3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2011 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
30 #include "operators.h"
31 #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 if (arg.info(info_flags::nonnegative))
197 if (is_ex_the_function(arg, abs))
200 return abs(arg).hold();
203 static void abs_print_latex(const ex & arg, const print_context & c)
205 c.s << "{|"; arg.print(c); c.s << "|}";
208 static void abs_print_csrc_float(const ex & arg, const print_context & c)
210 c.s << "fabs("; arg.print(c); c.s << ")";
213 static ex abs_conjugate(const ex & arg)
215 return abs(arg).hold();
218 static ex abs_real_part(const ex & arg)
220 return abs(arg).hold();
223 static ex abs_imag_part(const ex& arg)
228 static ex abs_power(const ex & arg, const ex & exp)
230 if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
231 || exp.info(info_flags::even)))
232 return power(arg, exp);
234 return power(abs(arg), exp).hold();
237 REGISTER_FUNCTION(abs, eval_func(abs_eval).
238 evalf_func(abs_evalf).
239 print_func<print_latex>(abs_print_latex).
240 print_func<print_csrc_float>(abs_print_csrc_float).
241 print_func<print_csrc_double>(abs_print_csrc_float).
242 conjugate_func(abs_conjugate).
243 real_part_func(abs_real_part).
244 imag_part_func(abs_imag_part).
245 power_func(abs_power));
251 static ex step_evalf(const ex & arg)
253 if (is_exactly_a<numeric>(arg))
254 return step(ex_to<numeric>(arg));
256 return step(arg).hold();
259 static ex step_eval(const ex & arg)
261 if (is_exactly_a<numeric>(arg))
262 return step(ex_to<numeric>(arg));
264 else if (is_exactly_a<mul>(arg) &&
265 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
266 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
269 // step(42*x) -> step(x)
270 return step(arg/oc).hold();
272 // step(-42*x) -> step(-x)
273 return step(-arg/oc).hold();
275 if (oc.real().is_zero()) {
277 // step(42*I*x) -> step(I*x)
278 return step(I*arg/oc).hold();
280 // step(-42*I*x) -> step(-I*x)
281 return step(-I*arg/oc).hold();
285 return step(arg).hold();
288 static ex step_series(const ex & arg,
289 const relational & rel,
293 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
294 if (arg_pt.info(info_flags::numeric)
295 && ex_to<numeric>(arg_pt).real().is_zero()
296 && !(options & series_options::suppress_branchcut))
297 throw (std::domain_error("step_series(): on imaginary axis"));
300 seq.push_back(expair(step(arg_pt), _ex0));
301 return pseries(rel,seq);
304 static ex step_conjugate(const ex& arg)
306 return step(arg).hold();
309 static ex step_real_part(const ex& arg)
311 return step(arg).hold();
314 static ex step_imag_part(const ex& arg)
319 REGISTER_FUNCTION(step, eval_func(step_eval).
320 evalf_func(step_evalf).
321 series_func(step_series).
322 conjugate_func(step_conjugate).
323 real_part_func(step_real_part).
324 imag_part_func(step_imag_part));
330 static ex csgn_evalf(const ex & arg)
332 if (is_exactly_a<numeric>(arg))
333 return csgn(ex_to<numeric>(arg));
335 return csgn(arg).hold();
338 static ex csgn_eval(const ex & arg)
340 if (is_exactly_a<numeric>(arg))
341 return csgn(ex_to<numeric>(arg));
343 else if (is_exactly_a<mul>(arg) &&
344 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
345 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
348 // csgn(42*x) -> csgn(x)
349 return csgn(arg/oc).hold();
351 // csgn(-42*x) -> -csgn(x)
352 return -csgn(arg/oc).hold();
354 if (oc.real().is_zero()) {
356 // csgn(42*I*x) -> csgn(I*x)
357 return csgn(I*arg/oc).hold();
359 // csgn(-42*I*x) -> -csgn(I*x)
360 return -csgn(I*arg/oc).hold();
364 return csgn(arg).hold();
367 static ex csgn_series(const ex & arg,
368 const relational & rel,
372 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
373 if (arg_pt.info(info_flags::numeric)
374 && ex_to<numeric>(arg_pt).real().is_zero()
375 && !(options & series_options::suppress_branchcut))
376 throw (std::domain_error("csgn_series(): on imaginary axis"));
379 seq.push_back(expair(csgn(arg_pt), _ex0));
380 return pseries(rel,seq);
383 static ex csgn_conjugate(const ex& arg)
385 return csgn(arg).hold();
388 static ex csgn_real_part(const ex& arg)
390 return csgn(arg).hold();
393 static ex csgn_imag_part(const ex& arg)
398 static ex csgn_power(const ex & arg, const ex & exp)
400 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
401 if (ex_to<numeric>(exp).is_odd())
402 return csgn(arg).hold();
404 return power(csgn(arg), _ex2).hold();
406 return power(csgn(arg), exp).hold();
410 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
411 evalf_func(csgn_evalf).
