3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2005 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 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
70 evalf_func(conjugate_evalf).
71 print_func<print_latex>(conjugate_print_latex).
72 conjugate_func(conjugate_conjugate).
73 set_name("conjugate","conjugate"));
79 static ex abs_evalf(const ex & arg)
81 if (is_exactly_a<numeric>(arg))
82 return abs(ex_to<numeric>(arg));
84 return abs(arg).hold();
87 static ex abs_eval(const ex & arg)
89 if (is_exactly_a<numeric>(arg))
90 return abs(ex_to<numeric>(arg));
92 return abs(arg).hold();
95 static void abs_print_latex(const ex & arg, const print_context & c)
97 c.s << "{|"; arg.print(c); c.s << "|}";
100 static void abs_print_csrc_float(const ex & arg, const print_context & c)
102 c.s << "fabs("; arg.print(c); c.s << ")";
105 static ex abs_conjugate(const ex & arg)
110 static ex abs_power(const ex & arg, const ex & exp)
112 if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
113 return power(arg, exp);
115 return power(abs(arg), exp).hold();
118 REGISTER_FUNCTION(abs, eval_func(abs_eval).
119 evalf_func(abs_evalf).
120 print_func<print_latex>(abs_print_latex).
121 print_func<print_csrc_float>(abs_print_csrc_float).
122 print_func<print_csrc_double>(abs_print_csrc_float).
123 conjugate_func(abs_conjugate).
124 power_func(abs_power));
130 static ex csgn_evalf(const ex & arg)
132 if (is_exactly_a<numeric>(arg))
133 return csgn(ex_to<numeric>(arg));
135 return csgn(arg).hold();
138 static ex csgn_eval(const ex & arg)
140 if (is_exactly_a<numeric>(arg))
141 return csgn(ex_to<numeric>(arg));
143 else if (is_exactly_a<mul>(arg) &&
144 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
145 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
148 // csgn(42*x) -> csgn(x)
149 return csgn(arg/oc).hold();
151 // csgn(-42*x) -> -csgn(x)
152 return -csgn(arg/oc).hold();
154 if (oc.real().is_zero()) {
156 // csgn(42*I*x) -> csgn(I*x)
157 return csgn(I*arg/oc).hold();
159 // csgn(-42*I*x) -> -csgn(I*x)
160 return -csgn(I*arg/oc).hold();
164 return csgn(arg).hold();
167 static ex csgn_series(const ex & arg,
168 const relational & rel,
172 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
173 if (arg_pt.info(info_flags::numeric)
174 && ex_to<numeric>(arg_pt).real().is_zero()
175 && !(options & series_options::suppress_branchcut))
176 throw (std::domain_error("csgn_series(): on imaginary axis"));
179 seq.push_back(expair(csgn(arg_pt), _ex0));
180 return pseries(rel,seq);
183 static ex csgn_conjugate(const ex& arg)
188 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
189 evalf_func(csgn_evalf).
190 series_func(csgn_series).
191 conjugate_func(csgn_conjugate));
195 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
196 // This function is closely related to the unwinding number K, sometimes found
197 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
200 static ex eta_evalf(const ex &x, const ex &y)
202 // It seems like we basically have to replicate the eval function here,
203 // since the expression might not be fully evaluated yet.
204 if (x.info(info_flags::positive) || y.info(info_flags::positive))
207 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
208 const numeric nx = ex_to<numeric>(x);
209 const numeric ny = ex_to<numeric>(y);
210 const numeric nxy = ex_to<numeric>(x*y);
212 if (nx.is_real() && nx.is_negative())
214 if (ny.is_real() && ny.is_negative())
216 if (nxy.is_real() && nxy.is_negative())
218 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
219 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
222 return eta(x,y).hold();
225 static ex eta_eval(const ex &x, const ex &y)
227 // trivial: eta(x,c) -> 0 if c is real and positive
228 if (x.info(info_flags::positive) || y.info(info_flags::positive))
231 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
232 // don't call eta_evalf here because it would call Pi.evalf()!
