag_part();
149 static void imag_part_print_latex(const ex & arg, const print_context & c)
151 c.s << "\\Im"; arg.print(c); c.s << "";
154 static ex imag_part_conjugate(const ex & arg)
156 return imag_part_function(arg).hold();
159 static ex imag_part_real_part(const ex & arg)
161 return imag_part_function(arg).hold();
164 static ex imag_part_imag_part(const ex & arg)
169 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
170 evalf_func(imag_part_evalf).
171 print_func<print_latex>(imag_part_print_latex).
172 conjugate_func(imag_part_conjugate).
173 real_part_func(imag_part_real_part).
174 imag_part_func(imag_part_imag_part).
175 set_name("imag_part","imag_part"));
181 static ex abs_evalf(const ex & arg)
183 if (is_exactly_a<numeric>(arg))
184 return abs(ex_to<numeric>(arg));
186 return abs(arg).hold();
189 static ex abs_eval(const ex & arg)
191 if (is_exactly_a<numeric>(arg))
192 return abs(ex_to<numeric>(arg));
194 return abs(arg).hold();
197 static void abs_print_latex(const ex & arg, const print_context & c)
199 c.s << "{|"; arg.print(c); c.s << "|}";
202 static void abs_print_csrc_float(const ex & arg, const print_context & c)
204 c.s << "fabs("; arg.print(c); c.s << ")";
207 static ex abs_conjugate(const ex & arg)
212 static ex abs_real_part(const ex & arg)
214 return abs(arg).hold();
217 static ex abs_imag_part(const ex& arg)
222 static ex abs_power(const ex & arg, const ex & exp)
224 if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
225 return power(arg, exp);
227 return power(abs(arg), exp).hold();
230 REGISTER_FUNCTION(abs, eval_func(abs_eval).
231 evalf_func(abs_evalf).
232 print_func<print_latex>(abs_print_latex).
233 print_func<print_csrc_float>(abs_print_csrc_float).
234 print_func<print_csrc_double>(abs_print_csrc_float).
235 conjugate_func(abs_conjugate).
236 real_part_func(abs_real_part).
237 imag_part_func(abs_imag_part).
238 power_func(abs_power));
244 static ex step_evalf(const ex & arg)
246 if (is_exactly_a<numeric>(arg))
247 return step(ex_to<numeric>(arg));
249 return step(arg).hold();
252 static ex step_eval(const ex & arg)
254 if (is_exactly_a<numeric>(arg))
255 return step(ex_to<numeric>(arg));
257 else if (is_exactly_a<mul>(arg) &&
258 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
259 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
262 // step(42*x) -> step(x)
263 return step(arg/oc).hold();
265 // step(-42*x) -> step(-x)
266 return step(-arg/oc).hold();
268 if (oc.real().is_zero()) {
270 // step(42*I*x) -> step(I*x)
271 return step(I*arg/oc).hold();
273 // step(-42*I*x) -> step(-I*x)
274 return step(-I*arg/oc).hold();
278 return step(arg).hold();
281 static ex step_series(const ex & arg,
282 const relational & rel,
286 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
287 if (arg_pt.info(info_flags::numeric)
288 && ex_to<numeric>(arg_pt).real().is_zero()
289 && !(options & series_options::suppress_branchcut))
290 throw (std::domain_error("step_series(): on imaginary axis"));
293 seq.push_back(expair(step(arg_pt), _ex0));
294 return pseries(rel,seq);
297 static ex step_conjugate(const ex& arg)
299 return step(arg).hold();
302 static ex step_real_part(const ex& arg)
304 return step(arg).hold();
307 static ex step_imag_part(const ex& arg)
312 REGISTER_FUNCTION(step, eval_func(step_eval).
313 evalf_func(step_evalf).
314 series_func(step_series).
315 conjugate_func(step_conjugate).
316 real_part_func(step_real_part).
