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 if (is_ex_the_function(arg, exp))
201 return exp(arg.op(0).real_part());
203 if (is_exactly_a<power>(arg)) {
204 const ex& base = arg.op(0);
205 const ex& exponent = arg.op(1);
206 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
207 return pow(abs(base), exponent.real_part());
210 if (is_ex_the_function(arg, conjugate_function))
211 return abs(arg.op(0));
213 if (is_ex_the_function(arg, step))
216 return abs(arg).hold();
219 static void abs_print_latex(const ex & arg, const print_context & c)
221 c.s << "{|"; arg.print(c); c.s << "|}";
224 static void abs_print_csrc_float(const ex & arg, const print_context & c)
226 c.s << "fabs("; arg.print(c); c.s << ")";
229 static ex abs_conjugate(const ex & arg)
231 return abs(arg).hold();
234 static ex abs_real_part(const ex & arg)
236 return abs(arg).hold();
239 static ex abs_imag_part(const ex& arg)
244 static ex abs_power(const ex & arg, const ex & exp)
246 if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
247 || exp.info(info_flags::even)))
248 return power(arg, exp);
250 return power(abs(arg), exp).hold();
253 REGISTER_FUNCTION(abs, eval_func(abs_eval).
254 evalf_func(abs_evalf).
255 print_func<print_latex>(abs_print_latex).
256 print_func<print_csrc_float>(abs_print_csrc_float).
257 print_func<print_csrc_double>(abs_print_csrc_float).
258 conjugate_func(abs_conjugate).
259 real_part_func(abs_real_part).
260 imag_part_func(abs_imag_part).
261 power_func(abs_power));
267 static ex step_evalf(const ex & arg)
269 if (is_exactly_a<numeric>(arg))
270 return step(ex_to<numeric>(arg));
272 return step(arg).hold();
275 static ex step_eval(const ex & arg)
277 if (is_exactly_a<numeric>(arg))
278 return step(ex_to<numeric>(arg));
280 else if (is_exactly_a<mul>(arg) &&
281 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
282 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
285 // step(42*x) -> step(x)
286 return step(arg/oc).hold();
288 // step(-42*x) -> step(-x)
289 return step(-arg/oc).hold();
291 if (oc.real().is_zero()) {
293 // step(42*I*x) -> step(I*x)
294 return step(I*arg/oc).hold();
296 // step(-42*I*x) -> step(-I*x)
297 return step(-I*arg/oc).hold();
301 return step(arg).hold();
304 static ex step_series(const ex & arg,
305 const relational & rel,
309 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
310 if (arg_pt.info(info_flags::numeric)
311 && ex_to<numeric>(arg_pt).real().is_zero()
312 && !(options & series_options::suppress_branchcut))
313 throw (std::domain_error("step_series(): on imaginary axis"));
316 seq.push_back(expair(step(arg_pt), _ex0));
317 return pseries(rel,seq);
320 static ex step_conjugate(const ex& arg)
322 return step(arg).hold();
325 static ex step_real_part(const ex& arg)
327 return step(arg).hold();
330 static ex step_imag_part(const ex& arg)
335 REGISTER_FUNCTION(step, eval_func(step_eval).
336 evalf_func(step_evalf).
337 series_func(step_series).
338 conjugate_func(step_conjugate).
339 real_part_func(step_real_part).
340 imag_part_func(step_imag_part));
346 static ex csgn_evalf(const ex & arg)
348 if (is_exactly_a<numeric>(arg))
349 return csgn(ex_to<numeric>(arg));
351 return csgn(arg).hold();
354 static ex csgn_eval(const ex & arg)
356 if (is_exactly_a<numeric>(arg))
357 return csgn(ex_to<numeric>(arg));
359 else if (is_exactly_a<mul>(arg) &&
360 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
361 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
364 // csgn(42*x) -> csgn(x)
365 return csgn(arg/oc).hold();
367 // csgn(-42*x) -> -csgn(x)
368 return -csgn(arg/oc).hold();
370 if (oc.real().is_zero()) {
372 // csgn(42*I*x) -> csgn(I*x)
373 return csgn(I*arg/oc).hold();
375 // csgn(-42*I*x) -> -csgn(I*x)
376 return -csgn(I*arg/oc).hold();
380 return csgn(arg).hold();
383 static ex csgn_series(const ex & arg,
384 const relational & rel,
388 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
389 if (arg_pt.info(info_flags::numeric)
390 && ex_to<numeric>(arg_pt).real().is_zero()
391 && !(options & series_options::suppress_branchcut))
392 throw (std::domain_error("csgn_series(): on imaginary axis"));
395 seq.push_back(expair(csgn(arg_pt), _ex0));
396 return pseries(rel,seq);
399 static ex csgn_conjugate(const ex& arg)
401 return csgn(arg).hold();
404 static ex csgn_real_part(const ex& arg)
406 return csgn(arg).hold();
409 static ex csgn_imag_part(const ex& arg)
414 static ex csgn_power(const ex & arg, const ex & exp)
416 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
417 if (ex_to<numeric>(exp).is_odd())
418 return csgn(arg).hold();
420 return power(csgn(arg), _ex2).hold();
422 return power(csgn(arg), exp).hold();
426 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
427 evalf_func(csgn_evalf).
