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 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 void abs_print_latex(const ex & arg, const print_context & c)
259 c.s << "{|"; arg.print(c); c.s << "|}";
262 static void abs_print_csrc_float(const ex & arg, const print_context & c)
264 c.s << "fabs("; arg.print(c); c.s << ")";
267 static ex abs_conjugate(const ex & arg)
269 return abs(arg).hold();
272 static ex abs_real_part(const ex & arg)
274 return abs(arg).hold();
277 static ex abs_imag_part(const ex& arg)
282 static ex abs_power(const ex & arg, const ex & exp)
284 if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
285 || exp.info(info_flags::even)))
286 return power(arg, exp);
288 return power(abs(arg), exp).hold();
291 bool abs_info(const ex & arg, unsigned inf)
294 case info_flags::integer:
295 case info_flags::even:
296 case info_flags::odd:
297 case info_flags::prime:
298 return arg.info(inf);
299 case info_flags::nonnegint:
300 return arg.info(info_flags::integer);
301 case info_flags::nonnegative:
302 case info_flags::real:
304 case info_flags::negative:
306 case info_flags::positive:
307 return arg.info(info_flags::positive) || arg.info(info_flags::negative);
308 case info_flags::has_indices: {
309 if (arg.info(info_flags::has_indices))
318 REGISTER_FUNCTION(abs, eval_func(abs_eval).
319 evalf_func(abs_evalf).
321 print_func<print_latex>(abs_print_latex).
322 print_func<print_csrc_float>(abs_print_csrc_float).
323 print_func<print_csrc_double>(abs_print_csrc_float).
324 conjugate_func(abs_conjugate).
325 real_part_func(abs_real_part).
326 imag_part_func(abs_imag_part).
327 power_func(abs_power));
333 static ex step_evalf(const ex & arg)
335 if (is_exactly_a<numeric>(arg))
336 return step(ex_to<numeric>(arg));
338 return step(arg).hold();
341 static ex step_eval(const ex & arg)
343 if (is_exactly_a<numeric>(arg))
344 return step(ex_to<numeric>(arg));
346 else if (is_exactly_a<mul>(arg) &&
347 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
348 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
351 // step(42*x) -> step(x)
352 return step(arg/oc).hold();
354 // step(-42*x) -> step(-x)
355 return step(-arg/oc).hold();
357 if (oc.real().is_zero()) {
359 // step(42*I*x) -> step(I*x)
360 return step(I*arg/oc).hold();
362 // step(-42*I*x) -> step(-I*x)
363 return step(-I*arg/oc).hold();
367 return step(arg).hold();
370 static ex step_series(const ex & arg,
371 const relational & rel,
375 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
376 if (arg_pt.info(info_flags::numeric)
377 && ex_to<numeric>(arg_pt).real().is_zero()
378 && !(options & series_options::suppress_branchcut))
379 throw (std::domain_error("step_series(): on imaginary axis"));
382 seq.push_back(expair(step(arg_pt), _ex0));
383 return pseries(rel,seq);
386 static ex step_conjugate(const ex& arg)
388 return step(arg).hold();
391 static ex step_real_part(const ex& arg)
393 return step(arg).hold();
396 static ex step_imag_part(const ex& arg)
401 REGISTER_FUNCTION(step, eval_func(step_eval).
402 evalf_func(step_evalf).
403 series_func(step_series).
404 conjugate_func(step_conjugate).
405 real_part_func(step_real_part).
406 imag_part_func(step_imag_part));
412 static ex csgn_evalf(const ex & arg)
414 if (is_exactly_a<numeric>(arg))
415 return csgn(ex_to<numeric>(arg));
417 return csgn(arg).hold();
420 static ex csgn_eval(const ex & arg)
422 if (is_exactly_a<numeric>(arg))
423 return csgn(ex_to<numeric>(arg));
425 else if (is_exactly_a<mul>(arg) &&
426 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
427 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
430 // csgn(42*x) -> csgn(x)
431 return csgn(arg/oc).hold();
433 // csgn(-42*x) -> -csgn(x)
434 return -csgn(arg/oc).hold();
436 if (oc.real().is_zero()) {
438 // csgn(42*I*x) -> csgn(I*x)
439 return csgn(I*arg/oc).hold();
441 // csgn(-42*I*x) -> -csgn(I*x)
442 return -csgn(I*arg/oc).hold();
446 return csgn(arg).hold();
449 static ex csgn_series(const ex & arg,
450 const relational & rel,
454 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
455 if (arg_pt.info(info_flags::numeric)
456 && ex_to<numeric>(arg_pt).real().is_zero()
457 && !(options & series_options::suppress_branchcut))
458 throw (std::domain_error("csgn_series(): on imaginary axis"));
461 seq.push_back(expair(csgn(arg_pt), _ex0));
462 return pseries(rel,seq);
465 static ex csgn_conjugate(const ex& arg)
467 return csgn(arg).hold();
470 static ex csgn_real_part(const ex& arg)
472 return csgn(arg).hold();
475 static ex csgn_imag_part(const ex& arg)
480 static ex csgn_power(const ex & arg, const ex & exp)
482 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
483 if (ex_to<numeric>(exp).is_odd())
484 return csgn(arg).hold();
486 return power(csgn(arg), _ex2).hold();
488 return power(csgn(arg), exp).hold();
492 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
493 evalf_func(csgn_evalf).