233 const numeric nx = ex_to<numeric>(x);
234 const numeric ny = ex_to<numeric>(y);
235 const numeric nxy = ex_to<numeric>(x*y);
237 if (nx.is_real() && nx.is_negative())
239 if (ny.is_real() && ny.is_negative())
241 if (nxy.is_real() && nxy.is_negative())
243 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
244 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
247 return eta(x,y).hold();
250 static ex eta_series(const ex & x, const ex & y,
251 const relational & rel,
255 const ex x_pt = x.subs(rel, subs_options::no_pattern);
256 const ex y_pt = y.subs(rel, subs_options::no_pattern);
257 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
258 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
259 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
260 throw (std::domain_error("eta_series(): on discontinuity"));
262 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
263 return pseries(rel,seq);
266 static ex eta_conjugate(const ex & x, const ex & y)
271 REGISTER_FUNCTION(eta, eval_func(eta_eval).
272 evalf_func(eta_evalf).
273 series_func(eta_series).
275 set_symmetry(sy_symm(0, 1)).
276 conjugate_func(eta_conjugate));
283 static ex Li2_evalf(const ex & x)
285 if (is_exactly_a<numeric>(x))
286 return Li2(ex_to<numeric>(x));
288 return Li2(x).hold();
291 static ex Li2_eval(const ex & x)
293 if (x.info(info_flags::numeric)) {
298 if (x.is_equal(_ex1))
299 return power(Pi,_ex2)/_ex6;
300 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
301 if (x.is_equal(_ex1_2))
302 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
303 // Li2(-1) -> -Pi^2/12
304 if (x.is_equal(_ex_1))
305 return -power(Pi,_ex2)/_ex12;
306 // Li2(I) -> -Pi^2/48+Catalan*I
308 return power(Pi,_ex2)/_ex_48 + Catalan*I;
309 // Li2(-I) -> -Pi^2/48-Catalan*I
311 return power(Pi,_ex2)/_ex_48 - Catalan*I;
313 if (!x.info(info_flags::crational))
314 return Li2(ex_to<numeric>(x));
317 return Li2(x).hold();
320 static ex Li2_deriv(const ex & x, unsigned deriv_param)
322 GINAC_ASSERT(deriv_param==0);
324 // d/dx Li2(x) -> -log(1-x)/x
325 return -log(_ex1-x)/x;
328 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
330 const ex x_pt = x.subs(rel, subs_options::no_pattern);
331 if (x_pt.info(info_flags::numeric)) {
332 // First special case: x==0 (derivatives have poles)
333 if (x_pt.is_zero()) {
335 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
336 // simply substitute x==0. The limit, however, exists: it is 1.
337 // We also know all higher derivatives' limits:
338 // (d/dx)^n Li2(x) == n!/n^2.
339 // So the primitive series expansion is
340 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
342 // We first construct such a primitive series expansion manually in
343 // a dummy symbol s and then insert the argument's series expansion
344 // for s. Reexpanding the resulting series returns the desired
348 // manually construct the primitive expansion
349 for (int i=1; i<order; ++i)
350 ser += pow(s,i) / pow(numeric(i), *_num2_p);
351 // substitute the argument's series expansion
352 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
353 // maybe that was terminating, so add a proper order term
355 nseq.push_back(expair(Order(_ex1), order));
356 ser += pseries(rel, nseq);
357 // reexpanding it will collapse the series again
358 return ser.series(rel, order);
359 // NB: Of course, this still does not allow us to compute anything
360 // like sin(Li2(x)).series(x==0,2), since then this code here is
361 // not reached and the derivative of sin(Li2(x)) doesn't allow the
362 // substitution x==0. Probably limits *are* needed for the general
363 // cases. In case L'Hospital's rule is implemented for limits and
364 // basic::series() takes care of this, this whole block is probably
367 // second special case: x==1 (branch point)
368 if (x_pt.is_equal(_ex1)) {
370 // construct series manually in a dummy symbol s
373 // manually construct the primitive expansion
374 for (int i=1; i<order; ++i)
375 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
376 // substitute the argument's series expansion
377 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
378 // maybe that was terminating, so add a proper order term
380 nseq.push_back(expair(Order(_ex1), order));
381 ser += pseries(rel, nseq);
382 // reexpanding it will collapse the series again
383 return ser.series(rel, order);
385 // third special case: x real, >=1 (branch cut)
386 if (!(options & series_options::suppress_branchcut) &&
387 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
389 // This is the branch cut: assemble the primitive series manually
390 // and then add the corresponding complex step function.
