3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2003 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 abs_evalf(const ex & arg)
48 if (is_exactly_a<numeric>(arg))
49 return abs(ex_to<numeric>(arg));
51 return abs(arg).hold();
54 static ex abs_eval(const ex & arg)
56 if (is_exactly_a<numeric>(arg))
57 return abs(ex_to<numeric>(arg));
59 return abs(arg).hold();
62 static void abs_print_latex(const ex & arg, const print_context & c)
64 c.s << "{|"; arg.print(c); c.s << "|}";
67 static void abs_print_csrc_float(const ex & arg, const print_context & c)
69 c.s << "fabs("; arg.print(c); c.s << ")";
72 REGISTER_FUNCTION(abs, eval_func(abs_eval).
73 evalf_func(abs_evalf).
74 print_func<print_latex>(abs_print_latex).
75 print_func<print_csrc_float>(abs_print_csrc_float).
76 print_func<print_csrc_double>(abs_print_csrc_float));
83 static ex csgn_evalf(const ex & arg)
85 if (is_exactly_a<numeric>(arg))
86 return csgn(ex_to<numeric>(arg));
88 return csgn(arg).hold();
91 static ex csgn_eval(const ex & arg)
93 if (is_exactly_a<numeric>(arg))
94 return csgn(ex_to<numeric>(arg));
96 else if (is_exactly_a<mul>(arg) &&
97 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
98 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
101 // csgn(42*x) -> csgn(x)
102 return csgn(arg/oc).hold();
104 // csgn(-42*x) -> -csgn(x)
105 return -csgn(arg/oc).hold();
107 if (oc.real().is_zero()) {
109 // csgn(42*I*x) -> csgn(I*x)
110 return csgn(I*arg/oc).hold();
112 // csgn(-42*I*x) -> -csgn(I*x)
113 return -csgn(I*arg/oc).hold();
117 return csgn(arg).hold();
120 static ex csgn_series(const ex & arg,
121 const relational & rel,
125 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
126 if (arg_pt.info(info_flags::numeric)
127 && ex_to<numeric>(arg_pt).real().is_zero()
128 && !(options & series_options::suppress_branchcut))
129 throw (std::domain_error("csgn_series(): on imaginary axis"));
132 seq.push_back(expair(csgn(arg_pt), _ex0));
133 return pseries(rel,seq);
136 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
137 evalf_func(csgn_evalf).
138 series_func(csgn_series));
142 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
143 // This function is closely related to the unwinding number K, sometimes found
144 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
147 static ex eta_evalf(const ex &x, const ex &y)
149 // It seems like we basically have to replicate the eval function here,
150 // since the expression might not be fully evaluated yet.
151 if (x.info(info_flags::positive) || y.info(info_flags::positive))
154 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
155 const numeric nx = ex_to<numeric>(x);
156 const numeric ny = ex_to<numeric>(y);
157 const numeric nxy = ex_to<numeric>(x*y);
159 if (nx.is_real() && nx.is_negative())
161 if (ny.is_real() && ny.is_negative())
163 if (nxy.is_real() && nxy.is_negative())
165 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
166 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
169 return eta(x,y).hold();
172 static ex eta_eval(const ex &x, const ex &y)
174 // trivial: eta(x,c) -> 0 if c is real and positive
175 if (x.info(info_flags::positive) || y.info(info_flags::positive))
178 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
179 // don't call eta_evalf here because it would call Pi.evalf()!
180 const numeric nx = ex_to<numeric>(x);
181 const numeric ny = ex_to<numeric>(y);
182 const numeric nxy = ex_to<numeric>(x*y);
184 if (nx.is_real() && nx.is_negative())
186 if (ny.is_real() && ny.is_negative())
188 if (nxy.is_real() && nxy.is_negative())
190 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
191 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
194 return eta(x,y).hold();
197 static ex eta_series(const ex & x, const ex & y,
198 const relational & rel,
202 const ex x_pt = x.subs(rel, subs_options::no_pattern);
203 const ex y_pt = y.subs(rel, subs_options::no_pattern);
204 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
205 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
206 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
207 throw (std::domain_error("eta_series(): on discontinuity"));
209 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
210 return pseries(rel,seq);
213 REGISTER_FUNCTION(eta, eval_func(eta_eval).
214 evalf_func(eta_evalf).