428 series_func(csgn_series).
429 conjugate_func(csgn_conjugate).
430 real_part_func(csgn_real_part).
431 imag_part_func(csgn_imag_part).
432 power_func(csgn_power));
436 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
437 // This function is closely related to the unwinding number K, sometimes found
438 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
441 static ex eta_evalf(const ex &x, const ex &y)
443 // It seems like we basically have to replicate the eval function here,
444 // since the expression might not be fully evaluated yet.
445 if (x.info(info_flags::positive) || y.info(info_flags::positive))
448 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
449 const numeric nx = ex_to<numeric>(x);
450 const numeric ny = ex_to<numeric>(y);
451 const numeric nxy = ex_to<numeric>(x*y);
453 if (nx.is_real() && nx.is_negative())
455 if (ny.is_real() && ny.is_negative())
457 if (nxy.is_real() && nxy.is_negative())
459 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
460 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
463 return eta(x,y).hold();
466 static ex eta_eval(const ex &x, const ex &y)
468 // trivial: eta(x,c) -> 0 if c is real and positive
469 if (x.info(info_flags::positive) || y.info(info_flags::positive))
472 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
473 // don't call eta_evalf here because it would call Pi.evalf()!
474 const numeric nx = ex_to<numeric>(x);
475 const numeric ny = ex_to<numeric>(y);
476 const numeric nxy = ex_to<numeric>(x*y);
478 if (nx.is_real() && nx.is_negative())
480 if (ny.is_real() && ny.is_negative())
482 if (nxy.is_real() && nxy.is_negative())
484 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
485 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
488 return eta(x,y).hold();
491 static ex eta_series(const ex & x, const ex & y,
492 const relational & rel,
496 const ex x_pt = x.subs(rel, subs_options::no_pattern);
497 const ex y_pt = y.subs(rel, subs_options::no_pattern);
498 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
499 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
500 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
501 throw (std::domain_error("eta_series(): on discontinuity"));
503 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
504 return pseries(rel,seq);
507 static ex eta_conjugate(const ex & x, const ex & y)
509 return -eta(x, y).hold();
512 static ex eta_real_part(const ex & x, const ex & y)
517 static ex eta_imag_part(const ex & x, const ex & y)
519 return -I*eta(x, y).hold();
522 REGISTER_FUNCTION(eta, eval_func(eta_eval).
523 evalf_func(eta_evalf).
524 series_func(eta_series).
526 set_symmetry(sy_symm(0, 1)).
527 conjugate_func(eta_conjugate).
528 real_part_func(eta_real_part).
529 imag_part_func(eta_imag_part));
536 static ex Li2_evalf(const ex & x)
538 if (is_exactly_a<numeric>(x))
539 return Li2(ex_to<numeric>(x));
541 return Li2(x).hold();
544 static ex Li2_eval(const ex & x)
546 if (x.info(info_flags::numeric)) {
551 if (x.is_equal(_ex1))
552 return power(Pi,_ex2)/_ex6;
553 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
554 if (x.is_equal(_ex1_2))
555 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
556 // Li2(-1) -> -Pi^2/12
557 if (x.is_equal(_ex_1))
558 return -power(Pi,_ex2)/_ex12;
559 // Li2(I) -> -Pi^2/48+Catalan*I
561 return power(Pi,_ex2)/_ex_48 + Catalan*I;
562 // Li2(-I) -> -Pi^2/48-Catalan*I
564 return power(Pi,_ex2)/_ex_48 - Catalan*I;
566 if (!x.info(info_flags::crational))
567 return Li2(ex_to<numeric>(x));
570 return Li2(x).hold();
573 static ex Li2_deriv(const ex & x, unsigned deriv_param)
575 GINAC_ASSERT(deriv_param==0);
577 // d/dx Li2(x) -> -log(1-x)/x
578 return -log(_ex1-x)/x;
581 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
583 const ex x_pt = x.subs(rel, subs_options::no_pattern);
584 if (x_pt.info(info_flags::numeric)) {
585 // First special case: x==0 (derivatives have poles)
586 if (x_pt.is_zero()) {
588 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
589 // simply substitute x==0. The limit, however, exists: it is 1.
