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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 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 REGISTER_FUNCTION(abs, eval_func(abs_eval).
111 evalf_func(abs_evalf).
112 print_func<print_latex>(abs_print_latex).
113 print_func<print_csrc_float>(abs_print_csrc_float).
114 print_func<print_csrc_double>(abs_print_csrc_float).
115 conjugate_func(abs_conjugate));
122 static ex csgn_evalf(const ex & arg)
124 if (is_exactly_a<numeric>(arg))
125 return csgn(ex_to<numeric>(arg));
127 return csgn(arg).hold();
130 static ex csgn_eval(const ex & arg)
132 if (is_exactly_a<numeric>(arg))
133 return csgn(ex_to<numeric>(arg));
135 else if (is_exactly_a<mul>(arg) &&
136 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
137 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
140 // csgn(42*x) -> csgn(x)
141 return csgn(arg/oc).hold();
143 // csgn(-42*x) -> -csgn(x)
144 return -csgn(arg/oc).hold();
146 if (oc.real().is_zero()) {
148 // csgn(42*I*x) -> csgn(I*x)
149 return csgn(I*arg/oc).hold();
151 // csgn(-42*I*x) -> -csgn(I*x)
152 return -csgn(I*arg/oc).hold();
156 return csgn(arg).hold();
159 static ex csgn_series(const ex & arg,
160 const relational & rel,
164 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
165 if (arg_pt.info(info_flags::numeric)
166 && ex_to<numeric>(arg_pt).real().is_zero()
167 && !(options & series_options::suppress_branchcut))
168 throw (std::domain_error("csgn_series(): on imaginary axis"));
171 seq.push_back(expair(csgn(arg_pt), _ex0));
172 return pseries(rel,seq);
175 static ex csgn_conjugate(const ex& arg)
180 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
181 evalf_func(csgn_evalf).
182 series_func(csgn_series).
183 conjugate_func(csgn_conjugate));
187 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
188 // This function is closely related to the unwinding number K, sometimes found
189 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
192 static ex eta_evalf(const ex &x, const ex &y)
194 // It seems like we basically have to replicate the eval function here,
195 // since the expression might not be fully evaluated yet.
196 if (x.info(info_flags::positive) || y.info(info_flags::positive))
199 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
200 const numeric nx = ex_to<numeric>(x);
201 const numeric ny = ex_to<numeric>(y);
202 const numeric nxy = ex_to<numeric>(x*y);
204 if (nx.is_real() && nx.is_negative())
206 if (ny.is_real() && ny.is_negative())
208 if (nxy.is_real() && nxy.is_negative())
210 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
211 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
214 return eta(x,y).hold();
217 static ex eta_eval(const ex &x, const ex &y)
219 // trivial: eta(x,c) -> 0 if c is real and positive
220 if (x.info(info_flags::positive) || y.info(info_flags::positive))
223 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
224 // don't call eta_evalf here because it would call Pi.evalf()!
225 const numeric nx = ex_to<numeric>(x);
226 const numeric ny = ex_to<numeric>(y);
227 const numeric nxy = ex_to<numeric>(x*y);
229 if (nx.is_real() && nx.is_negative())
231 if (ny.is_real() && ny.is_negative())
233 if (nxy.is_real() && nxy.is_negative())
235 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
236 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
239 return eta(x,y).hold();
242 static ex eta_series(const ex & x, const ex & y,
243 const relational & rel,
247 const ex x_pt = x.subs(rel, subs_options::no_pattern);
248 const ex y_pt = y.subs(rel, subs_options::no_pattern);
249 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
250 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
251 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
252 throw (std::domain_error("eta_series(): on discontinuity"));
254 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
255 return pseries(rel,seq);
258 static ex eta_conjugate(const ex & x, const ex & y)
263 REGISTER_FUNCTION(eta, eval_func(eta_eval).
264 evalf_func(eta_evalf).
265 series_func(eta_series).
267 set_symmetry(sy_symm(0, 1)).
