3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2008 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
33 #include "operators.h"
34 #include "relational.h"
46 static ex conjugate_evalf(const ex & arg)
48 if (is_exactly_a<numeric>(arg)) {
49 return ex_to<numeric>(arg).conjugate();
51 return conjugate_function(arg).hold();
54 static ex conjugate_eval(const ex & arg)
56 return arg.conjugate();
59 static void conjugate_print_latex(const ex & arg, const print_context & c)
61 c.s << "\\bar{"; arg.print(c); c.s << "}";
64 static ex conjugate_conjugate(const ex & arg)
69 static ex conjugate_real_part(const ex & arg)
71 return arg.real_part();
74 static ex conjugate_imag_part(const ex & arg)
76 return -arg.imag_part();
79 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
80 evalf_func(conjugate_evalf).
81 print_func<print_latex>(conjugate_print_latex).
82 conjugate_func(conjugate_conjugate).
83 real_part_func(conjugate_real_part).
84 imag_part_func(conjugate_imag_part).
85 set_name("conjugate","conjugate"));
91 static ex real_part_evalf(const ex & arg)
93 if (is_exactly_a<numeric>(arg)) {
94 return ex_to<numeric>(arg).real();
96 return real_part_function(arg).hold();
99 static ex real_part_eval(const ex & arg)
101 return arg.real_part();
104 static void real_part_print_latex(const ex & arg, const print_context & c)
106 c.s << "\\Re"; arg.print(c); c.s << "";
109 static ex real_part_conjugate(const ex & arg)
111 return real_part_function(arg).hold();
114 static ex real_part_real_part(const ex & arg)
116 return real_part_function(arg).hold();
119 static ex real_part_imag_part(const ex & arg)
124 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
125 evalf_func(real_part_evalf).
126 print_func<print_latex>(real_part_print_latex).
127 conjugate_func(real_part_conjugate).
128 real_part_func(real_part_real_part).
129 imag_part_func(real_part_imag_part).
130 set_name("real_part","real_part"));
136 static ex imag_part_evalf(const ex & arg)
138 if (is_exactly_a<numeric>(arg)) {
139 return ex_to<numeric>(arg).imag();
141 return imag_part_function(arg).hold();
144 static ex imag_part_eval(const ex & arg)
146 return arg.imag_part();
149 static void imag_part_print_latex(const ex & arg, const print_context & c)
151 c.s << "\\Im"; arg.print(c); c.s << "";
154 static ex imag_part_conjugate(const ex & arg)
156 return imag_part_function(arg).hold();
159 static ex imag_part_real_part(const ex & arg)
161 return imag_part_function(arg).hold();
164 static ex imag_part_imag_part(const ex & arg)
169 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
170 evalf_func(imag_part_evalf).
171 print_func<print_latex>(imag_part_print_latex).
172 conjugate_func(imag_part_conjugate).
173 real_part_func(imag_part_real_part).
174 imag_part_func(imag_part_imag_part).
175 set_name("imag_part","imag_part"));
181 static ex abs_evalf(const ex & arg)
183 if (is_exactly_a<numeric>(arg))
184 return abs(ex_to<numeric>(arg));
186 return abs(arg).hold();
189 static ex abs_eval(const ex & arg)
191 if (is_exactly_a<numeric>(arg))
192 return abs(ex_to<numeric>(arg));
194 if (arg.info(info_flags::nonnegative))
197 if (is_ex_the_function(arg, abs))
200 return abs(arg).hold();
203 static void abs_print_latex(const ex & arg, const print_context & c)
205 c.s << "{|"; arg.print(c); c.s << "|}";
208 static void abs_print_csrc_float(const ex & arg, const print_context & c)
210 c.s << "fabs("; arg.print(c); c.s << ")";
213 static ex abs_conjugate(const ex & arg)
218 static ex abs_real_part(const ex & arg)
220 return abs(arg).hold();
223 static ex abs_imag_part(const ex& arg)
228 static ex abs_power(const ex & arg, const ex & exp)
230 if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
231 return power(arg, exp);
233 return power(abs(arg), exp).hold();
236 REGISTER_FUNCTION(abs, eval_func(abs_eval).
237 evalf_func(abs_evalf).