590 // We also know all higher derivatives' limits:
591 // (d/dx)^n Li2(x) == n!/n^2.
592 // So the primitive series expansion is
593 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
595 // We first construct such a primitive series expansion manually in
596 // a dummy symbol s and then insert the argument's series expansion
597 // for s. Reexpanding the resulting series returns the desired
601 // manually construct the primitive expansion
602 for (int i=1; i<order; ++i)
603 ser += pow(s,i) / pow(numeric(i), *_num2_p);
604 // substitute the argument's series expansion
605 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
606 // maybe that was terminating, so add a proper order term
608 nseq.push_back(expair(Order(_ex1), order));
609 ser += pseries(rel, nseq);
610 // reexpanding it will collapse the series again
611 return ser.series(rel, order);
612 // NB: Of course, this still does not allow us to compute anything
613 // like sin(Li2(x)).series(x==0,2), since then this code here is
614 // not reached and the derivative of sin(Li2(x)) doesn't allow the
615 // substitution x==0. Probably limits *are* needed for the general
616 // cases. In case L'Hospital's rule is implemented for limits and
617 // basic::series() takes care of this, this whole block is probably
620 // second special case: x==1 (branch point)
621 if (x_pt.is_equal(_ex1)) {
623 // construct series manually in a dummy symbol s
626 // manually construct the primitive expansion
627 for (int i=1; i<order; ++i)
628 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
629 // substitute the argument's series expansion
630 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
631 // maybe that was terminating, so add a proper order term
633 nseq.push_back(expair(Order(_ex1), order));
634 ser += pseries(rel, nseq);
635 // reexpanding it will collapse the series again
636 return ser.series(rel, order);
638 // third special case: x real, >=1 (branch cut)
639 if (!(options & series_options::suppress_branchcut) &&
640 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
642 // This is the branch cut: assemble the primitive series manually
643 // and then add the corresponding complex step function.
644 const symbol &s = ex_to<symbol>(rel.lhs());
645 const ex point = rel.rhs();
648 // zeroth order term:
649 seq.push_back(expair(Li2(x_pt), _ex0));
650 // compute the intermediate terms:
651 ex replarg = series(Li2(x), s==foo, order);
652 for (size_t i=1; i<replarg.nops()-1; ++i)
653 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));
654 // append an order term:
655 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
656 return pseries(rel, seq);
659 // all other cases should be safe, by now:
660 throw do_taylor(); // caught by function::series()
663 static ex Li2_conjugate(const ex & x)
665 // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
666 // run along the positive real axis beginning at 1.
667 if (x.info(info_flags::negative)) {
668 return Li2(x).hold();
670 if (is_exactly_a<numeric>(x) &&
671 (!x.imag_part().is_zero() || x < *_num1_p)) {
672 return Li2(x.conjugate());
674 return conjugate_function(Li2(x)).hold();
677 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
678 evalf_func(Li2_evalf).
679 derivative_func(Li2_deriv).
680 series_func(Li2_series).
681 conjugate_func(Li2_conjugate).
682 latex_name("\\mathrm{Li}_2"));
688 static ex Li3_eval(const ex & x)
692 return Li3(x).hold();
695 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
696 latex_name("\\mathrm{Li}_3"));
699 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
702 static ex zetaderiv_eval(const ex & n, const ex & x)
704 if (n.info(info_flags::numeric)) {
705 // zetaderiv(0,x) -> zeta(x)
707 return zeta(x).hold();
710 return zetaderiv(n, x).hold();
713 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
715 GINAC_ASSERT(deriv_param<2);
717 if (deriv_param==0) {
719 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
722 return zetaderiv(n+1,x);
725 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
726 derivative_func(zetaderiv_deriv).