317 imag_part_func(step_imag_part));
323 static ex csgn_evalf(const ex & arg)
325 if (is_exactly_a<numeric>(arg))
326 return csgn(ex_to<numeric>(arg));
328 return csgn(arg).hold();
331 static ex csgn_eval(const ex & arg)
333 if (is_exactly_a<numeric>(arg))
334 return csgn(ex_to<numeric>(arg));
336 else if (is_exactly_a<mul>(arg) &&
337 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
338 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
341 // csgn(42*x) -> csgn(x)
342 return csgn(arg/oc).hold();
344 // csgn(-42*x) -> -csgn(x)
345 return -csgn(arg/oc).hold();
347 if (oc.real().is_zero()) {
349 // csgn(42*I*x) -> csgn(I*x)
350 return csgn(I*arg/oc).hold();
352 // csgn(-42*I*x) -> -csgn(I*x)
353 return -csgn(I*arg/oc).hold();
357 return csgn(arg).hold();
360 static ex csgn_series(const ex & arg,
361 const relational & rel,
365 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
366 if (arg_pt.info(info_flags::numeric)
367 && ex_to<numeric>(arg_pt).real().is_zero()
368 && !(options & series_options::suppress_branchcut))
369 throw (std::domain_error("csgn_series(): on imaginary axis"));
372 seq.push_back(expair(csgn(arg_pt), _ex0));
373 return pseries(rel,seq);
376 static ex csgn_conjugate(const ex& arg)
378 return csgn(arg).hold();
381 static ex csgn_real_part(const ex& arg)
383 return csgn(arg).hold();
386 static ex csgn_imag_part(const ex& arg)
391 static ex csgn_power(const ex & arg, const ex & exp)
393 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
394 if (ex_to<numeric>(exp).is_odd())
397 return power(csgn(arg), _ex2).hold();
399 return power(csgn(arg), exp).hold();
403 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
404 evalf_func(csgn_evalf).
405 series_func(csgn_series).
406 conjugate_func(csgn_conjugate).
407 real_part_func(csgn_real_part).
408 imag_part_func(csgn_imag_part).
409 power_func(csgn_power));
413 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
414 // This function is closely related to the unwinding number K, sometimes found
415 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
418 static ex eta_evalf(const ex &x, const ex &y)
420 // It seems like we basically have to replicate the eval function here,
421 // since the expression might not be fully evaluated yet.
422 if (x.info(info_flags::positive) || y.info(info_flags::positive))
425 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
426 const numeric nx = ex_to<numeric>(x);
427 const numeric ny = ex_to<numeric>(y);
428 const numeric nxy = ex_to<numeric>(x*y);
430 if (nx.is_real() && nx.is_negative())
432 if (ny.is_real() && ny.is_negative())
434 if (nxy.is_real() && nxy.is_negative())
436 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
437 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
440 return eta(x,y).hold();
443 static ex eta_eval(const ex &x, const ex &y)
445 // trivial: eta(x,c) -> 0 if c is real and positive
446 if (x.info(info_flags::positive) || y.info(info_flags::positive))
449 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
450 // don't call eta_evalf here because it would call Pi.evalf()!
451 const numeric nx = ex_to<numeric>(x);
452 const numeric ny = ex_to<numeric>(y);
453 const numeric nxy = ex_to<numeric>(x*y);
455 if (nx.is_real() && nx.is_negative())
457 if (ny.is_real() && ny.is_negative())
459 if (nxy.is_real() && nxy.is_negative())
461 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
462 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
465 return eta(x,y).hold();
468 static ex eta_series(const ex & x, const ex & y,
469 const relational & rel,
473 const ex x_pt = x.subs(rel, subs_options::no_pattern);
474 const ex y_pt = y.subs(rel, subs_options::no_pattern);
475 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
476 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
477 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
478 throw (std::domain_error("eta_series(): on discontinuity"));
480 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
481 return pseries(rel,seq);
484 static ex eta_conjugate(const ex & x, const ex & y)
489 static ex eta_real_part(const ex & x, const ex & y)
494 static ex eta_imag_part(const ex & x, const ex & y)
496 return -I*eta(x, y).hold();
499 REGISTER_FUNCTION(eta, eval_func(eta_eval).
500 evalf_func(eta_evalf).
501 series_func(eta_series).
503 set_symmetry(sy_symm(0, 1)).
504 conjugate_func(eta_conjugate).
505 real_part_func(eta_real_part).