391 const symbol &s = ex_to<symbol>(rel.lhs());
392 const ex point = rel.rhs();
395 // zeroth order term:
396 seq.push_back(expair(Li2(x_pt), _ex0));
397 // compute the intermediate terms:
398 ex replarg = series(Li2(x), s==foo, order);
399 for (size_t i=1; i<replarg.nops()-1; ++i)
400 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));
401 // append an order term:
402 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
403 return pseries(rel, seq);
406 // all other cases should be safe, by now:
407 throw do_taylor(); // caught by function::series()
410 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
411 evalf_func(Li2_evalf).
412 derivative_func(Li2_deriv).
413 series_func(Li2_series).
414 latex_name("\\mbox{Li}_2"));
420 static ex Li3_eval(const ex & x)
424 return Li3(x).hold();
427 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
428 latex_name("\\mbox{Li}_3"));
431 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
434 static ex zetaderiv_eval(const ex & n, const ex & x)
436 if (n.info(info_flags::numeric)) {
437 // zetaderiv(0,x) -> zeta(x)
442 return zetaderiv(n, x).hold();
445 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
447 GINAC_ASSERT(deriv_param<2);
449 if (deriv_param==0) {
451 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
454 return zetaderiv(n+1,x);
457 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
458 derivative_func(zetaderiv_deriv).
459 latex_name("\\zeta^\\prime"));
465 static ex factorial_evalf(const ex & x)
467 return factorial(x).hold();
470 static ex factorial_eval(const ex & x)
472 if (is_exactly_a<numeric>(x))
473 return factorial(ex_to<numeric>(x));
475 return factorial(x).hold();
478 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
480 if (is_exactly_a<symbol>(x) ||
481 is_exactly_a<constant>(x) ||
482 is_exactly_a<function>(x)) {
483 x.print(c); c.s << "!";
485 c.s << "("; x.print(c); c.s << ")!";
489 static ex factorial_conjugate(const ex & x)
494 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
495 evalf_func(factorial_evalf).
496 print_func<print_dflt>(factorial_print_dflt_latex).
497 print_func<print_latex>(factorial_print_dflt_latex).
498 conjugate_func(factorial_conjugate));
504 static ex binomial_evalf(const ex & x, const ex & y)
506 return binomial(x, y).hold();
509 static ex binomial_sym(const ex & x, const numeric & y)
511 if (y.is_integer()) {
512 if (y.is_nonneg_integer()) {
513 const unsigned N = y.to_int();
514 if (N == 0) return _ex0;
515 if (N == 1) return x;
517 for (unsigned i = 2; i <= N; ++i)
518 t = (t * (x + i - y - 1)).expand() / i;
524 return binomial(x, y).hold();
527 static ex binomial_eval(const ex & x, const ex &y)
529 if (is_exactly_a<numeric>(y)) {
530 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
531 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
533 return binomial_sym(x, ex_to<numeric>(y));
535 return binomial(x, y).hold();
538 // At the moment the numeric evaluation of a binomail function always
539 // gives a real number, but if this would be implemented using the gamma
540 // function, also complex conjugation should be changed (or rather, deleted).
541 static ex binomial_conjugate(const ex & x, const ex & y)
543 return binomial(x,y);
546 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
547 evalf_func(binomial_evalf).
548 conjugate_func(binomial_conjugate));
551 // Order term function (for truncated power series)
554 static ex Order_eval(const ex & x)
556 if (is_exactly_a<numeric>(x)) {
559 return Order(_ex1).hold();
562 } else if (is_exactly_a<mul>(x)) {
563 const mul &m = ex_to<mul>(x);
564 // O(c*expr) -> O(expr)
565 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
566 return Order(x / m.op(m.nops() - 1)).hold();
568 return Order(x).hold();
571 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
573 // Just wrap the function into a pseries object
575 GINAC_ASSERT(is_a<symbol>(r.lhs()));
576 const symbol &s = ex_to<symbol>(r.lhs());
577 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
578 return pseries(r, new_seq);
581 static ex Order_conjugate(const ex & x)
586 // Differentiation is handled in function::derivative because of its special requirements
588 REGISTER_FUNCTION(Order, eval_func(Order_eval).
589 series_func(Order_series).
590 latex_name("\\mathcal{O}").
