3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2004 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(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, eval_func(conjugate_eval).
70 evalf_func(conjugate_evalf).
71 print_func<print_latex>(conjugate_print_latex).
72 conjugate_func(conjugate_conjugate));
78 static ex abs_evalf(const ex & arg)
80 if (is_exactly_a<numeric>(arg))
81 return abs(ex_to<numeric>(arg));
83 return abs(arg).hold();
86 static ex abs_eval(const ex & arg)
88 if (is_exactly_a<numeric>(arg))
89 return abs(ex_to<numeric>(arg));
91 return abs(arg).hold();
94 static void abs_print_latex(const ex & arg, const print_context & c)
96 c.s << "{|"; arg.print(c); c.s << "|}";
99 static void abs_print_csrc_float(const ex & arg, const print_context & c)
101 c.s << "fabs("; arg.print(c); c.s << ")";
104 static ex abs_conjugate(const ex & arg)
109 REGISTER_FUNCTION(abs, eval_func(abs_eval).
110 evalf_func(abs_evalf).
111 print_func<print_latex>(abs_print_latex).
112 print_func<print_csrc_float>(abs_print_csrc_float).
113 print_func<print_csrc_double>(abs_print_csrc_float).
114 conjugate_func(abs_conjugate));
121 static ex csgn_evalf(const ex & arg)
123 if (is_exactly_a<numeric>(arg))
124 return csgn(ex_to<numeric>(arg));
126 return csgn(arg).hold();
129 static ex csgn_eval(const ex & arg)
131 if (is_exactly_a<numeric>(arg))
132 return csgn(ex_to<numeric>(arg));
134 else if (is_exactly_a<mul>(arg) &&
135 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
136 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
139 // csgn(42*x) -> csgn(x)
140 return csgn(arg/oc).hold();
142 // csgn(-42*x) -> -csgn(x)
143 return -csgn(arg/oc).hold();
145 if (oc.real().is_zero()) {
147 // csgn(42*I*x) -> csgn(I*x)
148 return csgn(I*arg/oc).hold();
150 // csgn(-42*I*x) -> -csgn(I*x)
151 return -csgn(I*arg/oc).hold();
155 return csgn(arg).hold();
158 static ex csgn_series(const ex & arg,
159 const relational & rel,
163 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
164 if (arg_pt.info(info_flags::numeric)
165 && ex_to<numeric>(arg_pt).real().is_zero()
166 && !(options & series_options::suppress_branchcut))
167 throw (std::domain_error("csgn_series(): on imaginary axis"));
170 seq.push_back(expair(csgn(arg_pt), _ex0));
171 return pseries(rel,seq);
174 static ex csgn_conjugate(const ex& arg)
179 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
180 evalf_func(csgn_evalf).
181 series_func(csgn_series).
182 conjugate_func(csgn_conjugate));
186 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
187 // This function is closely related to the unwinding number K, sometimes found
188 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
191 static ex eta_evalf(const ex &x, const ex &y)
193 // It seems like we basically have to replicate the eval function here,
194 // since the expression might not be fully evaluated yet.
195 if (x.info(info_flags::positive) || y.info(info_flags::positive))
198 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
199 const numeric nx = ex_to<numeric>(x);
200 const numeric ny = ex_to<numeric>(y);
201 const numeric nxy = ex_to<numeric>(x*y);
203 if (nx.is_real() && nx.is_negative())
205 if (ny.is_real() && ny.is_negative())
207 if (nxy.is_real() && nxy.is_negative())
209 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
210 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
213 return eta(x,y).hold();
216 static ex eta_eval(const ex &x, const ex &y)
218 // trivial: eta(x,c) -> 0 if c is real and positive
219 if (x.info(info_flags::positive) || y.info(info_flags::positive))
222 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
223 // don't call eta_evalf here because it would call Pi.evalf()!
224 const numeric nx = ex_to<numeric>(x);
225 const numeric ny = ex_to<numeric>(y);
226 const numeric nxy = ex_to<numeric>(x*y);
228 if (nx.is_real() && nx.is_negative())
230 if (ny.is_real() && ny.is_negative())
232 if (nxy.is_real() && nxy.is_negative())
234 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
235 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
238 return eta(x,y).hold();
241 static ex eta_series(const ex & x, const ex & y,
242 const relational & rel,
246 const ex x_pt = x.subs(rel, subs_options::no_pattern);
247 const ex y_pt = y.subs(rel, subs_options::no_pattern);
248 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
249 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
250 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
251 throw (std::domain_error("eta_series(): on discontinuity"));
253 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
254 return pseries(rel,seq);
257 static ex eta_conjugate(const ex & x, const ex & y)
262 REGISTER_FUNCTION(eta, eval_func(eta_eval).
263 evalf_func(eta_evalf).