506 imag_part_func(eta_imag_part));
513 static ex Li2_evalf(const ex & x)
515 if (is_exactly_a<numeric>(x))
516 return Li2(ex_to<numeric>(x));
518 return Li2(x).hold();
521 static ex Li2_eval(const ex & x)
523 if (x.info(info_flags::numeric)) {
528 if (x.is_equal(_ex1))
529 return power(Pi,_ex2)/_ex6;
530 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
531 if (x.is_equal(_ex1_2))
532 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
533 // Li2(-1) -> -Pi^2/12
534 if (x.is_equal(_ex_1))
535 return -power(Pi,_ex2)/_ex12;
536 // Li2(I) -> -Pi^2/48+Catalan*I
538 return power(Pi,_ex2)/_ex_48 + Catalan*I;
539 // Li2(-I) -> -Pi^2/48-Catalan*I
541 return power(Pi,_ex2)/_ex_48 - Catalan*I;
543 if (!x.info(info_flags::crational))
544 return Li2(ex_to<numeric>(x));
547 return Li2(x).hold();
550 static ex Li2_deriv(const ex & x, unsigned deriv_param)
552 GINAC_ASSERT(deriv_param==0);
554 // d/dx Li2(x) -> -log(1-x)/x
555 return -log(_ex1-x)/x;
558 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
560 const ex x_pt = x.subs(rel, subs_options::no_pattern);
561 if (x_pt.info(info_flags::numeric)) {
562 // First special case: x==0 (derivatives have poles)
563 if (x_pt.is_zero()) {
565 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
566 // simply substitute x==0. The limit, however, exists: it is 1.
567 // We also know all higher derivatives' limits:
568 // (d/dx)^n Li2(x) == n!/n^2.
569 // So the primitive series expansion is
570 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
572 // We first construct such a primitive series expansion manually in
573 // a dummy symbol s and then insert the argument's series expansion
574 // for s. Reexpanding the resulting series returns the desired
578 // manually construct the primitive expansion
579 for (int i=1; i<order; ++i)
580 ser += pow(s,i) / pow(numeric(i), *_num2_p);
581 // substitute the argument's series expansion
582 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
583 // maybe that was terminating, so add a proper order term
585 nseq.push_back(expair(Order(_ex1), order));
586 ser += pseries(rel, nseq);
587 // reexpanding it will collapse the series again
588 return ser.series(rel, order);
589 // NB: Of course, this still does not allow us to compute anything
590 // like sin(Li2(x)).series(x==0,2), since then this code here is
591 // not reached and the derivative of sin(Li2(x)) doesn't allow the
592 // substitution x==0. Probably limits *are* needed for the general
593 // cases. In case L'Hospital's rule is implemented for limits and
594 // basic::series() takes care of this, this whole block is probably
597 // second special case: x==1 (branch point)
598 if (x_pt.is_equal(_ex1)) {
600 // construct series manually in a dummy symbol s
603 // manually construct the primitive expansion
604 for (int i=1; i<order; ++i)
605 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
606 // substitute the argument's series expansion
607 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
608 // maybe that was terminating, so add a proper order term
610 nseq.push_back(expair(Order(_ex1), order));
611 ser += pseries(rel, nseq);
612 // reexpanding it will collapse the series again
613 return ser.series(rel, order);
615 // third special case: x real, >=1 (branch cut)
616 if (!(options & series_options::suppress_branchcut) &&
617 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
619 // This is the branch cut: assemble the primitive series manually
620 // and then add the corresponding complex step function.
621 const symbol &s = ex_to<symbol>(rel.lhs());
622 const ex point = rel.rhs();
625 // zeroth order term:
626 seq.push_back(expair(Li2(x_pt), _ex0));
627 // compute the intermediate terms:
628 ex replarg = series(Li2(x), s==foo, order);
629 for (size_t i=1; i<replarg.nops()-1; ++i)
630 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
631 // append an order term:
632 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
633 return pseries(rel, seq);
636 // all other cases should be safe, by now:
637 throw do_taylor(); // caught by function::series()
640 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
641 evalf_func(Li2_evalf).
642 derivative_func(Li2_deriv).
643 series_func(Li2_series).
644 latex_name("\\mbox{Li}_2"));
650 static ex Li3_eval(const ex & x)
654 return Li3(x).hold();
657 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
658 latex_name("\\mbox{Li}_3"));
661 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
664 static ex zetaderiv_eval(const ex & n, const ex & x)
666 if (n.info(info_flags::numeric)) {
667 // zetaderiv(0,x) -> zeta(x)
672 return zetaderiv(n, x).hold();
675 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
677 GINAC_ASSERT(deriv_param<2);
679 if (deriv_param==0) {
681 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
684 return zetaderiv(n+1,x);
687 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
688 derivative_func(zetaderiv_deriv).
