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 REGISTER_FUNCTION(abs, eval_func(abs_eval).
63 evalf_func(abs_evalf));
70 static ex csgn_evalf(const ex & arg)
72 if (is_exactly_a<numeric>(arg))
73 return csgn(ex_to<numeric>(arg));
75 return csgn(arg).hold();
78 static ex csgn_eval(const ex & arg)
80 if (is_exactly_a<numeric>(arg))
81 return csgn(ex_to<numeric>(arg));
83 else if (is_exactly_a<mul>(arg) &&
84 is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
85 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
88 // csgn(42*x) -> csgn(x)
89 return csgn(arg/oc).hold();
91 // csgn(-42*x) -> -csgn(x)
92 return -csgn(arg/oc).hold();
94 if (oc.real().is_zero()) {
96 // csgn(42*I*x) -> csgn(I*x)
97 return csgn(I*arg/oc).hold();
99 // csgn(-42*I*x) -> -csgn(I*x)
100 return -csgn(I*arg/oc).hold();
104 return csgn(arg).hold();
107 static ex csgn_series(const ex & arg,
108 const relational & rel,
112 const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
113 if (arg_pt.info(info_flags::numeric)
114 && ex_to<numeric>(arg_pt).real().is_zero()
115 && !(options & series_options::suppress_branchcut))
116 throw (std::domain_error("csgn_series(): on imaginary axis"));
119 seq.push_back(expair(csgn(arg_pt), _ex0));
120 return pseries(rel,seq);
123 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
124 evalf_func(csgn_evalf).
125 series_func(csgn_series));
129 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
130 // This function is closely related to the unwinding number K, sometimes found
131 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
134 static ex eta_evalf(const ex &x, const ex &y)
136 // It seems like we basically have to replicate the eval function here,
137 // since the expression might not be fully evaluated yet.
138 if (x.info(info_flags::positive) || y.info(info_flags::positive))
141 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
142 const numeric nx = ex_to<numeric>(x);
143 const numeric ny = ex_to<numeric>(y);
144 const numeric nxy = ex_to<numeric>(x*y);
146 if (nx.is_real() && nx.is_negative())
148 if (ny.is_real() && ny.is_negative())
150 if (nxy.is_real() && nxy.is_negative())
152 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
153 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
156 return eta(x,y).hold();
159 static ex eta_eval(const ex &x, const ex &y)
161 // trivial: eta(x,c) -> 0 if c is real and positive
162 if (x.info(info_flags::positive) || y.info(info_flags::positive))
165 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
166 // don't call eta_evalf here because it would call Pi.evalf()!
167 const numeric nx = ex_to<numeric>(x);
168 const numeric ny = ex_to<numeric>(y);
169 const numeric nxy = ex_to<numeric>(x*y);
171 if (nx.is_real() && nx.is_negative())
173 if (ny.is_real() && ny.is_negative())
175 if (nxy.is_real() && nxy.is_negative())
177 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
178 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
181 return eta(x,y).hold();
184 static ex eta_series(const ex & x, const ex & y,
185 const relational & rel,
189 const ex x_pt = x.subs(rel, subs_options::no_pattern);
190 const ex y_pt = y.subs(rel, subs_options::no_pattern);
191 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
192 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
193 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
194 throw (std::domain_error("eta_series(): on discontinuity"));
196 seq.push_back(expair(eta(x_pt,y_pt), _ex0));
197 return pseries(rel,seq);
200 REGISTER_FUNCTION(eta, eval_func(eta_eval).
201 evalf_func(eta_evalf).
202 series_func(eta_series).