238 print_func<print_latex>(abs_print_latex).
239 print_func<print_csrc_float>(abs_print_csrc_float).
240 print_func<print_csrc_double>(abs_print_csrc_float).
241 conjugate_func(abs_conjugate).
242 real_part_func(abs_real_part).
243 imag_part_func(abs_imag_part).
244 power_func(abs_power));
250 static ex step_evalf(const ex & arg)
252 if (is_exactly_a<numeric>(arg))
253 return step(ex_to<numeric>(arg));
255 return step(arg).hold();
258 static ex step_eval(const ex & arg)
260 if (is_exactly_a<numeric>(arg))
261 return step(ex_to<numeric>(arg));
263 else if (is_exactly_a<mul>(arg) &&
264 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
265 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
268 // step(42*x) -> step(x)
269 return step(arg/oc).hold();
271 // step(-42*x) -> step(-x)
272 return step(-arg/oc).hold();
274 if (oc.real().is_zero()) {
276 // step(42*I*x) -> step(I*x)
277 return step(I*arg/oc).hold();
279 // step(-42*I*x) -> step(-I*x)
280 return step(-I*arg/oc).hold();
284 return step(arg).hold();
287 static ex step_series(const ex & arg,
288 const relational & rel,
292 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
293 if (arg_pt.info(info_flags::numeric)
294 && ex_to<numeric>(arg_pt).real().is_zero()
295 && !(options & series_options::suppress_branchcut))
296 throw (std::domain_error("step_series(): on imaginary axis"));
299 seq.push_back(expair(step(arg_pt), _ex0));
300 return pseries(rel,seq);
303 static ex step_conjugate(const ex& arg)
305 return step(arg).hold();
308 static ex step_real_part(const ex& arg)
310 return step(arg).hold();
313 static ex step_imag_part(const ex& arg)
318 REGISTER_FUNCTION(step, eval_func(step_eval).
319 evalf_func(step_evalf).
320 series_func(step_series).
321 conjugate_func(step_conjugate).
322 real_part_func(step_real_part).
323 imag_part_func(step_imag_part));
329 static ex csgn_evalf(const ex & arg)
331 if (is_exactly_a<numeric>(arg))
332 return csgn(ex_to<numeric>(arg));
334 return csgn(arg).hold();
337 static ex csgn_eval(const ex & arg)
339 if (is_exactly_a<numeric>(arg))
340 return csgn(ex_to<numeric>(arg));
342 else if (is_exactly_a<mul>(arg) &&
343 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
344 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
347 // csgn(42*x) -> csgn(x)
348 return csgn(arg/oc).hold();
350 // csgn(-42*x) -> -csgn(x)
351 return -csgn(arg/oc).hold();
353 if (oc.real().is_zero()) {
355 // csgn(42*I*x) -> csgn(I*x)
356 return csgn(I*arg/oc).hold();
358 // csgn(-42*I*x) -> -csgn(I*x)
359 return -csgn(I*arg/oc).hold();
363 return csgn(arg).hold();
366 static ex csgn_series(const ex & arg,
367 const relational & rel,
371 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
372 if (arg_pt.info(info_flags::numeric)
373 && ex_to<numeric>(arg_pt).real().is_zero()
374 && !(options & series_options::suppress_branchcut))
375 throw (std::domain_error("csgn_series(): on imaginary axis"));
378 seq.push_back(expair(csgn(arg_pt), _ex0));
379 return pseries(rel,seq);
382 static ex csgn_conjugate(const ex& arg)
384 return csgn(arg).hold();
387 static ex csgn_real_part(const ex& arg)
389 return csgn(arg).hold();
392 static ex csgn_imag_part(const ex& arg)
397 static ex csgn_power(const ex & arg, const ex & exp)
399 if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
400 if (ex_to<numeric>(exp).is_odd())
403 return power(csgn(arg), _ex2).hold();
405 return power(csgn(arg), exp).hold();
409 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
410 evalf_func(csgn_evalf).
