3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2015 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
27 #include "fderivative.h"
31 #include "operators.h"
32 #include "relational.h"
47 static ex conjugate_evalf(const ex & arg)
49 if (is_exactly_a<numeric>(arg)) {
50 return ex_to<numeric>(arg).conjugate();
52 return conjugate_function(arg).hold();
55 static ex conjugate_eval(const ex & arg)
57 return arg.conjugate();
60 static void conjugate_print_latex(const ex & arg, const print_context & c)
62 c.s << "\\bar{"; arg.print(c); c.s << "}";
65 static ex conjugate_conjugate(const ex & arg)
70 // If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x)
71 // and conjugate((U+I*V).diff(x))
72 static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
74 if (s.info(info_flags::real))
75 return conjugate(arg.diff(s));
78 vec_arg.push_back(arg);
79 return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
83 static ex conjugate_real_part(const ex & arg)
85 return arg.real_part();
88 static ex conjugate_imag_part(const ex & arg)
90 return -arg.imag_part();
93 static bool func_arg_info(const ex & arg, unsigned inf)
95 // for some functions we can return the info() of its argument
96 // (think of conjugate())
98 case info_flags::polynomial:
99 case info_flags::integer_polynomial:
100 case info_flags::cinteger_polynomial:
101 case info_flags::rational_polynomial:
102 case info_flags::real:
103 case info_flags::rational:
104 case info_flags::integer:
105 case info_flags::crational:
106 case info_flags::cinteger:
107 case info_flags::even:
108 case info_flags::odd:
109 case info_flags::prime:
110 case info_flags::crational_polynomial:
111 case info_flags::rational_function:
112 case info_flags::algebraic:
113 case info_flags::positive:
114 case info_flags::negative:
115 case info_flags::nonnegative:
116 case info_flags::posint:
117 case info_flags::negint:
118 case info_flags::nonnegint:
119 case info_flags::has_indices:
120 return arg.info(inf);
125 static bool conjugate_info(const ex & arg, unsigned inf)
127 return func_arg_info(arg, inf);
130 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
131 evalf_func(conjugate_evalf).
132 expl_derivative_func(conjugate_expl_derivative).
133 info_func(conjugate_info).
134 print_func<print_latex>(conjugate_print_latex).
135 conjugate_func(conjugate_conjugate).
136 real_part_func(conjugate_real_part).
137 imag_part_func(conjugate_imag_part).
138 set_name("conjugate","conjugate"));
144 static ex real_part_evalf(const ex & arg)
146 if (is_exactly_a<numeric>(arg)) {
147 return ex_to<numeric>(arg).real();
149 return real_part_function(arg).hold();
152 static ex real_part_eval(const ex & arg)
154 return arg.real_part();
157 static void real_part_print_latex(const ex & arg, const print_context & c)
159 c.s << "\\Re"; arg.print(c); c.s << "";
162 static ex real_part_conjugate(const ex & arg)
164 return real_part_function(arg).hold();
167 static ex real_part_real_part(const ex & arg)
169 return real_part_function(arg).hold();
172 static ex real_part_imag_part(const ex & arg)
177 // If x is real then Re(e).diff(x) is equal to Re(e.diff(x))
178 static ex real_part_expl_derivative(const ex & arg, const symbol & s)
180 if (s.info(info_flags::real))
181 return real_part_function(arg.diff(s));
184 vec_arg.push_back(arg);
185 return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
189 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
190 evalf_func(real_part_evalf).
191 expl_derivative_func(real_part_expl_derivative).
192 print_func<print_latex>(real_part_print_latex).
193 conjugate_func(real_part_conjugate).
194 real_part_func(real_part_real_part).
195 imag_part_func(real_part_imag_part).
196 set_name("real_part","real_part"));
202 static ex imag_part_evalf(const ex & arg)
204 if (is_exactly_a<numeric>(arg)) {
205 return ex_to<numeric>(arg).imag();
207 return imag_part_function(arg).hold();
210 static ex imag_part_eval(const ex & arg)
212 return arg.imag_part();
215 static void imag_part_print_latex(const ex & arg, const print_context & c)
217 c.s << "\\Im"; arg.print(c); c.s << "";
220 static ex imag_part_conjugate(const ex & arg)
222 return imag_part_function(arg).hold();
225 static ex imag_part_real_part(const ex & arg)
227 return imag_part_function(arg).hold();
230 static ex imag_part_imag_part(const ex & arg)
235 // If x is real then Im(e).diff(x) is equal to Im(e.diff(x))
236 static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
238 if (s.info(info_flags::real))
239 return imag_part_function(arg.diff(s));
242 vec_arg.push_back(arg);
243 return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
247 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
248 evalf_func(imag_part_evalf).