727 latex_name("\\zeta^\\prime"));
733 static ex factorial_evalf(const ex & x)
735 return factorial(x).hold();
738 static ex factorial_eval(const ex & x)
740 if (is_exactly_a<numeric>(x))
741 return factorial(ex_to<numeric>(x));
743 return factorial(x).hold();
746 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
748 if (is_exactly_a<symbol>(x) ||
749 is_exactly_a<constant>(x) ||
750 is_exactly_a<function>(x)) {
751 x.print(c); c.s << "!";
753 c.s << "("; x.print(c); c.s << ")!";
757 static ex factorial_conjugate(const ex & x)
759 return factorial(x).hold();
762 static ex factorial_real_part(const ex & x)
764 return factorial(x).hold();
767 static ex factorial_imag_part(const ex & x)
772 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
773 evalf_func(factorial_evalf).
774 print_func<print_dflt>(factorial_print_dflt_latex).
775 print_func<print_latex>(factorial_print_dflt_latex).
776 conjugate_func(factorial_conjugate).
777 real_part_func(factorial_real_part).
778 imag_part_func(factorial_imag_part));
784 static ex binomial_evalf(const ex & x, const ex & y)
786 return binomial(x, y).hold();
789 static ex binomial_sym(const ex & x, const numeric & y)
791 if (y.is_integer()) {
792 if (y.is_nonneg_integer()) {
793 const unsigned N = y.to_int();
794 if (N == 0) return _ex1;
795 if (N == 1) return x;
797 for (unsigned i = 2; i <= N; ++i)
798 t = (t * (x + i - y - 1)).expand() / i;
804 return binomial(x, y).hold();
807 static ex binomial_eval(const ex & x, const ex &y)
809 if (is_exactly_a<numeric>(y)) {
810 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
811 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
813 return binomial_sym(x, ex_to<numeric>(y));
815 return binomial(x, y).hold();
818 // At the moment the numeric evaluation of a binomail function always
819 // gives a real number, but if this would be implemented using the gamma
820 // function, also complex conjugation should be changed (or rather, deleted).
821 static ex binomial_conjugate(const ex & x, const ex & y)
823 return binomial(x,y).hold();
826 static ex binomial_real_part(const ex & x, const ex & y)
828 return binomial(x,y).hold();
831 static ex binomial_imag_part(const ex & x, const ex & y)
836 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
837 evalf_func(binomial_evalf).
838 conjugate_func(binomial_conjugate).
839 real_part_func(binomial_real_part).
840 imag_part_func(binomial_imag_part));
843 // Order term function (for truncated power series)
846 static ex Order_eval(const ex & x)
848 if (is_exactly_a<numeric>(x)) {
851 return Order(_ex1).hold();
854 } else if (is_exactly_a<mul>(x)) {
855 const mul &m = ex_to<mul>(x);
856 // O(c*expr) -> O(expr)
857 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
858 return Order(x / m.op(m.nops() - 1)).hold();
860 return Order(x).hold();
863 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
865 // Just wrap the function into a pseries object
867 GINAC_ASSERT(is_a<symbol>(r.lhs()));
868 const symbol &s = ex_to<symbol>(r.lhs());
869 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
870 return pseries(r, new_seq);
873 static ex Order_conjugate(const ex & x)
875 return Order(x).hold();
878 static ex Order_real_part(const ex & x)
880 return Order(x).hold();
883 static ex Order_imag_part(const ex & x)
885 if(x.info(info_flags::real))
887 return Order(x).hold();
890 // Differentiation is handled in function::derivative because of its special requirements
892 REGISTER_FUNCTION(Order, eval_func(Order_eval).
893 series_func(Order_series).
894 latex_name("\\mathcal{O}").
895 conjugate_func(Order_conjugate).
896 real_part_func(Order_real_part).
