3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2014 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 static bool func_arg_info(const ex & arg, unsigned inf)
81 // for some functions we can return the info() of its argument
82 // (think of conjugate())
84 case info_flags::polynomial:
85 case info_flags::integer_polynomial:
86 case info_flags::cinteger_polynomial:
87 case info_flags::rational_polynomial:
88 case info_flags::real:
89 case info_flags::rational:
90 case info_flags::integer:
91 case info_flags::crational:
92 case info_flags::cinteger:
93 case info_flags::even:
95 case info_flags::prime:
96 case info_flags::crational_polynomial:
97 case info_flags::rational_function:
98 case info_flags::algebraic:
99 case info_flags::positive:
100 case info_flags::negative:
101 case info_flags::nonnegative:
102 case info_flags::posint:
103 case info_flags::negint:
104 case info_flags::nonnegint:
105 case info_flags::has_indices:
106 return arg.info(inf);
111 static bool conjugate_info(const ex & arg, unsigned inf)
113 return func_arg_info(arg, inf);
116 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
117 evalf_func(conjugate_evalf).
118 info_func(conjugate_info).
119 print_func<print_latex>(conjugate_print_latex).
120 conjugate_func(conjugate_conjugate).
121 real_part_func(conjugate_real_part).
122 imag_part_func(conjugate_imag_part).
123 set_name("conjugate","conjugate"));
129 static ex real_part_evalf(const ex & arg)
131 if (is_exactly_a<numeric>(arg)) {
132 return ex_to<numeric>(arg).real();
134 return real_part_function(arg).hold();
137 static ex real_part_eval(const ex & arg)
139 return arg.real_part();
142 static void real_part_print_latex(const ex & arg, const print_context & c)
144 c.s << "\\Re"; arg.print(c); c.s << "";
147 static ex real_part_conjugate(const ex & arg)
149 return real_part_function(arg).hold();
152 static ex real_part_real_part(const ex & arg)
154 return real_part_function(arg).hold();
157 static ex real_part_imag_part(const ex & arg)
162 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
163 evalf_func(real_part_evalf).
164 print_func<print_latex>(real_part_print_latex).
165 conjugate_func(real_part_conjugate).
166 real_part_func(real_part_real_part).
167 imag_part_func(real_part_imag_part).
168 set_name("real_part","real_part"));
174 static ex imag_part_evalf(const ex & arg)
176 if (is_exactly_a<numeric>(arg)) {
177 return ex_to<numeric>(arg).imag();
179 return imag_part_function(arg).hold();
182 static ex imag_part_eval(const ex & arg)
184 return arg.imag_part();
187 static void imag_part_print_latex(const ex & arg, const print_context & c)
189 c.s << "\\Im"; arg.print(c); c.s << "";
192 static ex imag_part_conjugate(const ex & arg)
194 return imag_part_function(arg).hold();
197 static ex imag_part_real_part(const ex & arg)
199 return imag_part_function(arg).hold();
202 static ex imag_part_imag_part(const ex & arg)
207 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
208 evalf_func(imag_part_evalf).
209 print_func<print_latex>(imag_part_print_latex).
210 conjugate_func(imag_part_conjugate).
211 real_part_func(imag_part_real_part).
212 imag_part_func(imag_part_imag_part).