494 series_func(csgn_series).
495 conjugate_func(csgn_conjugate).
496 real_part_func(csgn_real_part).
497 imag_part_func(csgn_imag_part).
498 power_func(csgn_power));
502 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
503 // This function is closely related to the unwinding number K, sometimes found
504 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
507 static ex eta_evalf(const ex &x, const ex &y)
509 // It seems like we basically have to replicate the eval function here,
510 // since the expression might not be fully evaluated yet.
511 if (x.info(info_flags::positive) || y.info(info_flags::positive))
514 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
515 const numeric nx = ex_to<numeric>(x);
516 const numeric ny = ex_to<numeric>(y);
517 const numeric nxy = ex_to<numeric>(x*y);
519 if (nx.is_real() && nx.is_negative())
521 if (ny.is_real() && ny.is_negative())
523 if (nxy.is_real() && nxy.is_negative())
525 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
526 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
529 return eta(x,y).hold();
532 static ex eta_eval(const ex &x, const ex &y)
534 // trivial: eta(x,c) -> 0 if c is real and positive
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 // don't call eta_evalf here because it would call Pi.evalf()!
540 const numeric nx = ex_to<numeric>(x);
541 const numeric ny = ex_to<numeric>(y);
542 const numeric nxy = ex_to<numeric>(x*y);
544 if (nx.is_real() && nx.is_negative())
546 if (ny.is_real() && ny.is_negative())
548 if (nxy.is_real() && nxy.is_negative())
550 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
551 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
554 return eta(x,y).hold();
557 static ex eta_series(const ex & x, const ex & y,
558 const relational & rel,
562 const ex x_pt = x.subs(rel, subs_options::no_pattern);
563 const ex y_pt = y.subs(rel, subs_options::no_pattern);
564 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
565 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
566 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
567 throw (std::domain_error("eta_series(): on discontinuity"));
569 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
570 return pseries(rel,seq);
573 static ex eta_conjugate(const ex & x, const ex & y)
575 return -eta(x, y).hold();
578 static ex eta_real_part(const ex & x, const ex & y)
583 static ex eta_imag_part(const ex & x, const ex & y)
585 return -I*eta(x, y).hold();
588 REGISTER_FUNCTION(eta, eval_func(eta_eval).
589 evalf_func(eta_evalf).
590 series_func(eta_series).
592 set_symmetry(sy_symm(0, 1)).
593 conjugate_func(eta_conjugate).
594 real_part_func(eta_real_part).
595 imag_part_func(eta_imag_part));
602 static ex Li2_evalf(const ex & x)
604 if (is_exactly_a<numeric>(x))
605 return Li2(ex_to<numeric>(x));
607 return Li2(x).hold();
610 static ex Li2_eval(const ex & x)
612 if (x.info(info_flags::numeric)) {
617 if (x.is_equal(_ex1))
618 return power(Pi,_ex2)/_ex6;
619 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
620 if (x.is_equal(_ex1_2))
621 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
622 // Li2(-1) -> -Pi^2/12
623 if (x.is_equal(_ex_1))
624 return -power(Pi,_ex2)/_ex12;
625 // Li2(I) -> -Pi^2/48+Catalan*I
627 return power(Pi,_ex2)/_ex_48 + Catalan*I;
628 // Li2(-I) -> -Pi^2/48-Catalan*I
630 return power(Pi,_ex2)/_ex_48 - Catalan*I;
632 if (!x.info(info_flags::crational))
633 return Li2(ex_to<numeric>(x));
636 return Li2(x).hold();
639 static ex Li2_deriv(const ex & x, unsigned deriv_param)
641 GINAC_ASSERT(deriv_param==0);
643 // d/dx Li2(x) -> -log(1-x)/x
644 return -log(_ex1-x)/x;
647 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
649 const ex x_pt = x.subs(rel, subs_options::no_pattern);
650 if (x_pt.info(info_flags::numeric)) {
651 // First special case: x==0 (derivatives have poles)
652 if (x_pt.is_zero()) {
654 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
655 // simply substitute x==0. The limit, however, exists: it is 1.