411 series_func(csgn_series).
412 conjugate_func(csgn_conjugate).
413 real_part_func(csgn_real_part).
414 imag_part_func(csgn_imag_part).
415 power_func(csgn_power));
419 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
420 // This function is closely related to the unwinding number K, sometimes found
421 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
424 static ex eta_evalf(const ex &x, const ex &y)
426 // It seems like we basically have to replicate the eval function here,
427 // since the expression might not be fully evaluated yet.
428 if (x.info(info_flags::positive) || y.info(info_flags::positive))
431 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
432 const numeric nx = ex_to<numeric>(x);
433 const numeric ny = ex_to<numeric>(y);
434 const numeric nxy = ex_to<numeric>(x*y);
436 if (nx.is_real() && nx.is_negative())
438 if (ny.is_real() && ny.is_negative())
440 if (nxy.is_real() && nxy.is_negative())
442 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
443 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
446 return eta(x,y).hold();
449 static ex eta_eval(const ex &x, const ex &y)
451 // trivial: eta(x,c) -> 0 if c is real and positive
452 if (x.info(info_flags::positive) || y.info(info_flags::positive))
455 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
456 // don't call eta_evalf here because it would call Pi.evalf()!
457 const numeric nx = ex_to<numeric>(x);
458 const numeric ny = ex_to<numeric>(y);
459 const numeric nxy = ex_to<numeric>(x*y);
461 if (nx.is_real() && nx.is_negative())
463 if (ny.is_real() && ny.is_negative())
465 if (nxy.is_real() && nxy.is_negative())
467 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
468 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
471 return eta(x,y).hold();
474 static ex eta_series(const ex & x, const ex & y,
475 const relational & rel,
479 const ex x_pt = x.subs(rel, subs_options::no_pattern);
480 const ex y_pt = y.subs(rel, subs_options::no_pattern);
481 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
482 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
483 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
484 throw (std::domain_error("eta_series(): on discontinuity"));
486 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
487 return pseries(rel,seq);
490 static ex eta_conjugate(const ex & x, const ex & y)
495 static ex eta_real_part(const ex & x, const ex & y)
500 static ex eta_imag_part(const ex & x, const ex & y)
502 return -I*eta(x, y).hold();
505 REGISTER_FUNCTION(eta, eval_func(eta_eval).
506 evalf_func(eta_evalf).
507 series_func(eta_series).
509 set_symmetry(sy_symm(0, 1)).
510 conjugate_func(eta_conjugate).
511 real_part_func(eta_real_part).
512 imag_part_func(eta_imag_part));
519 static ex Li2_evalf(const ex & x)
521 if (is_exactly_a<numeric>(x))
522 return Li2(ex_to<numeric>(x));
524 return Li2(x).hold();
527 static ex Li2_eval(const ex & x)
529 if (x.info(info_flags::numeric)) {
534 if (x.is_equal(_ex1))
535 return power(Pi,_ex2)/_ex6;
536 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
537 if (x.is_equal(_ex1_2))
538 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
539 // Li2(-1) -> -Pi^2/12
540 if (x.is_equal(_ex_1))
541 return -power(Pi,_ex2)/_ex12;
542 // Li2(I) -> -Pi^2/48+Catalan*I
544 return power(Pi,_ex2)/_ex_48 + Catalan*I;
545 // Li2(-I) -> -Pi^2/48-Catalan*I
547 return power(Pi,_ex2)/_ex_48 - Catalan*I;
549 if (!x.info(info_flags::crational))
550 return Li2(ex_to<numeric>(x));
553 return Li2(x).hold();
556 static ex Li2_deriv(const ex & x, unsigned deriv_param)
558 GINAC_ASSERT(deriv_param==0);
560 // d/dx Li2(x) -> -log(1-x)/x
561 return -log(_ex1-x)/x;
564 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
566 const ex x_pt = x.subs(rel, subs_options::no_pattern);
567 if (x_pt.info(info_flags::numeric)) {
568 // First special case: x==0 (derivatives have poles)
569 if (x_pt.is_zero()) {
571 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
572 // simply substitute x==0. The limit, however, exists: it is 1.