412 series_func(csgn_series).
413 conjugate_func(csgn_conjugate).
414 real_part_func(csgn_real_part).
415 imag_part_func(csgn_imag_part).
416 power_func(csgn_power));
420 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
421 // This function is closely related to the unwinding number K, sometimes found
422 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
425 static ex eta_evalf(const ex &x, const ex &y)
427 // It seems like we basically have to replicate the eval function here,
428 // since the expression might not be fully evaluated yet.
429 if (x.info(info_flags::positive) || y.info(info_flags::positive))
432 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
433 const numeric nx = ex_to<numeric>(x);
434 const numeric ny = ex_to<numeric>(y);
435 const numeric nxy = ex_to<numeric>(x*y);
437 if (nx.is_real() && nx.is_negative())
439 if (ny.is_real() && ny.is_negative())
441 if (nxy.is_real() && nxy.is_negative())
443 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
444 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
447 return eta(x,y).hold();
450 static ex eta_eval(const ex &x, const ex &y)
452 // trivial: eta(x,c) -> 0 if c is real and positive
453 if (x.info(info_flags::positive) || y.info(info_flags::positive))
456 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
457 // don't call eta_evalf here because it would call Pi.evalf()!
458 const numeric nx = ex_to<numeric>(x);
459 const numeric ny = ex_to<numeric>(y);
460 const numeric nxy = ex_to<numeric>(x*y);
462 if (nx.is_real() && nx.is_negative())
464 if (ny.is_real() && ny.is_negative())
466 if (nxy.is_real() && nxy.is_negative())
468 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
469 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
472 return eta(x,y).hold();
475 static ex eta_series(const ex & x, const ex & y,
476 const relational & rel,
480 const ex x_pt = x.subs(rel, subs_options::no_pattern);
481 const ex y_pt = y.subs(rel, subs_options::no_pattern);
482 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
483 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
484 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
485 throw (std::domain_error("eta_series(): on discontinuity"));
487 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
488 return pseries(rel,seq);
491 static ex eta_conjugate(const ex & x, const ex & y)
493 return -eta(x, y).hold();
496 static ex eta_real_part(const ex & x, const ex & y)
501 static ex eta_imag_part(const ex & x, const ex & y)
503 return -I*eta(x, y).hold();
506 REGISTER_FUNCTION(eta, eval_func(eta_eval).
507 evalf_func(eta_evalf).
508 series_func(eta_series).
510 set_symmetry(sy_symm(0, 1)).
511 conjugate_func(eta_conjugate).
512 real_part_func(eta_real_part).
513 imag_part_func(eta_imag_part));
520 static ex Li2_evalf(const ex & x)
522 if (is_exactly_a<numeric>(x))
523 return Li2(ex_to<numeric>(x));
525 return Li2(x).hold();
528 static ex Li2_eval(const ex & x)
530 if (x.info(info_flags::numeric)) {
535 if (x.is_equal(_ex1))
536 return power(Pi,_ex2)/_ex6;
537 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
538 if (x.is_equal(_ex1_2))
539 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
540 // Li2(-1) -> -Pi^2/12
541 if (x.is_equal(_ex_1))
542 return -power(Pi,_ex2)/_ex12;
543 // Li2(I) -> -Pi^2/48+Catalan*I
545 return power(Pi,_ex2)/_ex_48 + Catalan*I;
546 // Li2(-I) -> -Pi^2/48-Catalan*I
548 return power(Pi,_ex2)/_ex_48 - Catalan*I;
550 if (!x.info(info_flags::crational))
551 return Li2(ex_to<numeric>(x));
554 return Li2(x).hold();
557 static ex Li2_deriv(const ex & x, unsigned deriv_param)
559 GINAC_ASSERT(deriv_param==0);
561 // d/dx Li2(x) -> -log(1-x)/x
562 return -log(_ex1-x)/x;
565 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
567 const ex x_pt = x.subs(rel, subs_options::no_pattern);
568 if (x_pt.info(info_flags::numeric)) {
569 // First special case: x==0 (derivatives have poles)
570 if (x_pt.is_zero()) {
572 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
573 // simply substitute x==0. The limit, however, exists: it is 1.