656 // We also know all higher derivatives' limits:
657 // (d/dx)^n Li2(x) == n!/n^2.
658 // So the primitive series expansion is
659 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
661 // We first construct such a primitive series expansion manually in
662 // a dummy symbol s and then insert the argument's series expansion
663 // for s. Reexpanding the resulting series returns the desired
667 // manually construct the primitive expansion
668 for (int i=1; i<order; ++i)
669 ser += pow(s,i) / pow(numeric(i), *_num2_p);
670 // substitute the argument's series expansion
671 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
672 // maybe that was terminating, so add a proper order term
674 nseq.push_back(expair(Order(_ex1), order));
675 ser += pseries(rel, nseq);
676 // reexpanding it will collapse the series again
677 return ser.series(rel, order);
678 // NB: Of course, this still does not allow us to compute anything
679 // like sin(Li2(x)).series(x==0,2), since then this code here is
680 // not reached and the derivative of sin(Li2(x)) doesn't allow the
681 // substitution x==0. Probably limits *are* needed for the general
682 // cases. In case L'Hospital's rule is implemented for limits and
683 // basic::series() takes care of this, this whole block is probably
686 // second special case: x==1 (branch point)
687 if (x_pt.is_equal(_ex1)) {
689 // construct series manually in a dummy symbol s
692 // manually construct the primitive expansion
693 for (int i=1; i<order; ++i)
694 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
695 // substitute the argument's series expansion
696 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
697 // maybe that was terminating, so add a proper order term
699 nseq.push_back(expair(Order(_ex1), order));
700 ser += pseries(rel, nseq);
701 // reexpanding it will collapse the series again
702 return ser.series(rel, order);
704 // third special case: x real, >=1 (branch cut)
705 if (!(options & series_options::suppress_branchcut) &&
706 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
708 // This is the branch cut: assemble the primitive series manually
709 // and then add the corresponding complex step function.
710 const symbol &s = ex_to<symbol>(rel.lhs());
711 const ex point = rel.rhs();
714 // zeroth order term:
715 seq.push_back(expair(Li2(x_pt), _ex0));
716 // compute the intermediate terms:
717 ex replarg = series(Li2(x), s==foo, order);
718 for (size_t i=1; i<replarg.nops()-1; ++i)
719 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));
720 // append an order term:
721 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
722 return pseries(rel, seq);
725 // all other cases should be safe, by now:
726 throw do_taylor(); // caught by function::series()
729 static ex Li2_conjugate(const ex & x)
731 // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
732 // run along the positive real axis beginning at 1.
733 if (x.info(info_flags::negative)) {
734 return Li2(x).hold();
736 if (is_exactly_a<numeric>(x) &&
737 (!x.imag_part().is_zero() || x < *_num1_p)) {
738 return Li2(x.conjugate());
740 return conjugate_function(Li2(x)).hold();
743 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
744 evalf_func(Li2_evalf).
745 derivative_func(Li2_deriv).
746 series_func(Li2_series).
747 conjugate_func(Li2_conjugate).
748 latex_name("\\mathrm{Li}_2"));
754 static ex Li3_eval(const ex & x)
758 return Li3(x).hold();
761 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
762 latex_name("\\mathrm{Li}_3"));
765 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
768 static ex zetaderiv_eval(const ex & n, const ex & x)
770 if (n.info(info_flags::numeric)) {
771 // zetaderiv(0,x) -> zeta(x)
773 return zeta(x).hold();
776 return zetaderiv(n, x).hold();
779 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
781 GINAC_ASSERT(deriv_param<2);
783 if (deriv_param==0) {
785 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
788 return zetaderiv(n+1,x);
791 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
792 derivative_func(zetaderiv_deriv).
793 latex_name("\\zeta^\\prime"));
799 static ex factorial_evalf(const ex & x)
801 return factorial(x).hold();
804 static ex factorial_eval(const ex & x)
806 if (is_exactly_a<numeric>(x))
807 return factorial(ex_to<numeric>(x));
809 return factorial(x).hold();
812 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
814 if (is_exactly_a<symbol>(x) ||
815 is_exactly_a<constant>(x) ||
816 is_exactly_a<function>(x)) {
817 x.print(c); c.s << "!";
819 c.s << "("; x.print(c); c.s << ")!";
823 static ex factorial_conjugate(const ex & x)
825 return factorial(x).hold();
828 static ex factorial_real_part(const ex & x)
830 return factorial(x).hold();
833 static ex factorial_imag_part(const ex & x)
838 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
839 evalf_func(factorial_evalf).