573 // We also know all higher derivatives' limits:
574 // (d/dx)^n Li2(x) == n!/n^2.
575 // So the primitive series expansion is
576 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
578 // We first construct such a primitive series expansion manually in
579 // a dummy symbol s and then insert the argument's series expansion
580 // for s. Reexpanding the resulting series returns the desired
584 // manually construct the primitive expansion
585 for (int i=1; i<order; ++i)
586 ser += pow(s,i) / pow(numeric(i), *_num2_p);
587 // substitute the argument's series expansion
588 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
589 // maybe that was terminating, so add a proper order term
591 nseq.push_back(expair(Order(_ex1), order));
592 ser += pseries(rel, nseq);
593 // reexpanding it will collapse the series again
594 return ser.series(rel, order);
595 // NB: Of course, this still does not allow us to compute anything
596 // like sin(Li2(x)).series(x==0,2), since then this code here is
597 // not reached and the derivative of sin(Li2(x)) doesn't allow the
598 // substitution x==0. Probably limits *are* needed for the general
599 // cases. In case L'Hospital's rule is implemented for limits and
600 // basic::series() takes care of this, this whole block is probably
603 // second special case: x==1 (branch point)
604 if (x_pt.is_equal(_ex1)) {
606 // construct series manually in a dummy symbol s
609 // manually construct the primitive expansion
610 for (int i=1; i<order; ++i)
611 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
612 // substitute the argument's series expansion
613 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
614 // maybe that was terminating, so add a proper order term
616 nseq.push_back(expair(Order(_ex1), order));
617 ser += pseries(rel, nseq);
618 // reexpanding it will collapse the series again
619 return ser.series(rel, order);
621 // third special case: x real, >=1 (branch cut)
622 if (!(options & series_options::suppress_branchcut) &&
623 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
625 // This is the branch cut: assemble the primitive series manually
626 // and then add the corresponding complex step function.
627 const symbol &s = ex_to<symbol>(rel.lhs());
628 const ex point = rel.rhs();
631 // zeroth order term:
632 seq.push_back(expair(Li2(x_pt), _ex0));
633 // compute the intermediate terms:
634 ex replarg = series(Li2(x), s==foo, order);
635 for (size_t i=1; i<replarg.nops()-1; ++i)
636 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));
637 // append an order term:
638 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
639 return pseries(rel, seq);
642 // all other cases should be safe, by now:
643 throw do_taylor(); // caught by function::series()
646 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
647 evalf_func(Li2_evalf).
648 derivative_func(Li2_deriv).
649 series_func(Li2_series).
650 latex_name("\\mbox{Li}_2"));
656 static ex Li3_eval(const ex & x)
660 return Li3(x).hold();
663 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
664 latex_name("\\mbox{Li}_3"));
667 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
670 static ex zetaderiv_eval(const ex & n, const ex & x)
672 if (n.info(info_flags::numeric)) {
673 // zetaderiv(0,x) -> zeta(x)
678 return zetaderiv(n, x).hold();
681 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
683 GINAC_ASSERT(deriv_param<2);
685 if (deriv_param==0) {
687 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
690 return zetaderiv(n+1,x);
693 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
694 derivative_func(zetaderiv_deriv).