897 imag_part_func(Order_imag_part));
900 // Solve linear system
903 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
905 // solve a system of linear equations
906 if (eqns.info(info_flags::relation_equal)) {
907 if (!symbols.info(info_flags::symbol))
908 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
909 const ex sol = lsolve(lst(eqns),lst(symbols));
911 GINAC_ASSERT(sol.nops()==1);
912 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
914 return sol.op(0).op(1); // return rhs of first solution
918 if (!eqns.info(info_flags::list)) {
919 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
921 for (size_t i=0; i<eqns.nops(); i++) {
922 if (!eqns.op(i).info(info_flags::relation_equal)) {
923 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
926 if (!symbols.info(info_flags::list)) {
927 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
929 for (size_t i=0; i<symbols.nops(); i++) {
930 if (!symbols.op(i).info(info_flags::symbol)) {
931 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
935 // build matrix from equation system
936 matrix sys(eqns.nops(),symbols.nops());
937 matrix rhs(eqns.nops(),1);
938 matrix vars(symbols.nops(),1);
940 for (size_t r=0; r<eqns.nops(); r++) {
941 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
943 for (size_t c=0; c<symbols.nops(); c++) {
944 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
945 linpart -= co*symbols.op(c);
948 linpart = linpart.expand();
952 // test if system is linear and fill vars matrix
953 for (size_t i=0; i<symbols.nops(); i++) {
954 vars(i,0) = symbols.op(i);
955 if (sys.has(symbols.op(i)))
956 throw(std::logic_error("lsolve: system is not linear"));
957 if (rhs.has(symbols.op(i)))
958 throw(std::logic_error("lsolve: system is not linear"));
963 solution = sys.solve(vars,rhs,options);
964 } catch (const std::runtime_error & e) {
965 // Probably singular matrix or otherwise overdetermined system:
966 // It is consistent to return an empty list
969 GINAC_ASSERT(solution.cols()==1);
970 GINAC_ASSERT(solution.rows()==symbols.nops());
972 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
974 for (size_t i=0; i<symbols.nops(); i++)
975 sollist.append(symbols.op(i)==solution(i,0));
981 // Find real root of f(x) numerically
985 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
987 if (!x1.is_real() || !x2.is_real()) {
988 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
991 throw std::runtime_error("fsolve(): vanishing interval");
993 // xx[0] == left interval limit, xx[1] == right interval limit.
994 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
995 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
996 numeric xx[2] = { x1<x2 ? x1 : x2,
999 if (is_a<relational>(f_in)) {
1000 f = f_in.lhs()-f_in.rhs();
1004 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1005 f.subs(x==xx[1]).evalf() };
1006 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1007 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1009 numeric fx[2] = { ex_to<numeric>(fx_[0]),
1010 ex_to<numeric>(fx_[1]) };
1011 if (!fx[0].is_real() || !fx[1].is_real()) {
1012 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1014 if (fx[0]*fx[1]>=0) {
1015 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1018 // The Newton-Raphson method has quadratic convergence! Simply put, it
1019 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
1020 const ex ff = normal(-f/f.diff(x));
1021 int side = 0; // Start at left interval limit.
1027 ex dx_ = ff.subs(x == xx[side]).evalf();
1028 if (!is_a<numeric>(dx_))
1029 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1030 xx[side] += ex_to<numeric>(dx_);
1031 // Now check if Newton-Raphson method shot out of the interval
1032 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1033 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1035 // Compute f(x) only if new x is inside the interval.
1036 // The function might be difficult to compute numerically
1037 // or even ill defined outside the interval. Also it's
1038 // a small optimization.
1039 ex f_x = f.subs(x == xx[side]).evalf();
1040 if (!is_a<numeric>(f_x))
1041 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1042 fx[side] = ex_to<numeric>(f_x);
1045 // Oops, Newton-Raphson method shot out of the interval.
1046 // Restore, and try again with the other side instead!
1053 ex dx_ = ff.subs(x == xx[side]).evalf();
1054 if (!is_a<numeric>(dx_))
1055 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1056 xx[side] += ex_to<numeric>(dx_);
1058 ex f_x = f.subs(x==xx[side]).evalf();
1059 if (!is_a<numeric>(f_x))
1060 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1061 fx[side] = ex_to<numeric>(f_x);
1063 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1064 // Oops, the root isn't bracketed any more.
1065 // Restore, and perform a bisection!
1069 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1070 // they occur pretty often when Newton-Raphson arrives at an x too
1071 // close to the result on one side of the interval and
1072 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1073 // precision errors! Recall that this function does not have a
1074 // precision goal as one of its arguments but instead relies on
1075 // x converging to a fixed point. We speed up the (safe but slow)
1076 // bisection method by mixing in a dash of the (unsafer but faster)
1077 // secant method: Instead of splitting the interval at the
1078 // arithmetic mean (bisection), we split it nearer to the root as
1079 // determined by the secant between the values xx[0] and xx[1].
1080 // Don't set the secant_weight to one because that could disturb
1081 // the convergence in some corner cases!
1082 static const double secant_weight = 0.984375; // == 63/64 < 1
1083 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1084 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1085 ex fxmid_ = f.subs(x == xxmid).evalf();
1086 if (!is_a<numeric>(fxmid_))
1087 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1088 numeric fxmid = ex_to<numeric>(fxmid_);
1089 if (fxmid.is_zero()) {
1093 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1101 } while (xxprev!=xx[side]);
1106 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1107 * for static lib (so ginsh will see them). */
1108 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1109 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1111 } // namespace GiNaC