249 expl_derivative_func(imag_part_expl_derivative).
250 print_func<print_latex>(imag_part_print_latex).
251 conjugate_func(imag_part_conjugate).
252 real_part_func(imag_part_real_part).
253 imag_part_func(imag_part_imag_part).
254 set_name("imag_part","imag_part"));
260 static ex abs_evalf(const ex & arg)
262 if (is_exactly_a<numeric>(arg))
263 return abs(ex_to<numeric>(arg));
265 return abs(arg).hold();
268 static ex abs_eval(const ex & arg)
270 if (is_exactly_a<numeric>(arg))
271 return abs(ex_to<numeric>(arg));
273 if (arg.info(info_flags::nonnegative))
276 if (is_ex_the_function(arg, abs))
279 if (is_ex_the_function(arg, exp))
280 return exp(arg.op(0).real_part());
282 if (is_exactly_a<power>(arg)) {
283 const ex& base = arg.op(0);
284 const ex& exponent = arg.op(1);
285 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
286 return pow(abs(base), exponent.real_part());
289 if (is_ex_the_function(arg, conjugate_function))
290 return abs(arg.op(0));
292 if (is_ex_the_function(arg, step))
295 return abs(arg).hold();
298 static ex abs_expand(const ex & arg, unsigned options)
300 if ((options & expand_options::expand_transcendental)
301 && is_exactly_a<mul>(arg)) {
303 prodseq.reserve(arg.nops());
304 for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
305 if (options & expand_options::expand_function_args)
306 prodseq.push_back(abs(i->expand(options)));
308 prodseq.push_back(abs(*i));
310 return (new mul(prodseq))->setflag(status_flags::dynallocated | status_flags::expanded);
313 if (options & expand_options::expand_function_args)
314 return abs(arg.expand(options)).hold();
316 return abs(arg).hold();
319 static ex abs_expl_derivative(const ex & arg, const symbol & s)
321 ex diff_arg = arg.diff(s);
322 return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
325 static void abs_print_latex(const ex & arg, const print_context & c)
327 c.s << "{|"; arg.print(c); c.s << "|}";
330 static void abs_print_csrc_float(const ex & arg, const print_context & c)
332 c.s << "fabs("; arg.print(c); c.s << ")";
335 static ex abs_conjugate(const ex & arg)
337 return abs(arg).hold();
340 static ex abs_real_part(const ex & arg)
342 return abs(arg).hold();
345 static ex abs_imag_part(const ex& arg)
350 static ex abs_power(const ex & arg, const ex & exp)
352 if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
353 if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
354 return power(arg, exp);
356 return power(arg, exp/2)*power(arg.conjugate(), exp/2);
358 return power(abs(arg), exp).hold();
361 bool abs_info(const ex & arg, unsigned inf)
364 case info_flags::integer:
365 case info_flags::even:
366 case info_flags::odd:
367 case info_flags::prime:
368 return arg.info(inf);
369 case info_flags::nonnegint:
370 return arg.info(info_flags::integer);
371 case info_flags::nonnegative:
372 case info_flags::real:
374 case info_flags::negative:
376 case info_flags::positive:
377 return arg.info(info_flags::positive) || arg.info(info_flags::negative);
378 case info_flags::has_indices: {
379 if (arg.info(info_flags::has_indices))
388 REGISTER_FUNCTION(abs, eval_func(abs_eval).
389 evalf_func(abs_evalf).
390 expand_func(abs_expand).
391 expl_derivative_func(abs_expl_derivative).
393 print_func<print_latex>(abs_print_latex).
394 print_func<print_csrc_float>(abs_print_csrc_float).
395 print_func<print_csrc_double>(abs_print_csrc_float).
396 conjugate_func(abs_conjugate).
397 real_part_func(abs_real_part).