695 latex_name("\\zeta^\\prime"));
701 static ex factorial_evalf(const ex & x)
703 return factorial(x).hold();
706 static ex factorial_eval(const ex & x)
708 if (is_exactly_a<numeric>(x))
709 return factorial(ex_to<numeric>(x));
711 return factorial(x).hold();
714 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
716 if (is_exactly_a<symbol>(x) ||
717 is_exactly_a<constant>(x) ||
718 is_exactly_a<function>(x)) {
719 x.print(c); c.s << "!";
721 c.s << "("; x.print(c); c.s << ")!";
725 static ex factorial_conjugate(const ex & x)
727 return factorial(x).hold();
730 static ex factorial_real_part(const ex & x)
732 return factorial(x).hold();
735 static ex factorial_imag_part(const ex & x)
740 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
741 evalf_func(factorial_evalf).
742 print_func<print_dflt>(factorial_print_dflt_latex).
743 print_func<print_latex>(factorial_print_dflt_latex).
744 conjugate_func(factorial_conjugate).
745 real_part_func(factorial_real_part).
746 imag_part_func(factorial_imag_part));
752 static ex binomial_evalf(const ex & x, const ex & y)
754 return binomial(x, y).hold();
757 static ex binomial_sym(const ex & x, const numeric & y)
759 if (y.is_integer()) {
760 if (y.is_nonneg_integer()) {
761 const unsigned N = y.to_int();
762 if (N == 0) return _ex0;
763 if (N == 1) return x;
765 for (unsigned i = 2; i <= N; ++i)
766 t = (t * (x + i - y - 1)).expand() / i;
772 return binomial(x, y).hold();
775 static ex binomial_eval(const ex & x, const ex &y)
777 if (is_exactly_a<numeric>(y)) {
778 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
779 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
781 return binomial_sym(x, ex_to<numeric>(y));
783 return binomial(x, y).hold();
786 // At the moment the numeric evaluation of a binomail function always
787 // gives a real number, but if this would be implemented using the gamma
788 // function, also complex conjugation should be changed (or rather, deleted).
789 static ex binomial_conjugate(const ex & x, const ex & y)
791 return binomial(x,y).hold();
794 static ex binomial_real_part(const ex & x, const ex & y)
796 return binomial(x,y).hold();
799 static ex binomial_imag_part(const ex & x, const ex & y)
804 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
805 evalf_func(binomial_evalf).
806 conjugate_func(binomial_conjugate).
807 real_part_func(binomial_real_part).
808 imag_part_func(binomial_imag_part));
811 // Order term function (for truncated power series)
814 static ex Order_eval(const ex & x)
816 if (is_exactly_a<numeric>(x)) {
819 return Order(_ex1).hold();
822 } else if (is_exactly_a<mul>(x)) {
823 const mul &m = ex_to<mul>(x);
824 // O(c*expr) -> O(expr)
825 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
826 return Order(x / m.op(m.nops() - 1)).hold();
828 return Order(x).hold();
831 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
833 // Just wrap the function into a pseries object
835 GINAC_ASSERT(is_a<symbol>(r.lhs()));
836 const symbol &s = ex_to<symbol>(r.lhs());
837 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
838 return pseries(r, new_seq);
841 static ex Order_conjugate(const ex & x)
843 return Order(x).hold();
846 static ex Order_real_part(const ex & x)
848 return Order(x).hold();
851 static ex Order_imag_part(const ex & x)
853 if(x.info(info_flags::real))
855 return Order(x).hold();
858 // Differentiation is handled in function::derivative because of its special requirements
860 REGISTER_FUNCTION(Order, eval_func(Order_eval).
861 series_func(Order_series).
862 latex_name("\\mathcal{O}").
863 conjugate_func(Order_conjugate).
864 real_part_func(Order_real_part).