213 set_name("imag_part","imag_part"));
219 static ex abs_evalf(const ex & arg)
221 if (is_exactly_a<numeric>(arg))
222 return abs(ex_to<numeric>(arg));
224 return abs(arg).hold();
227 static ex abs_eval(const ex & arg)
229 if (is_exactly_a<numeric>(arg))
230 return abs(ex_to<numeric>(arg));
232 if (arg.info(info_flags::nonnegative))
235 if (is_ex_the_function(arg, abs))
238 if (is_ex_the_function(arg, exp))
239 return exp(arg.op(0).real_part());
241 if (is_exactly_a<power>(arg)) {
242 const ex& base = arg.op(0);
243 const ex& exponent = arg.op(1);
244 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
245 return pow(abs(base), exponent.real_part());
248 if (is_ex_the_function(arg, conjugate_function))
249 return abs(arg.op(0));
251 if (is_ex_the_function(arg, step))
254 return abs(arg).hold();
257 static ex abs_expand(const ex & arg, unsigned options)
259 if ((options & expand_options::expand_transcendental)
260 && is_exactly_a<mul>(arg)) {
262 prodseq.reserve(arg.nops());
263 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
264 if (options & expand_options::expand_function_args)
265 prodseq.push_back(abs(i->expand(options)));
267 prodseq.push_back(abs(*i));
269 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
272 if (options & expand_options::expand_function_args)
273 return abs(arg.expand(options)).hold();
275 return abs(arg).hold();
278 static void abs_print_latex(const ex & arg, const print_context & c)
280 c.s << "{|"; arg.print(c); c.s << "|}";
283 static void abs_print_csrc_float(const ex & arg, const print_context & c)
285 c.s << "fabs("; arg.print(c); c.s << ")";
288 static ex abs_conjugate(const ex & arg)
290 return abs(arg).hold();
293 static ex abs_real_part(const ex & arg)
295 return abs(arg).hold();
298 static ex abs_imag_part(const ex& arg)
303 static ex abs_power(const ex & arg, const ex & exp)
305 if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
306 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
307 return power(arg, exp);
309 return power(arg, exp/2)*power(arg.conjugate(), exp/2);
311 return power(abs(arg), exp).hold();
314 bool abs_info(const ex & arg, unsigned inf)
317 case info_flags::integer:
318 case info_flags::even:
319 case info_flags::odd:
320 case info_flags::prime:
321 return arg.info(inf);
322 case info_flags::nonnegint:
323 return arg.info(info_flags::integer);
324 case info_flags::nonnegative:
325 case info_flags::real:
327 case info_flags::negative:
329 case info_flags::positive:
330 return arg.info(info_flags::positive) || arg.info(info_flags::negative);
331 case info_flags::has_indices: {
332 if (arg.info(info_flags::has_indices))
341 REGISTER_FUNCTION(abs, eval_func(abs_eval).
342 evalf_func(abs_evalf).
343 expand_func(abs_expand).
345 print_func<print_latex>(abs_print_latex).
346 print_func<print_csrc_float>(abs_print_csrc_float).
347 print_func<print_csrc_double>(abs_print_csrc_float).
348 conjugate_func(abs_conjugate).
349 real_part_func(abs_real_part).
350 imag_part_func(abs_imag_part).
351 power_func(abs_power));
357 static ex step_evalf(const ex & arg)
359 if (is_exactly_a<numeric>(arg))
360 return step(ex_to<numeric>(arg));
362 return step(arg).hold();
365 static ex step_eval(const ex & arg)
367 if (is_exactly_a<numeric>(arg))
368 return step(ex_to<numeric>(arg));
370 else if (is_exactly_a<mul>(arg) &&
371 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
372 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
375 // step(42*x) -> step(x)
376 return step(arg/oc).hold();
378 // step(-42*x) -> step(-x)
379 return step(-arg/oc).hold();
381 if (oc.real().is_zero()) {
383 // step(42*I*x) -> step(I*x)
384 return step(I*arg/oc).hold();
386 // step(-42*I*x) -> step(-I*x)
387 return step(-I*arg/oc).hold();
391 return step(arg).hold();
394 static ex step_series(const ex & arg,
395 const relational & rel,
399 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
400 if (arg_pt.info(info_flags::numeric)
401 && ex_to<numeric>(arg_pt).real().is_zero()
402 && !(options & series_options::suppress_branchcut))
403 throw (std::domain_error("step_series(): on imaginary axis"));
406 seq.push_back(expair(step(arg_pt), _ex0));
407 return pseries(rel,seq);
410 static ex step_conjugate(const ex& arg)
412 return step(arg).hold();
415 static ex step_real_part(const ex& arg)
417 return step(arg).hold();
420 static ex step_imag_part(const ex& arg)
425 REGISTER_FUNCTION(step, eval_func(step_eval).
426 evalf_func(step_evalf).