268 conjugate_func(eta_conjugate));
275 static ex Li2_evalf(const ex & x)
277 if (is_exactly_a<numeric>(x))
278 return Li2(ex_to<numeric>(x));
280 return Li2(x).hold();
283 static ex Li2_eval(const ex & x)
285 if (x.info(info_flags::numeric)) {
290 if (x.is_equal(_ex1))
291 return power(Pi,_ex2)/_ex6;
292 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
293 if (x.is_equal(_ex1_2))
294 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
295 // Li2(-1) -> -Pi^2/12
296 if (x.is_equal(_ex_1))
297 return -power(Pi,_ex2)/_ex12;
298 // Li2(I) -> -Pi^2/48+Catalan*I
300 return power(Pi,_ex2)/_ex_48 + Catalan*I;
301 // Li2(-I) -> -Pi^2/48-Catalan*I
303 return power(Pi,_ex2)/_ex_48 - Catalan*I;
305 if (!x.info(info_flags::crational))
306 return Li2(ex_to<numeric>(x));
309 return Li2(x).hold();
312 static ex Li2_deriv(const ex & x, unsigned deriv_param)
314 GINAC_ASSERT(deriv_param==0);
316 // d/dx Li2(x) -> -log(1-x)/x
317 return -log(_ex1-x)/x;
320 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
322 const ex x_pt = x.subs(rel, subs_options::no_pattern);
323 if (x_pt.info(info_flags::numeric)) {
324 // First special case: x==0 (derivatives have poles)
325 if (x_pt.is_zero()) {
327 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
328 // simply substitute x==0. The limit, however, exists: it is 1.
329 // We also know all higher derivatives' limits:
330 // (d/dx)^n Li2(x) == n!/n^2.
331 // So the primitive series expansion is
332 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
334 // We first construct such a primitive series expansion manually in
335 // a dummy symbol s and then insert the argument's series expansion
336 // for s. Reexpanding the resulting series returns the desired
340 // manually construct the primitive expansion
341 for (int i=1; i<order; ++i)
342 ser += pow(s,i) / pow(numeric(i), _num2);
343 // substitute the argument's series expansion
344 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
345 // maybe that was terminating, so add a proper order term
347 nseq.push_back(expair(Order(_ex1), order));
348 ser += pseries(rel, nseq);
349 // reexpanding it will collapse the series again
350 return ser.series(rel, order);
351 // NB: Of course, this still does not allow us to compute anything
352 // like sin(Li2(x)).series(x==0,2), since then this code here is
353 // not reached and the derivative of sin(Li2(x)) doesn't allow the
354 // substitution x==0. Probably limits *are* needed for the general
355 // cases. In case L'Hospital's rule is implemented for limits and
356 // basic::series() takes care of this, this whole block is probably
359 // second special case: x==1 (branch point)
360 if (x_pt.is_equal(_ex1)) {
362 // construct series manually in a dummy symbol s
365 // manually construct the primitive expansion
366 for (int i=1; i<order; ++i)
367 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
368 // substitute the argument's series expansion
369 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
370 // maybe that was terminating, so add a proper order term
372 nseq.push_back(expair(Order(_ex1), order));
373 ser += pseries(rel, nseq);
374 // reexpanding it will collapse the series again
375 return ser.series(rel, order);
377 // third special case: x real, >=1 (branch cut)
378 if (!(options & series_options::suppress_branchcut) &&
379 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
381 // This is the branch cut: assemble the primitive series manually
382 // and then add the corresponding complex step function.
383 const symbol &s = ex_to<symbol>(rel.lhs());
384 const ex point = rel.rhs();
387 // zeroth order term:
388 seq.push_back(expair(Li2(x_pt), _ex0));
389 // compute the intermediate terms:
390 ex replarg = series(Li2(x), s==foo, order);
391 for (size_t i=1; i<replarg.nops()-1; ++i)
392 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));
393 // append an order term:
394 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
395 return pseries(rel, seq);
398 // all other cases should be safe, by now:
399 throw do_taylor(); // caught by function::series()
402 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
403 evalf_func(Li2_evalf).
404 derivative_func(Li2_deriv).
405 series_func(Li2_series).
406 latex_name("\\mbox{Li}_2"));
412 static ex Li3_eval(const ex & x)
416 return Li3(x).hold();
419 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
420 latex_name("\\mbox{Li}_3"));
423 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
426 static ex zetaderiv_eval(const ex & n, const ex & x)
428 if (n.info(info_flags::numeric)) {
429 // zetaderiv(0,x) -> zeta(x)
434 return zetaderiv(n, x).hold();
437 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
439 GINAC_ASSERT(deriv_param<2);
441 if (deriv_param==0) {
443 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
446 return zetaderiv(n+1,x);
449 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
450 derivative_func(zetaderiv_deriv).
451 latex_name("\\zeta^\\prime"));
457 static ex factorial_evalf(const ex & x)
459 return factorial(x).hold();
462 static ex factorial_eval(const ex & x)
464 if (is_exactly_a<numeric>(x))
465 return factorial(ex_to<numeric>(x));
467 return factorial(x).hold();
470 static ex factorial_conjugate(const ex & x)
475 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
476 evalf_func(factorial_evalf).