398 imag_part_func(abs_imag_part).
399 power_func(abs_power));
405 static ex step_evalf(const ex & arg)
407 if (is_exactly_a<numeric>(arg))
408 return step(ex_to<numeric>(arg));
410 return step(arg).hold();
413 static ex step_eval(const ex & arg)
415 if (is_exactly_a<numeric>(arg))
416 return step(ex_to<numeric>(arg));
418 else if (is_exactly_a<mul>(arg) &&
419 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
420 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
423 // step(42*x) -> step(x)
424 return step(arg/oc).hold();
426 // step(-42*x) -> step(-x)
427 return step(-arg/oc).hold();
429 if (oc.real().is_zero()) {
431 // step(42*I*x) -> step(I*x)
432 return step(I*arg/oc).hold();
434 // step(-42*I*x) -> step(-I*x)
435 return step(-I*arg/oc).hold();
439 return step(arg).hold();
442 static ex step_series(const ex & arg,
443 const relational & rel,
447 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
448 if (arg_pt.info(info_flags::numeric)
449 && ex_to<numeric>(arg_pt).real().is_zero()
450 && !(options & series_options::suppress_branchcut))
451 throw (std::domain_error("step_series(): on imaginary axis"));
454 seq.push_back(expair(step(arg_pt), _ex0));
455 return pseries(rel,seq);
458 static ex step_conjugate(const ex& arg)
460 return step(arg).hold();
463 static ex step_real_part(const ex& arg)
465 return step(arg).hold();
468 static ex step_imag_part(const ex& arg)
473 REGISTER_FUNCTION(step, eval_func(step_eval).
474 evalf_func(step_evalf).
475 series_func(step_series).
476 conjugate_func(step_conjugate).
477 real_part_func(step_real_part).
478 imag_part_func(step_imag_part));
484 static ex csgn_evalf(const ex & arg)
486 if (is_exactly_a<numeric>(arg))
487 return csgn(ex_to<numeric>(arg));
489 return csgn(arg).hold();
492 static ex csgn_eval(const ex & arg)
494 if (is_exactly_a<numeric>(arg))
495 return csgn(ex_to<numeric>(arg));
497 else if (is_exactly_a<mul>(arg) &&
498 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
499 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
502 // csgn(42*x) -> csgn(x)
503 return csgn(arg/oc).hold();
505 // csgn(-42*x) -> -csgn(x)
506 return -csgn(arg/oc).hold();
508 if (oc.real().is_zero()) {
510 // csgn(42*I*x) -> csgn(I*x)
511 return csgn(I*arg/oc).hold();
513 // csgn(-42*I*x) -> -csgn(I*x)
514 return -csgn(I*arg/oc).hold();
518 return csgn(arg).hold();
521 static ex csgn_series(const ex & arg,
522 const relational & rel,
526 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
527 if (arg_pt.info(info_flags::numeric)
528 && ex_to<numeric>(arg_pt).real().is_zero()
529 && !(options & series_options::suppress_branchcut))
530 throw (std::domain_error("csgn_series(): on imaginary axis"));
533 seq.push_back(expair(csgn(arg_pt), _ex0));
534 return pseries(rel,seq);
537 static ex csgn_conjugate(const ex& arg)
539 return csgn(arg).hold();
542 static ex csgn_real_part(const ex& arg)
544 return csgn(arg).hold();
547 static ex csgn_imag_part(const ex& arg)
552 static ex csgn_power(const ex & arg, const ex & exp)
554 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
555 if (ex_to<numeric>(exp).is_odd())
556 return csgn(arg).hold();
558 return power(csgn(arg), _ex2).hold();
560 return power(csgn(arg), exp).hold();
564 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
565 evalf_func(csgn_evalf).
566 series_func(csgn_series).
567 conjugate_func(csgn_conjugate).
568 real_part_func(csgn_real_part).
569 imag_part_func(csgn_imag_part).
570 power_func(csgn_power));
574 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
575 // This function is closely related to the unwinding number K, sometimes found
576 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
579 static ex eta_evalf(const ex &x, const ex &y)
581 // It seems like we basically have to replicate the eval function here,
582 // since the expression might not be fully evaluated yet.