427 series_func(step_series).
428 conjugate_func(step_conjugate).
429 real_part_func(step_real_part).
430 imag_part_func(step_imag_part));
436 static ex csgn_evalf(const ex & arg)
438 if (is_exactly_a<numeric>(arg))
439 return csgn(ex_to<numeric>(arg));
441 return csgn(arg).hold();
444 static ex csgn_eval(const ex & arg)
446 if (is_exactly_a<numeric>(arg))
447 return csgn(ex_to<numeric>(arg));
449 else if (is_exactly_a<mul>(arg) &&
450 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
451 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
454 // csgn(42*x) -> csgn(x)
455 return csgn(arg/oc).hold();
457 // csgn(-42*x) -> -csgn(x)
458 return -csgn(arg/oc).hold();
460 if (oc.real().is_zero()) {
462 // csgn(42*I*x) -> csgn(I*x)
463 return csgn(I*arg/oc).hold();
465 // csgn(-42*I*x) -> -csgn(I*x)
466 return -csgn(I*arg/oc).hold();
470 return csgn(arg).hold();
473 static ex csgn_series(const ex & arg,
474 const relational & rel,
478 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
479 if (arg_pt.info(info_flags::numeric)
480 && ex_to<numeric>(arg_pt).real().is_zero()
481 && !(options & series_options::suppress_branchcut))
482 throw (std::domain_error("csgn_series(): on imaginary axis"));
485 seq.push_back(expair(csgn(arg_pt), _ex0));
486 return pseries(rel,seq);
489 static ex csgn_conjugate(const ex& arg)
491 return csgn(arg).hold();
494 static ex csgn_real_part(const ex& arg)
496 return csgn(arg).hold();
499 static ex csgn_imag_part(const ex& arg)
504 static ex csgn_power(const ex & arg, const ex & exp)
506 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
507 if (ex_to<numeric>(exp).is_odd())
508 return csgn(arg).hold();
510 return power(csgn(arg), _ex2).hold();
512 return power(csgn(arg), exp).hold();
516 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
517 evalf_func(csgn_evalf).
518 series_func(csgn_series).
519 conjugate_func(csgn_conjugate).
520 real_part_func(csgn_real_part).
521 imag_part_func(csgn_imag_part).
522 power_func(csgn_power));
526 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
527 // This function is closely related to the unwinding number K, sometimes found
528 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
531 static ex eta_evalf(const ex &x, const ex &y)
533 // It seems like we basically have to replicate the eval function here,
534 // since the expression might not be fully evaluated yet.
535 if (x.info(info_flags::positive) || y.info(info_flags::positive))
538 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
539 const numeric nx = ex_to<numeric>(x);
540 const numeric ny = ex_to<numeric>(y);
541 const numeric nxy = ex_to<numeric>(x*y);
543 if (nx.is_real() && nx.is_negative())
545 if (ny.is_real() && ny.is_negative())
547 if (nxy.is_real() && nxy.is_negative())
549 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
550 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
553 return eta(x,y).hold();
556 static ex eta_eval(const ex &x, const ex &y)
558 // trivial: eta(x,c) -> 0 if c is real and positive
559 if (x.info(info_flags::positive) || y.info(info_flags::positive))
562 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
563 // don't call eta_evalf here because it would call Pi.evalf()!