574 // We also know all higher derivatives' limits:
575 // (d/dx)^n Li2(x) == n!/n^2.
576 // So the primitive series expansion is
577 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
579 // We first construct such a primitive series expansion manually in
580 // a dummy symbol s and then insert the argument's series expansion
581 // for s. Reexpanding the resulting series returns the desired
585 // manually construct the primitive expansion
586 for (int i=1; i<order; ++i)
587 ser += pow(s,i) / pow(numeric(i), *_num2_p);
588 // substitute the argument's series expansion
589 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
590 // maybe that was terminating, so add a proper order term
592 nseq.push_back(expair(Order(_ex1), order));
593 ser += pseries(rel, nseq);
594 // reexpanding it will collapse the series again
595 return ser.series(rel, order);
596 // NB: Of course, this still does not allow us to compute anything
597 // like sin(Li2(x)).series(x==0,2), since then this code here is
598 // not reached and the derivative of sin(Li2(x)) doesn't allow the
599 // substitution x==0. Probably limits *are* needed for the general
600 // cases. In case L'Hospital's rule is implemented for limits and
601 // basic::series() takes care of this, this whole block is probably
604 // second special case: x==1 (branch point)
605 if (x_pt.is_equal(_ex1)) {
607 // construct series manually in a dummy symbol s
610 // manually construct the primitive expansion
611 for (int i=1; i<order; ++i)
612 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
613 // substitute the argument's series expansion
614 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
615 // maybe that was terminating, so add a proper order term
617 nseq.push_back(expair(Order(_ex1), order));
618 ser += pseries(rel, nseq);
619 // reexpanding it will collapse the series again
620 return ser.series(rel, order);
622 // third special case: x real, >=1 (branch cut)
623 if (!(options & series_options::suppress_branchcut) &&
624 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
626 // This is the branch cut: assemble the primitive series manually
627 // and then add the corresponding complex step function.
628 const symbol &s = ex_to<symbol>(rel.lhs());
629 const ex point = rel.rhs();
632 // zeroth order term:
633 seq.push_back(expair(Li2(x_pt), _ex0));
634 // compute the intermediate terms:
635 ex replarg = series(Li2(x), s==foo, order);
636 for (size_t i=1; i<replarg.nops()-1; ++i)
637 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));
638 // append an order term:
639 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
640 return pseries(rel, seq);
643 // all other cases should be safe, by now:
644 throw do_taylor(); // caught by function::series()
647 static ex Li2_conjugate(const ex & x)
649 // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
650 // run along the positive real axis beginning at 1.
651 if (x.info(info_flags::negative)) {
652 return Li2(x).hold();
654 if (is_exactly_a<numeric>(x) &&
655 (!x.imag_part().is_zero() || x < *_num1_p)) {
656 return Li2(x.conjugate());
658 return conjugate_function(Li2(x)).hold();
661 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
662 evalf_func(Li2_evalf).
663 derivative_func(Li2_deriv).
664 series_func(Li2_series).
665 conjugate_func(Li2_conjugate).
666 latex_name("\\mathrm{Li}_2"));
672 static ex Li3_eval(const ex & x)
676 return Li3(x).hold();
679 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
680 latex_name("\\mathrm{Li}_3"));
683 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
686 static ex zetaderiv_eval(const ex & n, const ex & x)
688 if (n.info(info_flags::numeric)) {
689 // zetaderiv(0,x) -> zeta(x)
691 return zeta(x).hold();
694 return zetaderiv(n, x).hold();
697 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
699 GINAC_ASSERT(deriv_param<2);
701 if (deriv_param==0) {
703 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
706 return zetaderiv(n+1,x);
709 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
710 derivative_func(zetaderiv_deriv).
711 latex_name("\\zeta^\\prime"));
717 static ex factorial_evalf(const ex & x)
719 return factorial(x).hold();
722 static ex factorial_eval(const ex & x)
724 if (is_exactly_a<numeric>(x))
725 return factorial(ex_to<numeric>(x));
727 return factorial(x).hold();
730 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
732 if (is_exactly_a<symbol>(x) ||
733 is_exactly_a<constant>(x) ||
734 is_exactly_a<function>(x)) {
735 x.print(c); c.s << "!";
737 c.s << "("; x.print(c); c.s << ")!";
741 static ex factorial_conjugate(const ex & x)
743 return factorial(x).hold();
746 static ex factorial_real_part(const ex & x)
748 return factorial(x).hold();
751 static ex factorial_imag_part(const ex & x)
756 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
757 evalf_func(factorial_evalf).