591 conjugate_func(Order_conjugate));
594 // Solve linear system
597 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
599 // solve a system of linear equations
600 if (eqns.info(info_flags::relation_equal)) {
601 if (!symbols.info(info_flags::symbol))
602 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
603 const ex sol = lsolve(lst(eqns),lst(symbols));
605 GINAC_ASSERT(sol.nops()==1);
606 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
608 return sol.op(0).op(1); // return rhs of first solution
612 if (!eqns.info(info_flags::list)) {
613 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
615 for (size_t i=0; i<eqns.nops(); i++) {
616 if (!eqns.op(i).info(info_flags::relation_equal)) {
617 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
620 if (!symbols.info(info_flags::list)) {
621 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
623 for (size_t i=0; i<symbols.nops(); i++) {
624 if (!symbols.op(i).info(info_flags::symbol)) {
625 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
629 // build matrix from equation system
630 matrix sys(eqns.nops(),symbols.nops());
631 matrix rhs(eqns.nops(),1);
632 matrix vars(symbols.nops(),1);
634 for (size_t r=0; r<eqns.nops(); r++) {
635 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
637 for (size_t c=0; c<symbols.nops(); c++) {
638 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
639 linpart -= co*symbols.op(c);
642 linpart = linpart.expand();
646 // test if system is linear and fill vars matrix
647 for (size_t i=0; i<symbols.nops(); i++) {
648 vars(i,0) = symbols.op(i);
649 if (sys.has(symbols.op(i)))
650 throw(std::logic_error("lsolve: system is not linear"));
651 if (rhs.has(symbols.op(i)))
652 throw(std::logic_error("lsolve: system is not linear"));
657 solution = sys.solve(vars,rhs,options);
658 } catch (const std::runtime_error & e) {
659 // Probably singular matrix or otherwise overdetermined system:
660 // It is consistent to return an empty list
663 GINAC_ASSERT(solution.cols()==1);
664 GINAC_ASSERT(solution.rows()==symbols.nops());
666 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
668 for (size_t i=0; i<symbols.nops(); i++)
669 sollist.append(symbols.op(i)==solution(i,0));
675 // Find real root of f(x) numerically
679 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
681 if (!x1.is_real() || !x2.is_real()) {
682 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
685 throw std::runtime_error("fsolve(): vanishing interval");
687 // xx[0] == left interval limit, xx[1] == right interval limit.
688 // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
689 // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
690 numeric xx[2] = { x1<x2 ? x1 : x2,
693 if (is_a<relational>(f_in)) {
694 f = f_in.lhs()-f_in.rhs();
698 const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
699 f.subs(x==xx[1]).evalf() };
700 if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
701 throw std::runtime_error("fsolve(): function does not evaluate numerically");
703 numeric fx[2] = { ex_to<numeric>(fx_[0]),
704 ex_to<numeric>(fx_[1]) };
705 if (!fx[0].is_real() || !fx[1].is_real()) {
706 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
708 if (fx[0]*fx[1]>=0) {
709 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
712 // The Newton-Raphson method has quadratic convergence! Simply put, it
713 // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
714 const ex ff = normal(-f/f.diff(x));
715 int side = 0; // Start at left interval limit.
721 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
722 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
723 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
724 // Oops, Newton-Raphson method shot out of the interval.
725 // Restore, and try again with the other side instead!
731 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
732 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
734 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
735 // Oops, the root isn't bracketed any more.
736 // Restore, and perform a bisection!
740 // Ah, the bisection! Bisections converge linearly. Unfortunately,
741 // they occur pretty often when Newton-Raphson arrives at an x too
742 // close to the result on one side of the interval and
743 // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
744 // precision errors! Recall that this function does not have a
745 // precision goal as one of its arguments but instead relies on
746 // x converging to a fixed point. We speed up the (safe but slow)
747 // bisection method by mixing in a dash of the (unsafer but faster)
748 // secant method: Instead of splitting the interval at the
749 // arithmetic mean (bisection), we split it nearer to the root as
750 // determined by the secant between the values xx[0] and xx[1].
751 // Don't set the secant_weight to one because that could disturb
752 // the convergence in some corner cases!
753 static const double secant_weight = 0.984375; // == 63/64 < 1
754 numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
755 + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
756 numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
757 if (fxmid.is_zero()) {
761 if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
769 } while (xxprev!=xx[side]);
774 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
775 * for static lib (so ginsh will see them). */
776 unsigned force_include_tgamma = tgamma_SERIAL::serial;
777 unsigned force_include_zeta1 = zeta1_SERIAL::serial;