564 const numeric nx = ex_to<numeric>(x);
565 const numeric ny = ex_to<numeric>(y);
566 const numeric nxy = ex_to<numeric>(x*y);
568 if (nx.is_real() && nx.is_negative())
570 if (ny.is_real() && ny.is_negative())
572 if (nxy.is_real() && nxy.is_negative())
574 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
575 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
578 return eta(x,y).hold();
581 static ex eta_series(const ex & x, const ex & y,
582 const relational & rel,
586 const ex x_pt = x.subs(rel, subs_options::no_pattern);
587 const ex y_pt = y.subs(rel, subs_options::no_pattern);
588 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
589 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
590 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
591 throw (std::domain_error("eta_series(): on discontinuity"));
593 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
594 return pseries(rel,seq);
597 static ex eta_conjugate(const ex & x, const ex & y)
599 return -eta(x, y).hold();
602 static ex eta_real_part(const ex & x, const ex & y)
607 static ex eta_imag_part(const ex & x, const ex & y)
609 return -I*eta(x, y).hold();
612 REGISTER_FUNCTION(eta, eval_func(eta_eval).
613 evalf_func(eta_evalf).
614 series_func(eta_series).
616 set_symmetry(sy_symm(0, 1)).
617 conjugate_func(eta_conjugate).
618 real_part_func(eta_real_part).
619 imag_part_func(eta_imag_part));
626 static ex Li2_evalf(const ex & x)
628 if (is_exactly_a<numeric>(x))
629 return Li2(ex_to<numeric>(x));
631 return Li2(x).hold();
634 static ex Li2_eval(const ex & x)
636 if (x.info(info_flags::numeric)) {
641 if (x.is_equal(_ex1))
642 return power(Pi,_ex2)/_ex6;
643 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
644 if (x.is_equal(_ex1_2))
645 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
646 // Li2(-1) -> -Pi^2/12
647 if (x.is_equal(_ex_1))
648 return -power(Pi,_ex2)/_ex12;
649 // Li2(I) -> -Pi^2/48+Catalan*I
651 return power(Pi,_ex2)/_ex_48 + Catalan*I;
652 // Li2(-I) -> -Pi^2/48-Catalan*I
654 return power(Pi,_ex2)/_ex_48 - Catalan*I;
656 if (!x.info(info_flags::crational))
657 return Li2(ex_to<numeric>(x));
660 return Li2(x).hold();
663 static ex Li2_deriv(const ex & x, unsigned deriv_param)
665 GINAC_ASSERT(deriv_param==0);
667 // d/dx Li2(x) -> -log(1-x)/x
668 return -log(_ex1-x)/x;
671 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
673 const ex x_pt = x.subs(rel, subs_options::no_pattern);
674 if (x_pt.info(info_flags::numeric)) {
675 // First special case: x==0 (derivatives have poles)
676 if (x_pt.is_zero()) {
678 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
679 // simply substitute x==0. The limit, however, exists: it is 1.
680 // We also know all higher derivatives' limits:
681 // (d/dx)^n Li2(x) == n!/n^2.
682 // So the primitive series expansion is
683 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
685 // We first construct such a primitive series expansion manually in
686 // a dummy symbol s and then insert the argument's series expansion
687 // for s. Reexpanding the resulting series returns the desired
691 // manually construct the primitive expansion
692 for (int i=1; i<order; ++i)
693 ser += pow(s,i) / pow(numeric(i), *_num2_p);
694 // substitute the argument's series expansion
695 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
696 // maybe that was terminating, so add a proper order term
698 nseq.push_back(expair(Order(_ex1), order));
699 ser += pseries(rel, nseq);
700 // reexpanding it will collapse the series again
701 return ser.series(rel, order);
702 // NB: Of course, this still does not allow us to compute anything
703 // like sin(Li2(x)).series(x==0,2), since then this code here is
704 // not reached and the derivative of sin(Li2(x)) doesn't allow the
705 // substitution x==0. Probably limits *are* needed for the general
706 // cases. In case L'Hospital's rule is implemented for limits and
707 // basic::series() takes care of this, this whole block is probably
710 // second special case: x==1 (branch point)
711 if (x_pt.is_equal(_ex1)) {
713 // construct series manually in a dummy symbol s
716 // manually construct the primitive expansion
717 for (int i=1; i<order; ++i)
718 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
719 // substitute the argument's series expansion
720 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
721 // maybe that was terminating, so add a proper order term
723 nseq.push_back(expair(Order(_ex1), order));
724 ser += pseries(rel, nseq);
725 // reexpanding it will collapse the series again
726 return ser.series(rel, order);
728 // third special case: x real, >=1 (branch cut)
729 if (!(options & series_options::suppress_branchcut) &&
730 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
732 // This is the branch cut: assemble the primitive series manually
733 // and then add the corresponding complex step function.