840 print_func<print_dflt>(factorial_print_dflt_latex).
841 print_func<print_latex>(factorial_print_dflt_latex).
842 conjugate_func(factorial_conjugate).
843 real_part_func(factorial_real_part).
844 imag_part_func(factorial_imag_part));
850 static ex binomial_evalf(const ex & x, const ex & y)
852 return binomial(x, y).hold();
855 static ex binomial_sym(const ex & x, const numeric & y)
857 if (y.is_integer()) {
858 if (y.is_nonneg_integer()) {
859 const unsigned N = y.to_int();
860 if (N == 0) return _ex1;
861 if (N == 1) return x;
863 for (unsigned i = 2; i <= N; ++i)
864 t = (t * (x + i - y - 1)).expand() / i;
870 return binomial(x, y).hold();
873 static ex binomial_eval(const ex & x, const ex &y)
875 if (is_exactly_a<numeric>(y)) {
876 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
877 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
879 return binomial_sym(x, ex_to<numeric>(y));
881 return binomial(x, y).hold();
884 // At the moment the numeric evaluation of a binomail function always
885 // gives a real number, but if this would be implemented using the gamma
886 // function, also complex conjugation should be changed (or rather, deleted).
887 static ex binomial_conjugate(const ex & x, const ex & y)
889 return binomial(x,y).hold();
892 static ex binomial_real_part(const ex & x, const ex & y)
894 return binomial(x,y).hold();
897 static ex binomial_imag_part(const ex & x, const ex & y)
902 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
903 evalf_func(binomial_evalf).
904 conjugate_func(binomial_conjugate).
905 real_part_func(binomial_real_part).
906 imag_part_func(binomial_imag_part));
909 // Order term function (for truncated power series)
912 static ex Order_eval(const ex & x)
914 if (is_exactly_a<numeric>(x)) {
917 return Order(_ex1).hold();
920 } else if (is_exactly_a<mul>(x)) {
921 const mul &m = ex_to<mul>(x);
922 // O(c*expr) -> O(expr)
923 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
924 return Order(x / m.op(m.nops() - 1)).hold();
926 return Order(x).hold();
929 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
931 // Just wrap the function into a pseries object
933 GINAC_ASSERT(is_a<symbol>(r.lhs()));
934 const symbol &s = ex_to<symbol>(r.lhs());
935 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
936 return pseries(r, new_seq);
939 static ex Order_conjugate(const ex & x)
941 return Order(x).hold();
944 static ex Order_real_part(const ex & x)
946 return Order(x).hold();
949 static ex Order_imag_part(const ex & x)
951 if(x.info(info_flags::real))
953 return Order(x).hold();
956 // Differentiation is handled in function::derivative because of its special requirements
958 REGISTER_FUNCTION(Order, eval_func(Order_eval).
959 series_func(Order_series).
960 latex_name("\\mathcal{O}").
961 conjugate_func(Order_conjugate).
962 real_part_func(Order_real_part).