204 set_symmetry(sy_symm(0, 1)));
211 static ex Li2_evalf(const ex & x)
213 if (is_exactly_a<numeric>(x))
214 return Li2(ex_to<numeric>(x));
216 return Li2(x).hold();
219 static ex Li2_eval(const ex & x)
221 if (x.info(info_flags::numeric)) {
226 if (x.is_equal(_ex1))
227 return power(Pi,_ex2)/_ex6;
228 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
229 if (x.is_equal(_ex1_2))
230 return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
231 // Li2(-1) -> -Pi^2/12
232 if (x.is_equal(_ex_1))
233 return -power(Pi,_ex2)/_ex12;
234 // Li2(I) -> -Pi^2/48+Catalan*I
236 return power(Pi,_ex2)/_ex_48 + Catalan*I;
237 // Li2(-I) -> -Pi^2/48-Catalan*I
239 return power(Pi,_ex2)/_ex_48 - Catalan*I;
241 if (!x.info(info_flags::crational))
242 return Li2(ex_to<numeric>(x));
245 return Li2(x).hold();
248 static ex Li2_deriv(const ex & x, unsigned deriv_param)
250 GINAC_ASSERT(deriv_param==0);
252 // d/dx Li2(x) -> -log(1-x)/x
253 return -log(_ex1-x)/x;
256 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
258 const ex x_pt = x.subs(rel, subs_options::no_pattern);
259 if (x_pt.info(info_flags::numeric)) {
260 // First special case: x==0 (derivatives have poles)
261 if (x_pt.is_zero()) {
263 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
264 // simply substitute x==0. The limit, however, exists: it is 1.
265 // We also know all higher derivatives' limits:
266 // (d/dx)^n Li2(x) == n!/n^2.
267 // So the primitive series expansion is
268 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
270 // We first construct such a primitive series expansion manually in
271 // a dummy symbol s and then insert the argument's series expansion
272 // for s. Reexpanding the resulting series returns the desired
276 // manually construct the primitive expansion
277 for (int i=1; i<order; ++i)
278 ser += pow(s,i) / pow(numeric(i), _num2);
279 // substitute the argument's series expansion
280 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
281 // maybe that was terminating, so add a proper order term
283 nseq.push_back(expair(Order(_ex1), order));
284 ser += pseries(rel, nseq);
285 // reexpanding it will collapse the series again
286 return ser.series(rel, order);
287 // NB: Of course, this still does not allow us to compute anything
288 // like sin(Li2(x)).series(x==0,2), since then this code here is
289 // not reached and the derivative of sin(Li2(x)) doesn't allow the
290 // substitution x==0. Probably limits *are* needed for the general
291 // cases. In case L'Hospital's rule is implemented for limits and
292 // basic::series() takes care of this, this whole block is probably
295 // second special case: x==1 (branch point)
296 if (x_pt.is_equal(_ex1)) {
298 // construct series manually in a dummy symbol s
301 // manually construct the primitive expansion
302 for (int i=1; i<order; ++i)
303 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
304 // substitute the argument's series expansion
305 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
306 // maybe that was terminating, so add a proper order term
308 nseq.push_back(expair(Order(_ex1), order));
309 ser += pseries(rel, nseq);
310 // reexpanding it will collapse the series again
311 return ser.series(rel, order);
313 // third special case: x real, >=1 (branch cut)
314 if (!(options & series_options::suppress_branchcut) &&
315 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
317 // This is the branch cut: assemble the primitive series manually
318 // and then add the corresponding complex step function.
319 const symbol &s = ex_to<symbol>(rel.lhs());
320 const ex point = rel.rhs();
323 // zeroth order term:
324 seq.push_back(expair(Li2(x_pt), _ex0));
325 // compute the intermediate terms:
326 ex replarg = series(Li2(x), s==foo, order);
327 for (size_t i=1; i<replarg.nops()-1; ++i)
328 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));
329 // append an order term:
330 seq.push_back(expair(Order(_ex1), replarg.nops()-1));
331 return pseries(rel, seq);
334 // all other cases should be safe, by now:
335 throw do_taylor(); // caught by function::series()
338 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
339 evalf_func(Li2_evalf).
340 derivative_func(Li2_deriv).
341 series_func(Li2_series).