734 const symbol &s = ex_to<symbol>(rel.lhs());
735 const ex point = rel.rhs();
738 // zeroth order term:
739 seq.push_back(expair(Li2(x_pt), _ex0));
740 // compute the intermediate terms:
741 ex replarg = series(Li2(x), s==foo, order);
742 for (size_t i=1; i<replarg.nops()-1; ++i)
743 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));
744 // append an order term:
745 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
746 return pseries(rel, seq);
749 // all other cases should be safe, by now:
750 throw do_taylor(); // caught by function::series()
753 static ex Li2_conjugate(const ex & x)
755 // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
756 // run along the positive real axis beginning at 1.
757 if (x.info(info_flags::negative)) {
758 return Li2(x).hold();
760 if (is_exactly_a<numeric>(x) &&
761 (!x.imag_part().is_zero() || x < *_num1_p)) {
762 return Li2(x.conjugate());
764 return conjugate_function(Li2(x)).hold();
767 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
768 evalf_func(Li2_evalf).
769 derivative_func(Li2_deriv).
770 series_func(Li2_series).
771 conjugate_func(Li2_conjugate).
772 latex_name("\\mathrm{Li}_2"));
778 static ex Li3_eval(const ex & x)
782 return Li3(x).hold();
785 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
786 latex_name("\\mathrm{Li}_3"));
789 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
792 static ex zetaderiv_eval(const ex & n, const ex & x)
794 if (n.info(info_flags::numeric)) {
795 // zetaderiv(0,x) -> zeta(x)
797 return zeta(x).hold();
800 return zetaderiv(n, x).hold();
803 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
805 GINAC_ASSERT(deriv_param<2);
807 if (deriv_param==0) {
809 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
812 return zetaderiv(n+1,x);
815 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
816 derivative_func(zetaderiv_deriv).
817 latex_name("\\zeta^\\prime"));
823 static ex factorial_evalf(const ex & x)
825 return factorial(x).hold();
828 static ex factorial_eval(const ex & x)
830 if (is_exactly_a<numeric>(x))
831 return factorial(ex_to<numeric>(x));
833 return factorial(x).hold();
836 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
838 if (is_exactly_a<symbol>(x) ||
839 is_exactly_a<constant>(x) ||
840 is_exactly_a<function>(x)) {
841 x.print(c); c.s << "!";
843 c.s << "("; x.print(c); c.s << ")!";
847 static ex factorial_conjugate(const ex & x)
849 return factorial(x).hold();
852 static ex factorial_real_part(const ex & x)
854 return factorial(x).hold();
857 static ex factorial_imag_part(const ex & x)
862 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
863 evalf_func(factorial_evalf).
864 print_func<print_dflt>(factorial_print_dflt_latex).
865 print_func<print_latex>(factorial_print_dflt_latex).
866 conjugate_func(factorial_conjugate).
867 real_part_func(factorial_real_part).
868 imag_part_func(factorial_imag_part));
874 static ex binomial_evalf(const ex & x, const ex & y)
876 return binomial(x, y).hold();
879 static ex binomial_sym(const ex & x, const numeric & y)
881 if (y.is_integer()) {
882 if (y.is_nonneg_integer()) {
883 const unsigned N = y.to_int();
884 if (N == 0) return _ex1;
885 if (N == 1) return x;
887 for (unsigned i = 2; i <= N; ++i)
888 t = (t * (x + i - y - 1)).expand() / i;
894 return binomial(x, y).hold();
897 static ex binomial_eval(const ex & x, const ex &y)
899 if (is_exactly_a<numeric>(y)) {
900 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
901 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
903 return binomial_sym(x, ex_to<numeric>(y));
905 return binomial(x, y).hold();
908 // At the moment the numeric evaluation of a binomail function always
909 // gives a real number, but if this would be implemented using the gamma
910 // function, also complex conjugation should be changed (or rather, deleted).