477 conjugate_func(factorial_conjugate));
483 static ex binomial_evalf(const ex & x, const ex & y)
485 return binomial(x, y).hold();
488 static ex binomial_sym(const ex & x, const numeric & y)
490 if (y.is_integer()) {
491 if (y.is_nonneg_integer()) {
492 const unsigned N = y.to_int();
493 if (N == 0) return _num0;
494 if (N == 1) return x;
496 for (unsigned i = 2; i <= N; ++i)
497 t = (t * (x + i - y - 1)).expand() / i;
503 return binomial(x, y).hold();
506 static ex binomial_eval(const ex & x, const ex &y)
508 if (is_exactly_a<numeric>(y)) {
509 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
510 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
512 return binomial_sym(x, ex_to<numeric>(y));
514 return binomial(x, y).hold();
517 // At the moment the numeric evaluation of a binomail function always
518 // gives a real number, but if this would be implemented using the gamma
519 // function, also complex conjugation should be changed (or rather, deleted).
520 static ex binomial_conjugate(const ex & x, const ex & y)
522 return binomial(x,y);
525 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
526 evalf_func(binomial_evalf).
527 conjugate_func(binomial_conjugate));
530 // Order term function (for truncated power series)
533 static ex Order_eval(const ex & x)
535 if (is_exactly_a<numeric>(x)) {
538 return Order(_ex1).hold();
541 } else if (is_exactly_a<mul>(x)) {
542 const mul &m = ex_to<mul>(x);
543 // O(c*expr) -> O(expr)
544 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
545 return Order(x / m.op(m.nops() - 1)).hold();
547 return Order(x).hold();
550 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
552 // Just wrap the function into a pseries object
554 GINAC_ASSERT(is_a<symbol>(r.lhs()));
555 const symbol &s = ex_to<symbol>(r.lhs());
556 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
557 return pseries(r, new_seq);
560 static ex Order_conjugate(const ex & x)
565 // Differentiation is handled in function::derivative because of its special requirements
567 REGISTER_FUNCTION(Order, eval_func(Order_eval).
568 series_func(Order_series).
569 latex_name("\\mathcal{O}").
570 conjugate_func(Order_conjugate));
573 // Solve linear system
576 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
578 // solve a system of linear equations
579 if (eqns.info(info_flags::relation_equal)) {
580 if (!symbols.info(info_flags::symbol))
581 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
582 const ex sol = lsolve(lst(eqns),lst(symbols));
584 GINAC_ASSERT(sol.nops()==1);
585 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
587 return sol.op(0).op(1); // return rhs of first solution
591 if (!eqns.info(info_flags::list)) {
592 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
594 for (size_t i=0; i<eqns.nops(); i++) {
595 if (!eqns.op(i).info(info_flags::relation_equal)) {
596 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
599 if (!symbols.info(info_flags::list)) {
600 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
602 for (size_t i=0; i<symbols.nops(); i++) {
603 if (!symbols.op(i).info(info_flags::symbol)) {
604 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
608 // build matrix from equation system
609 matrix sys(eqns.nops(),symbols.nops());
610 matrix rhs(eqns.nops(),1);
611 matrix vars(symbols.nops(),1);
613 for (size_t r=0; r<eqns.nops(); r++) {
614 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
616 for (size_t c=0; c<symbols.nops(); c++) {
617 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
618 linpart -= co*symbols.op(c);
621 linpart = linpart.expand();
625 // test if system is linear and fill vars matrix
626 for (size_t i=0; i<symbols.nops(); i++) {
627 vars(i,0) = symbols.op(i);
628 if (sys.has(symbols.op(i)))
629 throw(std::logic_error("lsolve: system is not linear"));
630 if (rhs.has(symbols.op(i)))
631 throw(std::logic_error("lsolve: system is not linear"));
636 solution = sys.solve(vars,rhs,options);
637 } catch (const std::runtime_error & e) {
638 // Probably singular matrix or otherwise overdetermined system:
639 // It is consistent to return an empty list
642 GINAC_ASSERT(solution.cols()==1);
643 GINAC_ASSERT(solution.rows()==symbols.nops());
645 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
647 for (size_t i=0; i<symbols.nops(); i++)
648 sollist.append(symbols.op(i)==solution(i,0));
653 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
654 * for static lib (so ginsh will see them). */
655 unsigned force_include_tgamma = tgamma_SERIAL::serial;
656 unsigned force_include_zeta1 = zeta1_SERIAL::serial;