583 if (x.info(info_flags::positive) || y.info(info_flags::positive))
586 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
587 const numeric nx = ex_to<numeric>(x);
588 const numeric ny = ex_to<numeric>(y);
589 const numeric nxy = ex_to<numeric>(x*y);
591 if (nx.is_real() && nx.is_negative())
593 if (ny.is_real() && ny.is_negative())
595 if (nxy.is_real() && nxy.is_negative())
597 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
598 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
601 return eta(x,y).hold();
604 static ex eta_eval(const ex &x, const ex &y)
606 // trivial: eta(x,c) -> 0 if c is real and positive
607 if (x.info(info_flags::positive) || y.info(info_flags::positive))
610 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
611 // don't call eta_evalf here because it would call Pi.evalf()!
612 const numeric nx = ex_to<numeric>(x);
613 const numeric ny = ex_to<numeric>(y);
614 const numeric nxy = ex_to<numeric>(x*y);
616 if (nx.is_real() && nx.is_negative())
618 if (ny.is_real() && ny.is_negative())
620 if (nxy.is_real() && nxy.is_negative())
622 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
623 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
626 return eta(x,y).hold();
629 static ex eta_series(const ex & x, const ex & y,
630 const relational & rel,
634 const ex x_pt = x.subs(rel, subs_options::no_pattern);
635 const ex y_pt = y.subs(rel, subs_options::no_pattern);
636 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
637 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
638 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
639 throw (std::domain_error("eta_series(): on discontinuity"));
641 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
642 return pseries(rel,seq);
645 static ex eta_conjugate(const ex & x, const ex & y)
647 return -eta(x, y).hold();
650 static ex eta_real_part(const ex & x, const ex & y)
655 static ex eta_imag_part(const ex & x, const ex & y)
657 return -I*eta(x, y).hold();
660 REGISTER_FUNCTION(eta, eval_func(eta_eval).
661 evalf_func(eta_evalf).
662 series_func(eta_series).
664 set_symmetry(sy_symm(0, 1)).
665 conjugate_func(eta_conjugate).
666 real_part_func(eta_real_part).
667 imag_part_func(eta_imag_part));
674 static ex Li2_evalf(const ex & x)
676 if (is_exactly_a<numeric>(x))
677 return Li2(ex_to<numeric>(x));
679 return Li2(x).hold();
682 static ex Li2_eval(const ex & x)
684 if (x.info(info_flags::numeric)) {
689 if (x.is_equal(_ex1))
690 return power(Pi,_ex2)/_ex6;
691 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
692 if (x.is_equal(_ex1_2))
693 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
694 // Li2(-1) -> -Pi^2/12
695 if (x.is_equal(_ex_1))
696 return -power(Pi,_ex2)/_ex12;
697 // Li2(I) -> -Pi^2/48+Catalan*I
699 return power(Pi,_ex2)/_ex_48 + Catalan*I;
700 // Li2(-I) -> -Pi^2/48-Catalan*I
702 return power(Pi,_ex2)/_ex_48 - Catalan*I;
704 if (!x.info(info_flags::crational))
705 return Li2(ex_to<numeric>(x));
708 return Li2(x).hold();
711 static ex Li2_deriv(const ex & x, unsigned deriv_param)
713 GINAC_ASSERT(deriv_param==0);
715 // d/dx Li2(x) -> -log(1-x)/x
716 return -log(_ex1-x)/x;
719 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
721 const ex x_pt = x.subs(rel, subs_options::no_pattern);
722 if (x_pt.info(info_flags::numeric)) {
723 // First special case: x==0 (derivatives have poles)
724 if (x_pt.is_zero()) {
726 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
727 // simply substitute x==0. The limit, however, exists: it is 1.
728 // We also know all higher derivatives' limits:
729 // (d/dx)^n Li2(x) == n!/n^2.