911 static ex binomial_conjugate(const ex & x, const ex & y)
913 return binomial(x,y).hold();
916 static ex binomial_real_part(const ex & x, const ex & y)
918 return binomial(x,y).hold();
921 static ex binomial_imag_part(const ex & x, const ex & y)
926 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
927 evalf_func(binomial_evalf).
928 conjugate_func(binomial_conjugate).
929 real_part_func(binomial_real_part).
930 imag_part_func(binomial_imag_part));
933 // Order term function (for truncated power series)
936 static ex Order_eval(const ex & x)
938 if (is_exactly_a<numeric>(x)) {
941 return Order(_ex1).hold();
944 } else if (is_exactly_a<mul>(x)) {
945 const mul &m = ex_to<mul>(x);
946 // O(c*expr) -> O(expr)
947 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
948 return Order(x / m.op(m.nops() - 1)).hold();
950 return Order(x).hold();
953 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
955 // Just wrap the function into a pseries object
957 GINAC_ASSERT(is_a<symbol>(r.lhs()));
958 const symbol &s = ex_to<symbol>(r.lhs());
959 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
960 return pseries(r, new_seq);
963 static ex Order_conjugate(const ex & x)
965 return Order(x).hold();
968 static ex Order_real_part(const ex & x)
970 return Order(x).hold();
973 static ex Order_imag_part(const ex & x)
975 if(x.info(info_flags::real))
977 return Order(x).hold();
980 // Differentiation is handled in function::derivative because of its special requirements
982 REGISTER_FUNCTION(Order, eval_func(Order_eval).
983 series_func(Order_series).
984 latex_name("\\mathcal{O}").
985 conjugate_func(Order_conjugate).
986 real_part_func(Order_real_part).
987 imag_part_func(Order_imag_part));
990 // Solve linear system
993 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
995 // solve a system of linear equations
996 if (eqns.info(info_flags::relation_equal)) {
997 if (!symbols.info(info_flags::symbol))
998 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
999 const ex sol = lsolve(lst(eqns),lst(symbols));
1001 GINAC_ASSERT(sol.nops()==1);
1002 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1004 return sol.op(0).op(1); // return rhs of first solution
1008 if (!eqns.info(info_flags::list)) {
1009 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1011 for (size_t i=0; i<eqns.nops(); i++) {
1012 if (!eqns.op(i).info(info_flags::relation_equal)) {
1013 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1016 if (!symbols.info(info_flags::list)) {
1017 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1019 for (size_t i=0; i<symbols.nops(); i++) {
1020 if (!symbols.op(i).info(info_flags::symbol)) {
1021 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1025 // build matrix from equation system
1026 matrix sys(eqns.nops(),symbols.nops());
1027 matrix rhs(eqns.nops(),1);
1028 matrix vars(symbols.nops(),1);
1030 for (size_t r=0; r<eqns.nops(); r++) {
1031 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1033 for (size_t c=0; c<symbols.nops(); c++) {
1034 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1035 linpart -= co*symbols.op(c);
1038 linpart = linpart.expand();
1039 rhs(r,0) = -linpart;
1042 // test if system is linear and fill vars matrix
1043 for (size_t i=0; i<symbols.nops(); i++) {
1044 vars(i,0) = symbols.op(i);
1045 if (sys.has(symbols.op(i)))
1046 throw(std::logic_error("lsolve: system is not linear"));
1047 if (rhs.has(symbols.op(i)))
1048 throw(std::logic_error("lsolve: system is not linear"));
1053 solution = sys.solve(vars,rhs,options);
1054 } catch (const std::runtime_error & e) {
1055 // Probably singular matrix or otherwise overdetermined system:
1056 // It is consistent to return an empty list
1059 GINAC_ASSERT(solution.cols()==1);
1060 GINAC_ASSERT(solution.rows()==symbols.nops());
1062 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1064 for (size_t i=0; i<symbols.nops(); i++)
1065 sollist.append(symbols.op(i)==solution(i,0));
1071 // Find real root of f(x) numerically
1075 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1077 if (!x1.is_real() || !x2.is_real()) {
1078 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1081 throw std::runtime_error("fsolve(): vanishing interval");
1083 // xx[0] == left interval limit, xx[1] == right interval limit.