689 latex_name("\\zeta^\\prime"));
695 static ex factorial_evalf(const ex & x)
697 return factorial(x).hold();
700 static ex factorial_eval(const ex & x)
702 if (is_exactly_a<numeric>(x))
703 return factorial(ex_to<numeric>(x));
705 return factorial(x).hold();
708 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
710 if (is_exactly_a<symbol>(x) ||
711 is_exactly_a<constant>(x) ||
712 is_exactly_a<function>(x)) {
713 x.print(c); c.s << "!";
715 c.s << "("; x.print(c); c.s << ")!";
719 static ex factorial_conjugate(const ex & x)
721 return factorial(x).hold();
724 static ex factorial_real_part(const ex & x)
726 return factorial(x).hold();
729 static ex factorial_imag_part(const ex & x)
734 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
735 evalf_func(factorial_evalf).
736 print_func<print_dflt>(factorial_print_dflt_latex).
737 print_func<print_latex>(factorial_print_dflt_latex).
738 conjugate_func(factorial_conjugate).
739 real_part_func(factorial_real_part).
740 imag_part_func(factorial_imag_part));
746 static ex binomial_evalf(const ex & x, const ex & y)
748 return binomial(x, y).hold();
751 static ex binomial_sym(const ex & x, const numeric & y)
753 if (y.is_integer()) {
754 if (y.is_nonneg_integer()) {
755 const unsigned N = y.to_int();
756 if (N == 0) return _ex0;
757 if (N == 1) return x;
759 for (unsigned i = 2; i <= N; ++i)
760 t = (t * (x + i - y - 1)).expand() / i;
766 return binomial(x, y).hold();
769 static ex binomial_eval(const ex & x, const ex &y)
771 if (is_exactly_a<numeric>(y)) {
772 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
773 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
775 return binomial_sym(x, ex_to<numeric>(y));
777 return binomial(x, y).hold();
780 // At the moment the numeric evaluation of a binomail function always
781 // gives a real number, but if this would be implemented using the gamma
782 // function, also complex conjugation should be changed (or rather, deleted).
783 static ex binomial_conjugate(const ex & x, const ex & y)
785 return binomial(x,y).hold();
788 static ex binomial_real_part(const ex & x, const ex & y)
790 return binomial(x,y).hold();
793 static ex binomial_imag_part(const ex & x, const ex & y)
798 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
799 evalf_func(binomial_evalf).
800 conjugate_func(binomial_conjugate).
801 real_part_func(binomial_real_part).
802 imag_part_func(binomial_imag_part));
805 // Order term function (for truncated power series)
808 static ex Order_eval(const ex & x)
810 if (is_exactly_a<numeric>(x)) {
813 return Order(_ex1).hold();
816 } else if (is_exactly_a<mul>(x)) {
817 const mul &m = ex_to<mul>(x);
818 // O(c*expr) -> O(expr)
819 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
820 return Order(x / m.op(m.nops() - 1)).hold();
822 return Order(x).hold();
825 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
827 // Just wrap the function into a pseries object
829 GINAC_ASSERT(is_a<symbol>(r.lhs()));
830 const symbol &s = ex_to<symbol>(r.lhs());
831 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
832 return pseries(r, new_seq);
835 static ex Order_conjugate(const ex & x)
837 return Order(x).hold();
840 static ex Order_real_part(const ex & x)
842 return Order(x).hold();
845 static ex Order_imag_part(const ex & x)
847 if(x.info(info_flags::real))
849 return Order(x).hold();
852 // Differentiation is handled in function::derivative because of its special requirements
854 REGISTER_FUNCTION(Order, eval_func(Order_eval).
855 series_func(Order_series).
856 latex_name("\\mathcal{O}").
857 conjugate_func(Order_conjugate).
858 real_part_func(Order_real_part).