758 print_func<print_dflt>(factorial_print_dflt_latex).
759 print_func<print_latex>(factorial_print_dflt_latex).
760 conjugate_func(factorial_conjugate).
761 real_part_func(factorial_real_part).
762 imag_part_func(factorial_imag_part));
768 static ex binomial_evalf(const ex & x, const ex & y)
770 return binomial(x, y).hold();
773 static ex binomial_sym(const ex & x, const numeric & y)
775 if (y.is_integer()) {
776 if (y.is_nonneg_integer()) {
777 const unsigned N = y.to_int();
778 if (N == 0) return _ex1;
779 if (N == 1) return x;
781 for (unsigned i = 2; i <= N; ++i)
782 t = (t * (x + i - y - 1)).expand() / i;
788 return binomial(x, y).hold();
791 static ex binomial_eval(const ex & x, const ex &y)
793 if (is_exactly_a<numeric>(y)) {
794 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
795 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
797 return binomial_sym(x, ex_to<numeric>(y));
799 return binomial(x, y).hold();
802 // At the moment the numeric evaluation of a binomail function always
803 // gives a real number, but if this would be implemented using the gamma
804 // function, also complex conjugation should be changed (or rather, deleted).
805 static ex binomial_conjugate(const ex & x, const ex & y)
807 return binomial(x,y).hold();
810 static ex binomial_real_part(const ex & x, const ex & y)
812 return binomial(x,y).hold();
815 static ex binomial_imag_part(const ex & x, const ex & y)
820 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
821 evalf_func(binomial_evalf).
822 conjugate_func(binomial_conjugate).
823 real_part_func(binomial_real_part).
824 imag_part_func(binomial_imag_part));
827 // Order term function (for truncated power series)
830 static ex Order_eval(const ex & x)
832 if (is_exactly_a<numeric>(x)) {
835 return Order(_ex1).hold();
838 } else if (is_exactly_a<mul>(x)) {
839 const mul &m = ex_to<mul>(x);
840 // O(c*expr) -> O(expr)
841 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
842 return Order(x / m.op(m.nops() - 1)).hold();
844 return Order(x).hold();
847 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
849 // Just wrap the function into a pseries object
851 GINAC_ASSERT(is_a<symbol>(r.lhs()));
852 const symbol &s = ex_to<symbol>(r.lhs());
853 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
854 return pseries(r, new_seq);
857 static ex Order_conjugate(const ex & x)
859 return Order(x).hold();
862 static ex Order_real_part(const ex & x)
864 return Order(x).hold();
867 static ex Order_imag_part(const ex & x)
869 if(x.info(info_flags::real))
871 return Order(x).hold();
874 // Differentiation is handled in function::derivative because of its special requirements
876 REGISTER_FUNCTION(Order, eval_func(Order_eval).
877 series_func(Order_series).
878 latex_name("\\mathcal{O}").
879 conjugate_func(Order_conjugate).
880 real_part_func(Order_real_part).