1084 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1085 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1086 numeric xx[2] = { x1<x2 ? x1 : x2,
1089 if (is_a<relational>(f_in)) {
1090 f = f_in.lhs()-f_in.rhs();
1094 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1095 f.subs(x==xx[1]).evalf() };
1096 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1097 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1099 numeric fx[2] = { ex_to<numeric>(fx_[0]),
1100 ex_to<numeric>(fx_[1]) };
1101 if (!fx[0].is_real() || !fx[1].is_real()) {
1102 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1104 if (fx[0]*fx[1]>=0) {
1105 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1108 // The Newton-Raphson method has quadratic convergence! Simply put, it
1109 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
1110 const ex ff = normal(-f/f.diff(x));
1111 int side = 0; // Start at left interval limit.
1117 ex dx_ = ff.subs(x == xx[side]).evalf();
1118 if (!is_a<numeric>(dx_))
1119 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1120 xx[side] += ex_to<numeric>(dx_);
1121 // Now check if Newton-Raphson method shot out of the interval
1122 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1123 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1125 // Compute f(x) only if new x is inside the interval.
1126 // The function might be difficult to compute numerically
1127 // or even ill defined outside the interval. Also it's
1128 // a small optimization.
1129 ex f_x = f.subs(x == xx[side]).evalf();
1130 if (!is_a<numeric>(f_x))
1131 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1132 fx[side] = ex_to<numeric>(f_x);
1135 // Oops, Newton-Raphson method shot out of the interval.
1136 // Restore, and try again with the other side instead!
1143 ex dx_ = ff.subs(x == xx[side]).evalf();
1144 if (!is_a<numeric>(dx_))
1145 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1146 xx[side] += ex_to<numeric>(dx_);
1148 ex f_x = f.subs(x==xx[side]).evalf();
1149 if (!is_a<numeric>(f_x))
1150 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1151 fx[side] = ex_to<numeric>(f_x);
1153 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1154 // Oops, the root isn't bracketed any more.
1155 // Restore, and perform a bisection!
1159 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1160 // they occur pretty often when Newton-Raphson arrives at an x too
1161 // close to the result on one side of the interval and
1162 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1163 // precision errors! Recall that this function does not have a
1164 // precision goal as one of its arguments but instead relies on
1165 // x converging to a fixed point. We speed up the (safe but slow)
1166 // bisection method by mixing in a dash of the (unsafer but faster)
1167 // secant method: Instead of splitting the interval at the
1168 // arithmetic mean (bisection), we split it nearer to the root as
1169 // determined by the secant between the values xx[0] and xx[1].
1170 // Don't set the secant_weight to one because that could disturb
1171 // the convergence in some corner cases!
1172 static const double secant_weight = 0.984375; // == 63/64 < 1
1173 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1174 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1175 ex fxmid_ = f.subs(x == xxmid).evalf();
1176 if (!is_a<numeric>(fxmid_))
1177 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1178 numeric fxmid = ex_to<numeric>(fxmid_);
1179 if (fxmid.is_zero()) {
1183 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1191 } while (xxprev!=xx[side]);
1196 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1197 * for static lib (so ginsh will see them). */
1198 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1199 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1201 } // namespace GiNaC