963 imag_part_func(Order_imag_part));
966 // Solve linear system
969 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
971 // solve a system of linear equations
972 if (eqns.info(info_flags::relation_equal)) {
973 if (!symbols.info(info_flags::symbol))
974 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
975 const ex sol = lsolve(lst(eqns),lst(symbols));
977 GINAC_ASSERT(sol.nops()==1);
978 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
980 return sol.op(0).op(1); // return rhs of first solution
984 if (!eqns.info(info_flags::list)) {
985 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
987 for (size_t i=0; i<eqns.nops(); i++) {
988 if (!eqns.op(i).info(info_flags::relation_equal)) {
989 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
992 if (!symbols.info(info_flags::list)) {
993 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
995 for (size_t i=0; i<symbols.nops(); i++) {
996 if (!symbols.op(i).info(info_flags::symbol)) {
997 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
1001 // build matrix from equation system
1002 matrix sys(eqns.nops(),symbols.nops());
1003 matrix rhs(eqns.nops(),1);
1004 matrix vars(symbols.nops(),1);
1006 for (size_t r=0; r<eqns.nops(); r++) {
1007 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
1009 for (size_t c=0; c<symbols.nops(); c++) {
1010 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
1011 linpart -= co*symbols.op(c);
1014 linpart = linpart.expand();
1015 rhs(r,0) = -linpart;
1018 // test if system is linear and fill vars matrix
1019 for (size_t i=0; i<symbols.nops(); i++) {
1020 vars(i,0) = symbols.op(i);
1021 if (sys.has(symbols.op(i)))
1022 throw(std::logic_error("lsolve: system is not linear"));
1023 if (rhs.has(symbols.op(i)))
1024 throw(std::logic_error("lsolve: system is not linear"));
1029 solution = sys.solve(vars,rhs,options);
1030 } catch (const std::runtime_error & e) {
1031 // Probably singular matrix or otherwise overdetermined system:
1032 // It is consistent to return an empty list
1035 GINAC_ASSERT(solution.cols()==1);
1036 GINAC_ASSERT(solution.rows()==symbols.nops());
1038 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
1040 for (size_t i=0; i<symbols.nops(); i++)
1041 sollist.append(symbols.op(i)==solution(i,0));
1047 // Find real root of f(x) numerically
1051 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
1053 if (!x1.is_real() || !x2.is_real()) {
1054 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
1057 throw std::runtime_error("fsolve(): vanishing interval");
1059 // xx[0] == left interval limit, xx[1] == right interval limit.
1060 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
1061 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
1062 numeric xx[2] = { x1<x2 ? x1 : x2,
1065 if (is_a<relational>(f_in)) {
1066 f = f_in.lhs()-f_in.rhs();
1070 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
1071 f.subs(x==xx[1]).evalf() };
1072 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1073 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1075 numeric fx[2] = { ex_to<numeric>(fx_[0]),
1076 ex_to<numeric>(fx_[1]) };
1077 if (!fx[0].is_real() || !fx[1].is_real()) {
1078 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1080 if (fx[0]*fx[1]>=0) {
1081 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1084 // The Newton-Raphson method has quadratic convergence! Simply put, it
1085 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
1086 const ex ff = normal(-f/f.diff(x));
1087 int side = 0; // Start at left interval limit.
1093 ex dx_ = ff.subs(x == xx[side]).evalf();
1094 if (!is_a<numeric>(dx_))
1095 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1096 xx[side] += ex_to<numeric>(dx_);
1097 // Now check if Newton-Raphson method shot out of the interval
1098 bool bad_shot = (side == 0 && xx[0] < xxprev) ||
1099 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1101 // Compute f(x) only if new x is inside the interval.
1102 // The function might be difficult to compute numerically
1103 // or even ill defined outside the interval. Also it's
1104 // a small optimization.
1105 ex f_x = f.subs(x == xx[side]).evalf();
1106 if (!is_a<numeric>(f_x))
1107 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1108 fx[side] = ex_to<numeric>(f_x);
1111 // Oops, Newton-Raphson method shot out of the interval.
1112 // Restore, and try again with the other side instead!
1119 ex dx_ = ff.subs(x == xx[side]).evalf();
1120 if (!is_a<numeric>(dx_))
1121 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1122 xx[side] += ex_to<numeric>(dx_);
1124 ex f_x = f.subs(x==xx[side]).evalf();
1125 if (!is_a<numeric>(f_x))
1126 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1127 fx[side] = ex_to<numeric>(f_x);
1129 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1130 // Oops, the root isn't bracketed any more.
1131 // Restore, and perform a bisection!
1135 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1136 // they occur pretty often when Newton-Raphson arrives at an x too
1137 // close to the result on one side of the interval and
1138 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1139 // precision errors! Recall that this function does not have a
1140 // precision goal as one of its arguments but instead relies on
1141 // x converging to a fixed point. We speed up the (safe but slow)
1142 // bisection method by mixing in a dash of the (unsafer but faster)
1143 // secant method: Instead of splitting the interval at the
1144 // arithmetic mean (bisection), we split it nearer to the root as
1145 // determined by the secant between the values xx[0] and xx[1].
1146 // Don't set the secant_weight to one because that could disturb
1147 // the convergence in some corner cases!
1148 static const double secant_weight = 0.984375; // == 63/64 < 1
1149 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1150 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1151 ex fxmid_ = f.subs(x == xxmid).evalf();
1152 if (!is_a<numeric>(fxmid_))
1153 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1154 numeric fxmid = ex_to<numeric>(fxmid_);
1155 if (fxmid.is_zero()) {
1159 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1167 } while (xxprev!=xx[side]);
1172 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1173 * for static lib (so ginsh will see them). */
1174 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1175 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1177 } // namespace GiNaC