264 series_func(eta_series).
266 set_symmetry(sy_symm(0, 1)).
267 conjugate_func(eta_conjugate));
274 static ex Li2_evalf(const ex & x)
276 if (is_exactly_a<numeric>(x))
277 return Li2(ex_to<numeric>(x));
279 return Li2(x).hold();
282 static ex Li2_eval(const ex & x)
284 if (x.info(info_flags::numeric)) {
289 if (x.is_equal(_ex1))
290 return power(Pi,_ex2)/_ex6;
291 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
292 if (x.is_equal(_ex1_2))
293 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
294 // Li2(-1) -> -Pi^2/12
295 if (x.is_equal(_ex_1))
296 return -power(Pi,_ex2)/_ex12;
297 // Li2(I) -> -Pi^2/48+Catalan*I
299 return power(Pi,_ex2)/_ex_48 + Catalan*I;
300 // Li2(-I) -> -Pi^2/48-Catalan*I
302 return power(Pi,_ex2)/_ex_48 - Catalan*I;
304 if (!x.info(info_flags::crational))
305 return Li2(ex_to<numeric>(x));
308 return Li2(x).hold();
311 static ex Li2_deriv(const ex & x, unsigned deriv_param)
313 GINAC_ASSERT(deriv_param==0);
315 // d/dx Li2(x) -> -log(1-x)/x
316 return -log(_ex1-x)/x;
319 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
321 const ex x_pt = x.subs(rel, subs_options::no_pattern);
322 if (x_pt.info(info_flags::numeric)) {
323 // First special case: x==0 (derivatives have poles)
324 if (x_pt.is_zero()) {
326 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
327 // simply substitute x==0. The limit, however, exists: it is 1.
328 // We also know all higher derivatives' limits:
329 // (d/dx)^n Li2(x) == n!/n^2.
330 // So the primitive series expansion is
331 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
333 // We first construct such a primitive series expansion manually in
334 // a dummy symbol s and then insert the argument's series expansion
335 // for s. Reexpanding the resulting series returns the desired
339 // manually construct the primitive expansion
340 for (int i=1; i<order; ++i)
341 ser += pow(s,i) / pow(numeric(i), _num2);
342 // substitute the argument's series expansion
343 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
344 // maybe that was terminating, so add a proper order term
346 nseq.push_back(expair(Order(_ex1), order));
347 ser += pseries(rel, nseq);
348 // reexpanding it will collapse the series again
349 return ser.series(rel, order);
350 // NB: Of course, this still does not allow us to compute anything
351 // like sin(Li2(x)).series(x==0,2), since then this code here is
352 // not reached and the derivative of sin(Li2(x)) doesn't allow the
353 // substitution x==0. Probably limits *are* needed for the general
354 // cases. In case L'Hospital's rule is implemented for limits and
355 // basic::series() takes care of this, this whole block is probably
358 // second special case: x==1 (branch point)
359 if (x_pt.is_equal(_ex1)) {
361 // construct series manually in a dummy symbol s
364 // manually construct the primitive expansion
365 for (int i=1; i<order; ++i)
366 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
367 // substitute the argument's series expansion
368 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
369 // maybe that was terminating, so add a proper order term
371 nseq.push_back(expair(Order(_ex1), order));
372 ser += pseries(rel, nseq);
373 // reexpanding it will collapse the series again
374 return ser.series(rel, order);
376 // third special case: x real, >=1 (branch cut)
377 if (!(options & series_options::suppress_branchcut) &&
378 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
380 // This is the branch cut: assemble the primitive series manually
381 // and then add the corresponding complex step function.
382 const symbol &s = ex_to<symbol>(rel.lhs());
383 const ex point = rel.rhs();
386 // zeroth order term:
387 seq.push_back(expair(Li2(x_pt), _ex0));
388 // compute the intermediate terms:
389 ex replarg = series(Li2(x), s==foo, order);
390 for (size_t i=1; i<replarg.nops()-1; ++i)
391 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));
392 // append an order term:
393 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
394 return pseries(rel, seq);
397 // all other cases should be safe, by now:
398 throw do_taylor(); // caught by function::series()
401 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
402 evalf_func(Li2_evalf).
403 derivative_func(Li2_deriv).
404 series_func(Li2_series).
405 latex_name("\\mbox{Li}_2"));
411 static ex Li3_eval(const ex & x)
415 return Li3(x).hold();
418 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
419 latex_name("\\mbox{Li}_3"));
422 // Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
425 static ex zetaderiv_eval(const ex & n, const ex & x)
427 if (n.info(info_flags::numeric)) {
428 // zetaderiv(0,x) -> zeta(x)
433 return zetaderiv(n, x).hold();
436 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
438 GINAC_ASSERT(deriv_param<2);
440 if (deriv_param==0) {
442 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
445 return zetaderiv(n+1,x);
448 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
449 derivative_func(zetaderiv_deriv).