881 imag_part_func(Order_imag_part));
884 // Solve linear system
887 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
889 // solve a system of linear equations
890 if (eqns.info(info_flags::relation_equal)) {
891 if (!symbols.info(info_flags::symbol))
892 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
893 const ex sol = lsolve(lst(eqns),lst(symbols));
895 GINAC_ASSERT(sol.nops()==1);
896 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
898 return sol.op(0).op(1); // return rhs of first solution
902 if (!eqns.info(info_flags::list)) {
903 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
905 for (size_t i=0; i<eqns.nops(); i++) {
906 if (!eqns.op(i).info(info_flags::relation_equal)) {
907 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
910 if (!symbols.info(info_flags::list)) {
911 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
913 for (size_t i=0; i<symbols.nops(); i++) {
914 if (!symbols.op(i).info(info_flags::symbol)) {
915 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
919 // build matrix from equation system
920 matrix sys(eqns.nops(),symbols.nops());
921 matrix rhs(eqns.nops(),1);
922 matrix vars(symbols.nops(),1);
924 for (size_t r=0; r<eqns.nops(); r++) {
925 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
927 for (size_t c=0; c<symbols.nops(); c++) {
928 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
929 linpart -= co*symbols.op(c);
932 linpart = linpart.expand();
936 // test if system is linear and fill vars matrix
937 for (size_t i=0; i<symbols.nops(); i++) {
938 vars(i,0) = symbols.op(i);
939 if (sys.has(symbols.op(i)))
940 throw(std::logic_error("lsolve: system is not linear"));
941 if (rhs.has(symbols.op(i)))
942 throw(std::logic_error("lsolve: system is not linear"));
947 solution = sys.solve(vars,rhs,options);
948 } catch (const std::runtime_error & e) {
949 // Probably singular matrix or otherwise overdetermined system:
950 // It is consistent to return an empty list
953 GINAC_ASSERT(solution.cols()==1);
954 GINAC_ASSERT(solution.rows()==symbols.nops());
956 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
958 for (size_t i=0; i<symbols.nops(); i++)
959 sollist.append(symbols.op(i)==solution(i,0));
965 // Find real root of f(x) numerically
969 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
971 if (!x1.is_real() || !x2.is_real()) {
972 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
975 throw std::runtime_error("fsolve(): vanishing interval");
977 // xx[0] == left interval limit, xx[1] == right interval limit.
978 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
979 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
980 numeric xx[2] = { x1<x2 ? x1 : x2,
983 if (is_a<relational>(f_in)) {
984 f = f_in.lhs()-f_in.rhs();
988 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
989 f.subs(x==xx[1]).evalf() };
990 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
991 throw std::runtime_error("fsolve(): function does not evaluate numerically");
993 numeric fx[2] = { ex_to<numeric>(fx_[0]),
994 ex_to<numeric>(fx_[1]) };
995 if (!fx[0].is_real() || !fx[1].is_real()) {
996 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
998 if (fx[0]*fx[1]>=0) {
999 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1002 // The Newton-Raphson method has quadratic convergence! Simply put, it
1003 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
1004 const ex ff = normal(-f/f.diff(x));
1005 int side = 0; // Start at left interval limit.
1011 ex dx_ = ff.subs(x == xx[side]).evalf();
1012 if (!is_a<numeric>(dx_))
1013 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1014 xx[side] += ex_to<numeric>(dx_);
1015 // Now check if Newton-Raphson method shot out of the interval
1016 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1017 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1019 // Compute f(x) only if new x is inside the interval.
1020 // The function might be difficult to compute numerically
1021 // or even ill defined outside the interval. Also it's
1022 // a small optimization.
1023 ex f_x = f.subs(x == xx[side]).evalf();
1024 if (!is_a<numeric>(f_x))
1025 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1026 fx[side] = ex_to<numeric>(f_x);
1029 // Oops, Newton-Raphson method shot out of the interval.
1030 // Restore, and try again with the other side instead!
1037 ex dx_ = ff.subs(x == xx[side]).evalf();
1038 if (!is_a<numeric>(dx_))
1039 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1040 xx[side] += ex_to<numeric>(dx_);
1042 ex f_x = f.subs(x==xx[side]).evalf();
1043 if (!is_a<numeric>(f_x))
1044 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1045 fx[side] = ex_to<numeric>(f_x);
1047 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1048 // Oops, the root isn't bracketed any more.
1049 // Restore, and perform a bisection!
1053 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1054 // they occur pretty often when Newton-Raphson arrives at an x too
1055 // close to the result on one side of the interval and
1056 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1057 // precision errors! Recall that this function does not have a
1058 // precision goal as one of its arguments but instead relies on
1059 // x converging to a fixed point. We speed up the (safe but slow)
1060 // bisection method by mixing in a dash of the (unsafer but faster)
1061 // secant method: Instead of splitting the interval at the
1062 // arithmetic mean (bisection), we split it nearer to the root as
1063 // determined by the secant between the values xx[0] and xx[1].
1064 // Don't set the secant_weight to one because that could disturb
1065 // the convergence in some corner cases!
1066 static const double secant_weight = 0.984375; // == 63/64 < 1
1067 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1068 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1069 ex fxmid_ = f.subs(x == xxmid).evalf();
1070 if (!is_a<numeric>(fxmid_))
1071 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1072 numeric fxmid = ex_to<numeric>(fxmid_);
1073 if (fxmid.is_zero()) {
1077 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1085 } while (xxprev!=xx[side]);
1090 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1091 * for static lib (so ginsh will see them). */
1092 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1093 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1095 } // namespace GiNaC