865 imag_part_func(Order_imag_part));
868 // Solve linear system
871 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
873 // solve a system of linear equations
874 if (eqns.info(info_flags::relation_equal)) {
875 if (!symbols.info(info_flags::symbol))
876 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
877 const ex sol = lsolve(lst(eqns),lst(symbols));
879 GINAC_ASSERT(sol.nops()==1);
880 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
882 return sol.op(0).op(1); // return rhs of first solution
886 if (!eqns.info(info_flags::list)) {
887 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
889 for (size_t i=0; i<eqns.nops(); i++) {
890 if (!eqns.op(i).info(info_flags::relation_equal)) {
891 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
894 if (!symbols.info(info_flags::list)) {
895 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
897 for (size_t i=0; i<symbols.nops(); i++) {
898 if (!symbols.op(i).info(info_flags::symbol)) {
899 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
903 // build matrix from equation system
904 matrix sys(eqns.nops(),symbols.nops());
905 matrix rhs(eqns.nops(),1);
906 matrix vars(symbols.nops(),1);
908 for (size_t r=0; r<eqns.nops(); r++) {
909 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
911 for (size_t c=0; c<symbols.nops(); c++) {
912 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
913 linpart -= co*symbols.op(c);
916 linpart = linpart.expand();
920 // test if system is linear and fill vars matrix
921 for (size_t i=0; i<symbols.nops(); i++) {
922 vars(i,0) = symbols.op(i);
923 if (sys.has(symbols.op(i)))
924 throw(std::logic_error("lsolve: system is not linear"));
925 if (rhs.has(symbols.op(i)))
926 throw(std::logic_error("lsolve: system is not linear"));
931 solution = sys.solve(vars,rhs,options);
932 } catch (const std::runtime_error & e) {
933 // Probably singular matrix or otherwise overdetermined system:
934 // It is consistent to return an empty list
937 GINAC_ASSERT(solution.cols()==1);
938 GINAC_ASSERT(solution.rows()==symbols.nops());
940 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
942 for (size_t i=0; i<symbols.nops(); i++)
943 sollist.append(symbols.op(i)==solution(i,0));
949 // Find real root of f(x) numerically
953 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
955 if (!x1.is_real() || !x2.is_real()) {
956 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
959 throw std::runtime_error("fsolve(): vanishing interval");
961 // xx[0] == left interval limit, xx[1] == right interval limit.
962 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
963 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
964 numeric xx[2] = { x1<x2 ? x1 : x2,
967 if (is_a<relational>(f_in)) {
968 f = f_in.lhs()-f_in.rhs();
972 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
973 f.subs(x==xx[1]).evalf() };
974 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
975 throw std::runtime_error("fsolve(): function does not evaluate numerically");
977 numeric fx[2] = { ex_to<numeric>(fx_[0]),
978 ex_to<numeric>(fx_[1]) };
979 if (!fx[0].is_real() || !fx[1].is_real()) {
980 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
982 if (fx[0]*fx[1]>=0) {
983 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
986 // The Newton-Raphson method has quadratic convergence! Simply put, it
987 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
988 const ex ff = normal(-f/f.diff(x));
989 int side = 0; // Start at left interval limit.
995 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
996 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
997 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
998 // Oops, Newton-Raphson method shot out of the interval.
999 // Restore, and try again with the other side instead!
1005 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
1006 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
1008 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1009 // Oops, the root isn't bracketed any more.
1010 // Restore, and perform a bisection!
1014 // Ah, the bisection! Bisections converge linearly. Unfortunately,
1015 // they occur pretty often when Newton-Raphson arrives at an x too
1016 // close to the result on one side of the interval and
1017 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1018 // precision errors! Recall that this function does not have a
1019 // precision goal as one of its arguments but instead relies on
1020 // x converging to a fixed point. We speed up the (safe but slow)
1021 // bisection method by mixing in a dash of the (unsafer but faster)
1022 // secant method: Instead of splitting the interval at the
1023 // arithmetic mean (bisection), we split it nearer to the root as
1024 // determined by the secant between the values xx[0] and xx[1].
1025 // Don't set the secant_weight to one because that could disturb
1026 // the convergence in some corner cases!
1027 static const double secant_weight = 0.984375; // == 63/64 < 1
1028 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1029 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1030 numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
1031 if (fxmid.is_zero()) {
1035 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1043 } while (xxprev!=xx[side]);
1048 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1049 * for static lib (so ginsh will see them). */
1050 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1051 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1053 } // namespace GiNaC