450 latex_name("\\zeta^\\prime"));
456 static ex factorial_evalf(const ex & x)
458 return factorial(x).hold();
461 static ex factorial_eval(const ex & x)
463 if (is_exactly_a<numeric>(x))
464 return factorial(ex_to<numeric>(x));
466 return factorial(x).hold();
469 static ex factorial_conjugate(const ex & x)
474 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
475 evalf_func(factorial_evalf).
476 conjugate_func(factorial_conjugate));
482 static ex binomial_evalf(const ex & x, const ex & y)
484 return binomial(x, y).hold();
487 static ex binomial_eval(const ex & x, const ex &y)
489 if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y))
490 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
492 return binomial(x, y).hold();
495 // At the moment the numeric evaluation of a binomail function always
496 // gives a real number, but if this would be implemented using the gamma
497 // function, also complex conjugation should be changed (or rather, deleted).
498 static ex binomial_conjugate(const ex & x, const ex & y)
500 return binomial(x,y);
503 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
504 evalf_func(binomial_evalf).
505 conjugate_func(binomial_conjugate));
508 // Order term function (for truncated power series)
511 static ex Order_eval(const ex & x)
513 if (is_exactly_a<numeric>(x)) {
516 return Order(_ex1).hold();
519 } else if (is_exactly_a<mul>(x)) {
520 const mul &m = ex_to<mul>(x);
521 // O(c*expr) -> O(expr)
522 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
523 return Order(x / m.op(m.nops() - 1)).hold();
525 return Order(x).hold();
528 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
530 // Just wrap the function into a pseries object
532 GINAC_ASSERT(is_a<symbol>(r.lhs()));
533 const symbol &s = ex_to<symbol>(r.lhs());
534 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
535 return pseries(r, new_seq);
538 static ex Order_conjugate(const ex & x)
543 // Differentiation is handled in function::derivative because of its special requirements
545 REGISTER_FUNCTION(Order, eval_func(Order_eval).
546 series_func(Order_series).
547 latex_name("\\mathcal{O}").
548 conjugate_func(Order_conjugate));
551 // Solve linear system
554 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
556 // solve a system of linear equations
557 if (eqns.info(info_flags::relation_equal)) {
558 if (!symbols.info(info_flags::symbol))
559 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
560 const ex sol = lsolve(lst(eqns),lst(symbols));
562 GINAC_ASSERT(sol.nops()==1);
563 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
565 return sol.op(0).op(1); // return rhs of first solution
569 if (!eqns.info(info_flags::list)) {
570 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
572 for (size_t i=0; i<eqns.nops(); i++) {
573 if (!eqns.op(i).info(info_flags::relation_equal)) {
574 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
577 if (!symbols.info(info_flags::list)) {
578 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
580 for (size_t i=0; i<symbols.nops(); i++) {
581 if (!symbols.op(i).info(info_flags::symbol)) {
582 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
586 // build matrix from equation system
587 matrix sys(eqns.nops(),symbols.nops());
588 matrix rhs(eqns.nops(),1);
589 matrix vars(symbols.nops(),1);
591 for (size_t r=0; r<eqns.nops(); r++) {
592 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
594 for (size_t c=0; c<symbols.nops(); c++) {
595 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
596 linpart -= co*symbols.op(c);
599 linpart = linpart.expand();
603 // test if system is linear and fill vars matrix
604 for (size_t i=0; i<symbols.nops(); i++) {
605 vars(i,0) = symbols.op(i);
606 if (sys.has(symbols.op(i)))
607 throw(std::logic_error("lsolve: system is not linear"));
608 if (rhs.has(symbols.op(i)))
609 throw(std::logic_error("lsolve: system is not linear"));
614 solution = sys.solve(vars,rhs,options);
615 } catch (const std::runtime_error & e) {
616 // Probably singular matrix or otherwise overdetermined system:
617 // It is consistent to return an empty list
620 GINAC_ASSERT(solution.cols()==1);
621 GINAC_ASSERT(solution.rows()==symbols.nops());
623 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
625 for (size_t i=0; i<symbols.nops(); i++)
626 sollist.append(symbols.op(i)==solution(i,0));
631 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
632 * for static lib (so ginsh will see them). */
633 unsigned force_include_tgamma = tgamma_SERIAL::serial;
634 unsigned force_include_zeta1 = zeta1_SERIAL::serial;