859 imag_part_func(Order_imag_part));
862 // Solve linear system
865 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
867 // solve a system of linear equations
868 if (eqns.info(info_flags::relation_equal)) {
869 if (!symbols.info(info_flags::symbol))
870 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
871 const ex sol = lsolve(lst(eqns),lst(symbols));
873 GINAC_ASSERT(sol.nops()==1);
874 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
876 return sol.op(0).op(1); // return rhs of first solution
880 if (!eqns.info(info_flags::list)) {
881 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
883 for (size_t i=0; i<eqns.nops(); i++) {
884 if (!eqns.op(i).info(info_flags::relation_equal)) {
885 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
888 if (!symbols.info(info_flags::list)) {
889 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
891 for (size_t i=0; i<symbols.nops(); i++) {
892 if (!symbols.op(i).info(info_flags::symbol)) {
893 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
897 // build matrix from equation system
898 matrix sys(eqns.nops(),symbols.nops());
899 matrix rhs(eqns.nops(),1);
900 matrix vars(symbols.nops(),1);
902 for (size_t r=0; r<eqns.nops(); r++) {
903 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
905 for (size_t c=0; c<symbols.nops(); c++) {
906 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
907 linpart -= co*symbols.op(c);
910 linpart = linpart.expand();
914 // test if system is linear and fill vars matrix
915 for (size_t i=0; i<symbols.nops(); i++) {
916 vars(i,0) = symbols.op(i);
917 if (sys.has(symbols.op(i)))
918 throw(std::logic_error("lsolve: system is not linear"));
919 if (rhs.has(symbols.op(i)))
920 throw(std::logic_error("lsolve: system is not linear"));
925 solution = sys.solve(vars,rhs,options);
926 } catch (const std::runtime_error & e) {
927 // Probably singular matrix or otherwise overdetermined system:
928 // It is consistent to return an empty list
931 GINAC_ASSERT(solution.cols()==1);
932 GINAC_ASSERT(solution.rows()==symbols.nops());
934 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
936 for (size_t i=0; i<symbols.nops(); i++)
937 sollist.append(symbols.op(i)==solution(i,0));
943 // Find real root of f(x) numerically
947 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
949 if (!x1.is_real() || !x2.is_real()) {
950 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
953 throw std::runtime_error("fsolve(): vanishing interval");
955 // xx[0] == left interval limit, xx[1] == right interval limit.
956 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
957 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
958 numeric xx[2] = { x1<x2 ? x1 : x2,
961 if (is_a<relational>(f_in)) {
962 f = f_in.lhs()-f_in.rhs();
966 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
967 f.subs(x==xx[1]).evalf() };
968 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
969 throw std::runtime_error("fsolve(): function does not evaluate numerically");
971 numeric fx[2] = { ex_to<numeric>(fx_[0]),
972 ex_to<numeric>(fx_[1]) };
973 if (!fx[0].is_real() || !fx[1].is_real()) {
974 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
976 if (fx[0]*fx[1]>=0) {
977 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
980 // The Newton-Raphson method has quadratic convergence! Simply put, it
981 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
982 const ex ff = normal(-f/f.diff(x));
983 int side = 0; // Start at left interval limit.
989 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
990 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
991 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
992 // Oops, Newton-Raphson method shot out of the interval.
993 // Restore, and try again with the other side instead!
999 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
1000 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
1002 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1003 // Oops, the root isn't bracketed any more.
1004 // Restore, and perform a bisection!
1008 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1009 // they occur pretty often when Newton-Raphson arrives at an x too
1010 // close to the result on one side of the interval and
1011 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1012 // precision errors! Recall that this function does not have a
1013 // precision goal as one of its arguments but instead relies on
1014 // x converging to a fixed point. We speed up the (safe but slow)
1015 // bisection method by mixing in a dash of the (unsafer but faster)
1016 // secant method: Instead of splitting the interval at the
1017 // arithmetic mean (bisection), we split it nearer to the root as
1018 // determined by the secant between the values xx[0] and xx[1].
1019 // Don't set the secant_weight to one because that could disturb
1020 // the convergence in some corner cases!
1021 static const double secant_weight = 0.984375; // == 63/64 < 1
1022 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1023 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1024 numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
1025 if (fxmid.is_zero()) {
1029 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1037 } while (xxprev!=xx[side]);
1042 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1043 * for static lib (so ginsh will see them). */
1044 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1045 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1047 } // namespace GiNaC