730 // So the primitive series expansion is
731 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
733 // We first construct such a primitive series expansion manually in
734 // a dummy symbol s and then insert the argument's series expansion
735 // for s. Reexpanding the resulting series returns the desired
739 // manually construct the primitive expansion
740 for (int i=1; i<order; ++i)
741 ser += pow(s,i) / pow(numeric(i), *_num2_p);
742 // substitute the argument's series expansion
743 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
744 // maybe that was terminating, so add a proper order term
746 nseq.push_back(expair(Order(_ex1), order));
747 ser += pseries(rel, nseq);
748 // reexpanding it will collapse the series again
749 return ser.series(rel, order);
750 // NB: Of course, this still does not allow us to compute anything
751 // like sin(Li2(x)).series(x==0,2), since then this code here is
752 // not reached and the derivative of sin(Li2(x)) doesn't allow the
753 // substitution x==0. Probably limits *are* needed for the general
754 // cases. In case L'Hospital's rule is implemented for limits and
755 // basic::series() takes care of this, this whole block is probably
758 // second special case: x==1 (branch point)
759 if (x_pt.is_equal(_ex1)) {
761 // construct series manually in a dummy symbol s
764 // manually construct the primitive expansion
765 for (int i=1; i<order; ++i)
766 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
767 // substitute the argument's series expansion
768 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
769 // maybe that was terminating, so add a proper order term
771 nseq.push_back(expair(Order(_ex1), order));
772 ser += pseries(rel, nseq);
773 // reexpanding it will collapse the series again
774 return ser.series(rel, order);
776 // third special case: x real, >=1 (branch cut)
777 if (!(options & series_options::suppress_branchcut) &&
778 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
780 // This is the branch cut: assemble the primitive series manually
781 // and then add the corresponding complex step function.
782 const symbol &s = ex_to<symbol>(rel.lhs());
783 const ex point = rel.rhs();
786 // zeroth order term:
787 seq.push_back(expair(Li2(x_pt), _ex0));
788 // compute the intermediate terms:
789 ex replarg = series(Li2(x), s==foo, order);
790 for (size_t i=1; i<replarg.nops()-1; ++i)
791 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));
792 // append an order term:
793 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
794 return pseries(rel, seq);
797 // all other cases should be safe, by now:
798 throw do_taylor(); // caught by function::series()
801 static ex Li2_conjugate(const ex & x)
803 // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
804 // run along the positive real axis beginning at 1.
805 if (x.info(info_flags::negative)) {
806 return Li2(x).hold();
808 if (is_exactly_a<numeric>(x) &&
809 (!x.imag_part().is_zero() || x < *_num1_p)) {
810 return Li2(x.conjugate());
812 return conjugate_function(Li2(x)).hold();
815 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
816 evalf_func(Li2_evalf).
817 derivative_func(Li2_deriv).
818 series_func(Li2_series).
819 conjugate_func(Li2_conjugate).
820 latex_name("\\mathrm{Li}_2"));
826 static ex Li3_eval(const ex & x)
830 return Li3(x).hold();
833 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
834 latex_name("\\mathrm{Li}_3"));
837 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
840 static ex zetaderiv_eval(const ex & n, const ex & x)
842 if (n.info(info_flags::numeric)) {
843 // zetaderiv(0,x) -> zeta(x)
845 return zeta(x).hold();
848 return zetaderiv(n, x).hold();
851 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
853 GINAC_ASSERT(deriv_param<2);
855 if (deriv_param==0) {
857 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
860 return zetaderiv(n+1,x);
863 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
864 derivative_func(zetaderiv_deriv).
865 latex_name("\\zeta^\\prime"));
871 static ex factorial_evalf(const ex & x)
873 return factorial(x).hold();
876 static ex factorial_eval(const ex & x)
878 if (is_exactly_a<numeric>(x))
879 return factorial(ex_to<numeric>(x));
881 return factorial(x).hold();
884 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
886 if (is_exactly_a<symbol>(x) ||
887 is_exactly_a<constant>(x) ||
888 is_exactly_a<function>(x)) {
889 x.print(c); c.s << "!";
891 c.s << "("; x.print(c); c.s << ")!";
895 static ex factorial_conjugate(const ex & x)
897 return factorial(x).hold();
900 static ex factorial_real_part(const ex & x)
902 return factorial(x).hold();
905 static ex factorial_imag_part(const ex & x)
910 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
911 evalf_func(factorial_evalf).
912 print_func<print_dflt>(factorial_print_dflt_latex).
913 print_func<print_latex>(factorial_print_dflt_latex).
914 conjugate_func(factorial_conjugate).
915 real_part_func(factorial_real_part).