215 series_func(eta_series).
217 set_symmetry(sy_symm(0, 1)));
224 static ex Li2_evalf(const ex & x)
226 if (is_exactly_a<numeric>(x))
227 return Li2(ex_to<numeric>(x));
229 return Li2(x).hold();
232 static ex Li2_eval(const ex & x)
234 if (x.info(info_flags::numeric)) {
239 if (x.is_equal(_ex1))
240 return power(Pi,_ex2)/_ex6;
241 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
242 if (x.is_equal(_ex1_2))
243 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
244 // Li2(-1) -> -Pi^2/12
245 if (x.is_equal(_ex_1))
246 return -power(Pi,_ex2)/_ex12;
247 // Li2(I) -> -Pi^2/48+Catalan*I
249 return power(Pi,_ex2)/_ex_48 + Catalan*I;
250 // Li2(-I) -> -Pi^2/48-Catalan*I
252 return power(Pi,_ex2)/_ex_48 - Catalan*I;
254 if (!x.info(info_flags::crational))
255 return Li2(ex_to<numeric>(x));
258 return Li2(x).hold();
261 static ex Li2_deriv(const ex & x, unsigned deriv_param)
263 GINAC_ASSERT(deriv_param==0);
265 // d/dx Li2(x) -> -log(1-x)/x
266 return -log(_ex1-x)/x;
269 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
271 const ex x_pt = x.subs(rel, subs_options::no_pattern);
272 if (x_pt.info(info_flags::numeric)) {
273 // First special case: x==0 (derivatives have poles)
274 if (x_pt.is_zero()) {
276 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
277 // simply substitute x==0. The limit, however, exists: it is 1.
278 // We also know all higher derivatives' limits:
279 // (d/dx)^n Li2(x) == n!/n^2.
280 // So the primitive series expansion is
281 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
283 // We first construct such a primitive series expansion manually in
284 // a dummy symbol s and then insert the argument's series expansion
285 // for s. Reexpanding the resulting series returns the desired
289 // manually construct the primitive expansion
290 for (int i=1; i<order; ++i)
291 ser += pow(s,i) / pow(numeric(i), _num2);
292 // substitute the argument's series expansion
293 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
294 // maybe that was terminating, so add a proper order term
296 nseq.push_back(expair(Order(_ex1), order));
297 ser += pseries(rel, nseq);
298 // reexpanding it will collapse the series again
299 return ser.series(rel, order);
300 // NB: Of course, this still does not allow us to compute anything
301 // like sin(Li2(x)).series(x==0,2), since then this code here is
302 // not reached and the derivative of sin(Li2(x)) doesn't allow the
303 // substitution x==0. Probably limits *are* needed for the general
304 // cases. In case L'Hospital's rule is implemented for limits and
305 // basic::series() takes care of this, this whole block is probably
308 // second special case: x==1 (branch point)
309 if (x_pt.is_equal(_ex1)) {
311 // construct series manually in a dummy symbol s
314 // manually construct the primitive expansion
315 for (int i=1; i<order; ++i)
316 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
317 // substitute the argument's series expansion
318 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
319 // maybe that was terminating, so add a proper order term
321 nseq.push_back(expair(Order(_ex1), order));
322 ser += pseries(rel, nseq);
323 // reexpanding it will collapse the series again
324 return ser.series(rel, order);
326 // third special case: x real, >=1 (branch cut)
327 if (!(options & series_options::suppress_branchcut) &&
328 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
330 // This is the branch cut: assemble the primitive series manually
331 // and then add the corresponding complex step function.
332 const symbol &s = ex_to<symbol>(rel.lhs());
333 const ex point = rel.rhs();
336 // zeroth order term:
337 seq.push_back(expair(Li2(x_pt), _ex0));
338 // compute the intermediate terms:
339 ex replarg = series(Li2(x), s==foo, order);
340 for (size_t i=1; i<replarg.nops()-1; ++i)
341 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));
342 // append an order term:
343 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
344 return pseries(rel, seq);
347 // all other cases should be safe, by now:
348 throw do_taylor(); // caught by function::series()
351 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
352 evalf_func(Li2_evalf).