342 latex_name("\\mbox{Li}_2"));
348 static ex Li3_eval(const ex & x)
352 return Li3(x).hold();
355 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
356 latex_name("\\mbox{Li}_3"));
362 static ex factorial_evalf(const ex & x)
364 return factorial(x).hold();
367 static ex factorial_eval(const ex & x)
369 if (is_exactly_a<numeric>(x))
370 return factorial(ex_to<numeric>(x));
372 return factorial(x).hold();
375 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
376 evalf_func(factorial_evalf));
382 static ex binomial_evalf(const ex & x, const ex & y)
384 return binomial(x, y).hold();
387 static ex binomial_eval(const ex & x, const ex &y)
389 if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y))
390 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
392 return binomial(x, y).hold();
395 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
396 evalf_func(binomial_evalf));
399 // Order term function (for truncated power series)
402 static ex Order_eval(const ex & x)
404 if (is_exactly_a<numeric>(x)) {
407 return Order(_ex1).hold();
410 } else if (is_exactly_a<mul>(x)) {
411 const mul &m = ex_to<mul>(x);
412 // O(c*expr) -> O(expr)
413 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
414 return Order(x / m.op(m.nops() - 1)).hold();
416 return Order(x).hold();
419 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
421 // Just wrap the function into a pseries object
423 GINAC_ASSERT(is_a<symbol>(r.lhs()));
424 const symbol &s = ex_to<symbol>(r.lhs());
425 new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
426 return pseries(r, new_seq);
429 // Differentiation is handled in function::derivative because of its special requirements
431 REGISTER_FUNCTION(Order, eval_func(Order_eval).
432 series_func(Order_series).
433 latex_name("\\mathcal{O}"));
436 // Solve linear system
439 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
441 // solve a system of linear equations
442 if (eqns.info(info_flags::relation_equal)) {
443 if (!symbols.info(info_flags::symbol))
444 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
445 const ex sol = lsolve(lst(eqns),lst(symbols));
447 GINAC_ASSERT(sol.nops()==1);
448 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
450 return sol.op(0).op(1); // return rhs of first solution
454 if (!eqns.info(info_flags::list)) {
455 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
457 for (size_t i=0; i<eqns.nops(); i++) {
458 if (!eqns.op(i).info(info_flags::relation_equal)) {
459 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
462 if (!symbols.info(info_flags::list)) {
463 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
465 for (size_t i=0; i<symbols.nops(); i++) {
466 if (!symbols.op(i).info(info_flags::symbol)) {
467 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
471 // build matrix from equation system
472 matrix sys(eqns.nops(),symbols.nops());
473 matrix rhs(eqns.nops(),1);
474 matrix vars(symbols.nops(),1);
476 for (size_t r=0; r<eqns.nops(); r++) {
477 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
479 for (size_t c=0; c<symbols.nops(); c++) {
480 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
481 linpart -= co*symbols.op(c);
484 linpart = linpart.expand();
488 // test if system is linear and fill vars matrix
489 for (size_t i=0; i<symbols.nops(); i++) {
490 vars(i,0) = symbols.op(i);
491 if (sys.has(symbols.op(i)))
492 throw(std::logic_error("lsolve: system is not linear"));
493 if (rhs.has(symbols.op(i)))
494 throw(std::logic_error("lsolve: system is not linear"));
499 solution = sys.solve(vars,rhs,options);
500 } catch (const std::runtime_error & e) {
501 // Probably singular matrix or otherwise overdetermined system:
502 // It is consistent to return an empty list
505 GINAC_ASSERT(solution.cols()==1);
506 GINAC_ASSERT(solution.rows()==symbols.nops());
508 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
510 for (size_t i=0; i<symbols.nops(); i++)
511 sollist.append(symbols.op(i)==solution(i,0));
516 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
517 * for static lib (so ginsh will see them). */
518 unsigned force_include_tgamma = tgamma_SERIAL::serial;
519 unsigned force_include_zeta1 = zeta1_SERIAL::serial;