916 imag_part_func(factorial_imag_part));
922 static ex binomial_evalf(const ex & x, const ex & y)
924 return binomial(x, y).hold();
927 static ex binomial_sym(const ex & x, const numeric & y)
929 if (y.is_integer()) {
930 if (y.is_nonneg_integer()) {
931 const unsigned N = y.to_int();
932 if (N == 0) return _ex1;
933 if (N == 1) return x;
935 for (unsigned i = 2; i <= N; ++i)
936 t = (t * (x + i - y - 1)).expand() / i;
942 return binomial(x, y).hold();
945 static ex binomial_eval(const ex & x, const ex &y)
947 if (is_exactly_a<numeric>(y)) {
948 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
949 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
951 return binomial_sym(x, ex_to<numeric>(y));
953 return binomial(x, y).hold();
956 // At the moment the numeric evaluation of a binomail function always
957 // gives a real number, but if this would be implemented using the gamma
958 // function, also complex conjugation should be changed (or rather, deleted).
959 static ex binomial_conjugate(const ex & x, const ex & y)
961 return binomial(x,y).hold();
964 static ex binomial_real_part(const ex & x, const ex & y)
966 return binomial(x,y).hold();
969 static ex binomial_imag_part(const ex & x, const ex & y)
974 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
975 evalf_func(binomial_evalf).
976 conjugate_func(binomial_conjugate).
977 real_part_func(binomial_real_part).
978 imag_part_func(binomial_imag_part));
981 // Order term function (for truncated power series)
984 static ex Order_eval(const ex & x)
986 if (is_exactly_a<numeric>(x)) {
989 return Order(_ex1).hold();
992 } else if (is_exactly_a<mul>(x)) {
993 const mul &m = ex_to<mul>(x);
994 // O(c*expr) -> O(expr)
995 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
996 return Order(x / m.op(m.nops() - 1)).hold();
998 return Order(x).hold();
1001 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
1003 // Just wrap the function into a pseries object
1005 GINAC_ASSERT(is_a<symbol>(r.lhs()));
1006 const symbol &s = ex_to<symbol>(r.lhs());
1007 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
1008 return pseries(r, new_seq);
1011 static ex Order_conjugate(const ex & x)
1013 return Order(x).hold();
1016 static ex Order_real_part(const ex & x)
1018 return Order(x).hold();
1021 static ex Order_imag_part(const ex & x)
1023 if(x.info(info_flags::real))
1025 return Order(x).hold();
1028 static ex Order_expl_derivative(const ex & arg, const symbol & s)
1030 return Order(arg.diff(s));
1033 REGISTER_FUNCTION(Order, eval_func(Order_eval).
1034 series_func(Order_series).
1035 latex_name("\\mathcal{O}").
1036 expl_derivative_func(Order_expl_derivative).
1037 conjugate_func(Order_conjugate).
1038 real_part_func(Order_real_part).
1039 imag_part_func(Order_imag_part));
1042 // Solve linear system
1045 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
1047 // solve a system of linear equations
1048 if (eqns.info(info_flags::relation_equal)) {
1049 if (!symbols.info(info_flags::symbol))
1050 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
1051 const ex sol = lsolve(lst(eqns),lst(symbols));
1053 GINAC_ASSERT(sol.nops()==1);
1054 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
1056 return sol.op(0).op(1); // return rhs of first solution
1060 if (!eqns.info(info_flags::list)) {
1061 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
1063 for (size_t i=0; i<eqns.nops(); i++) {
1064 if (!eqns.op(i).info(info_flags::relation_equal)) {
1065 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
1068 if (!symbols.info(info_flags::list)) {
1069 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
1071 for (size_t i=0; i<symbols.nops(); i++) {
1072 if (!symbols.op(i).info(info_flags::symbol)) {
1073 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1077 // build matrix from equation system
1078 matrix sys(eqns.nops(),symbols.nops());
1079 matrix rhs(eqns.nops(),1);
1080 matrix vars(symbols.nops(),1);
1082 for (size_t r=0; r<eqns.nops(); r++) {
1083 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1085 for (size_t c=0; c<symbols.nops(); c++) {
1086 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1087 linpart -= co*symbols.op(c);
1090 linpart = linpart.expand();
1091 rhs(r,0) = -linpart;
1094 // test if system is linear and fill vars matrix
1095 for (size_t i=0; i<symbols.nops(); i++) {
1096 vars(i,0) = symbols.op(i);
1097 if (sys.has(symbols.op(i)))
1098 throw(std::logic_error("lsolve: system is not linear"));
1099 if (rhs.has(symbols.op(i)))
1100 throw(std::logic_error("lsolve: system is not linear"));
1105 solution = sys.solve(vars,rhs,options);
1106 } catch (const std::runtime_error & e) {
1107 // Probably singular matrix or otherwise overdetermined system:
1108 // It is consistent to return an empty list
1111 GINAC_ASSERT(solution.cols()==1);
1112 GINAC_ASSERT(solution.rows()==symbols.nops());
1114 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1116 for (size_t i=0; i<symbols.nops(); i++)
1117 sollist.append(symbols.op(i)==solution(i,0));
1123 // Find real root of f(x) numerically
1127 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1129 if (!x1.is_real() || !x2.is_real()) {
1130 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1133 throw std::runtime_error("fsolve(): vanishing interval");
1135 // xx[0] == left interval limit, xx[1] == right interval limit.