353 derivative_func(Li2_deriv).
354 series_func(Li2_series).
355 latex_name("\\mbox{Li}_2"));
361 static ex Li3_eval(const ex & x)
365 return Li3(x).hold();
368 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
369 latex_name("\\mbox{Li}_3"));
375 static ex factorial_evalf(const ex & x)
377 return factorial(x).hold();
380 static ex factorial_eval(const ex & x)
382 if (is_exactly_a<numeric>(x))
383 return factorial(ex_to<numeric>(x));
385 return factorial(x).hold();
388 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
389 evalf_func(factorial_evalf));
395 static ex binomial_evalf(const ex & x, const ex & y)
397 return binomial(x, y).hold();
400 static ex binomial_eval(const ex & x, const ex &y)
402 if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y))
403 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
405 return binomial(x, y).hold();
408 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
409 evalf_func(binomial_evalf));
412 // Order term function (for truncated power series)
415 static ex Order_eval(const ex & x)
417 if (is_exactly_a<numeric>(x)) {
420 return Order(_ex1).hold();
423 } else if (is_exactly_a<mul>(x)) {
424 const mul &m = ex_to<mul>(x);
425 // O(c*expr) -> O(expr)
426 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
427 return Order(x / m.op(m.nops() - 1)).hold();
429 return Order(x).hold();
432 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
434 // Just wrap the function into a pseries object
436 GINAC_ASSERT(is_a<symbol>(r.lhs()));
437 const symbol &s = ex_to<symbol>(r.lhs());
438 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
439 return pseries(r, new_seq);
442 // Differentiation is handled in function::derivative because of its special requirements
444 REGISTER_FUNCTION(Order, eval_func(Order_eval).
445 series_func(Order_series).
446 latex_name("\\mathcal{O}"));
449 // Solve linear system
452 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
454 // solve a system of linear equations
455 if (eqns.info(info_flags::relation_equal)) {
456 if (!symbols.info(info_flags::symbol))
457 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
458 const ex sol = lsolve(lst(eqns),lst(symbols));
460 GINAC_ASSERT(sol.nops()==1);
461 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
463 return sol.op(0).op(1); // return rhs of first solution
467 if (!eqns.info(info_flags::list)) {
468 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
470 for (size_t i=0; i<eqns.nops(); i++) {
471 if (!eqns.op(i).info(info_flags::relation_equal)) {
472 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
475 if (!symbols.info(info_flags::list)) {
476 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
478 for (size_t i=0; i<symbols.nops(); i++) {
479 if (!symbols.op(i).info(info_flags::symbol)) {
480 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
484 // build matrix from equation system
485 matrix sys(eqns.nops(),symbols.nops());
486 matrix rhs(eqns.nops(),1);
487 matrix vars(symbols.nops(),1);
489 for (size_t r=0; r<eqns.nops(); r++) {
490 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
492 for (size_t c=0; c<symbols.nops(); c++) {
493 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
494 linpart -= co*symbols.op(c);
497 linpart = linpart.expand();
501 // test if system is linear and fill vars matrix
502 for (size_t i=0; i<symbols.nops(); i++) {
503 vars(i,0) = symbols.op(i);
504 if (sys.has(symbols.op(i)))
505 throw(std::logic_error("lsolve: system is not linear"));
506 if (rhs.has(symbols.op(i)))
507 throw(std::logic_error("lsolve: system is not linear"));
512 solution = sys.solve(vars,rhs,options);
513 } catch (const std::runtime_error & e) {
514 // Probably singular matrix or otherwise overdetermined system:
515 // It is consistent to return an empty list
518 GINAC_ASSERT(solution.cols()==1);
519 GINAC_ASSERT(solution.rows()==symbols.nops());
521 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
523 for (size_t i=0; i<symbols.nops(); i++)
524 sollist.append(symbols.op(i)==solution(i,0));
529 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
530 * for static lib (so ginsh will see them). */
531 unsigned force_include_tgamma = tgamma_SERIAL::serial;
532 unsigned force_include_zeta1 = zeta1_SERIAL::serial;