1136 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1137 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1138 numeric xx[2] = { x1<x2 ? x1 : x2,
1141 if (is_a<relational>(f_in)) {
1142 f = f_in.lhs()-f_in.rhs();
1146 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1147 f.subs(x==xx[1]).evalf() };
1148 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1149 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1151 numeric fx[2] = { ex_to<numeric>(fx_[0]),
1152 ex_to<numeric>(fx_[1]) };
1153 if (!fx[0].is_real() || !fx[1].is_real()) {
1154 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1156 if (fx[0]*fx[1]>=0) {
1157 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1160 // The Newton-Raphson method has quadratic convergence! Simply put, it
1161 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
1162 const ex ff = normal(-f/f.diff(x));
1163 int side = 0; // Start at left interval limit.
1169 ex dx_ = ff.subs(x == xx[side]).evalf();
1170 if (!is_a<numeric>(dx_))
1171 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1172 xx[side] += ex_to<numeric>(dx_);
1173 // Now check if Newton-Raphson method shot out of the interval
1174 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1175 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1177 // Compute f(x) only if new x is inside the interval.
1178 // The function might be difficult to compute numerically
1179 // or even ill defined outside the interval. Also it's
1180 // a small optimization.
1181 ex f_x = f.subs(x == xx[side]).evalf();
1182 if (!is_a<numeric>(f_x))
1183 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1184 fx[side] = ex_to<numeric>(f_x);
1187 // Oops, Newton-Raphson method shot out of the interval.
1188 // Restore, and try again with the other side instead!
1195 ex dx_ = ff.subs(x == xx[side]).evalf();
1196 if (!is_a<numeric>(dx_))
1197 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1198 xx[side] += ex_to<numeric>(dx_);
1200 ex f_x = f.subs(x==xx[side]).evalf();
1201 if (!is_a<numeric>(f_x))
1202 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1203 fx[side] = ex_to<numeric>(f_x);
1205 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1206 // Oops, the root isn't bracketed any more.
1207 // Restore, and perform a bisection!
1211 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1212 // they occur pretty often when Newton-Raphson arrives at an x too
1213 // close to the result on one side of the interval and
1214 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1215 // precision errors! Recall that this function does not have a
1216 // precision goal as one of its arguments but instead relies on
1217 // x converging to a fixed point. We speed up the (safe but slow)
1218 // bisection method by mixing in a dash of the (unsafer but faster)
1219 // secant method: Instead of splitting the interval at the
1220 // arithmetic mean (bisection), we split it nearer to the root as
1221 // determined by the secant between the values xx[0] and xx[1].
1222 // Don't set the secant_weight to one because that could disturb
1223 // the convergence in some corner cases!
1224 static const double secant_weight = 0.984375; // == 63/64 < 1
1225 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1226 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1227 ex fxmid_ = f.subs(x == xxmid).evalf();
1228 if (!is_a<numeric>(fxmid_))
1229 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1230 numeric fxmid = ex_to<numeric>(fxmid_);
1231 if (fxmid.is_zero()) {
1235 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1243 } while (xxprev!=xx[side]);
1248 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1249 * for static lib (so ginsh will see them). */
1250 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1251 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1253 } // namespace GiNaC