3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2001 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 "relational.h"
44 static ex abs_evalf(const ex & arg)
47 TYPECHECK(arg,numeric)
48 END_TYPECHECK(abs(arg))
50 return abs(ex_to<numeric>(arg));
53 static ex abs_eval(const ex & arg)
55 if (is_ex_exactly_of_type(arg, numeric))
56 return abs(ex_to<numeric>(arg));
58 return abs(arg).hold();
61 REGISTER_FUNCTION(abs, eval_func(abs_eval).
62 evalf_func(abs_evalf));
69 static ex csgn_evalf(const ex & arg)
72 TYPECHECK(arg,numeric)
73 END_TYPECHECK(csgn(arg))
75 return csgn(ex_to<numeric>(arg));
78 static ex csgn_eval(const ex & arg)
80 if (is_ex_exactly_of_type(arg, numeric))
81 return csgn(ex_to<numeric>(arg));
83 else if (is_ex_of_type(arg, mul) &&
84 is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
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);
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).
132 static ex eta_evalf(const ex &x, const ex &y)
134 // It seems like we basically have to replicate the eval function here,
135 // since the expression might not be fully evaluated yet.
136 if (x.info(info_flags::positive) || y.info(info_flags::positive))
139 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
140 const numeric nx = ex_to<numeric>(x);
141 const numeric ny = ex_to<numeric>(y);
142 const numeric nxy = ex_to<numeric>(x*y);
144 if (nx.is_real() && nx.is_negative())
146 if (ny.is_real() && ny.is_negative())
148 if (nxy.is_real() && nxy.is_negative())
150 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
151 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
154 return eta(x,y).hold();
157 static ex eta_eval(const ex &x, const ex &y)
159 // trivial: eta(x,c) -> 0 if c is real and positive
160 if (x.info(info_flags::positive) || y.info(info_flags::positive))
163 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
164 // don't call eta_evalf here because it would call Pi.evalf()!
165 const numeric nx = ex_to<numeric>(x);
166 const numeric ny = ex_to<numeric>(y);
167 const numeric nxy = ex_to<numeric>(x*y);
169 if (nx.is_real() && nx.is_negative())
171 if (ny.is_real() && ny.is_negative())
173 if (nxy.is_real() && nxy.is_negative())
175 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
176 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
179 return eta(x,y).hold();
182 static ex eta_series(const ex & x, const ex & y,
183 const relational & rel,
187 const ex x_pt = x.subs(rel);
188 const ex y_pt = y.subs(rel);
189 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
190 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
191 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
192 throw (std::domain_error("eta_series(): on discontinuity"));
194 seq.push_back(expair(eta(x_pt,y_pt), _ex0()));
195 return pseries(rel,seq);
198 REGISTER_FUNCTION(eta, eval_func(eta_eval).
199 evalf_func(eta_evalf).
200 series_func(eta_series).
201 latex_name("\\eta"));
208 static ex Li2_evalf(const ex & x)
212 END_TYPECHECK(Li2(x))
214 return Li2(ex_to<numeric>(x)); // -> numeric Li2(numeric)
217 static ex Li2_eval(const ex & x)
219 if (x.info(info_flags::numeric)) {
224 if (x.is_equal(_ex1()))
225 return power(Pi,_ex2())/_ex6();
226 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
227 if (x.is_equal(_ex1_2()))
228 return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
229 // Li2(-1) -> -Pi^2/12
230 if (x.is_equal(_ex_1()))
231 return -power(Pi,_ex2())/_ex12();
232 // Li2(I) -> -Pi^2/48+Catalan*I
234 return power(Pi,_ex2())/_ex_48() + Catalan*I;
235 // Li2(-I) -> -Pi^2/48-Catalan*I
237 return power(Pi,_ex2())/_ex_48() - Catalan*I;
239 if (!x.info(info_flags::crational))
243 return Li2(x).hold();
246 static ex Li2_deriv(const ex & x, unsigned deriv_param)
248 GINAC_ASSERT(deriv_param==0);
250 // d/dx Li2(x) -> -log(1-x)/x
254 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
256 const ex x_pt = x.subs(rel);
257 if (x_pt.info(info_flags::numeric)) {
258 // First special case: x==0 (derivatives have poles)
259 if (x_pt.is_zero()) {
261 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
262 // simply substitute x==0. The limit, however, exists: it is 1.
263 // We also know all higher derivatives' limits:
264 // (d/dx)^n Li2(x) == n!/n^2.
265 // So the primitive series expansion is
266 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
268 // We first construct such a primitive series expansion manually in
269 // a dummy symbol s and then insert the argument's series expansion
270 // for s. Reexpanding the resulting series returns the desired
274 // manually construct the primitive expansion
275 for (int i=1; i<order; ++i)
276 ser += pow(s,i) / pow(numeric(i), _num2());
277 // substitute the argument's series expansion
278 ser = ser.subs(s==x.series(rel, order));
279 // maybe that was terminating, so add a proper order term
281 nseq.push_back(expair(Order(_ex1()), order));
282 ser += pseries(rel, nseq);
283 // reexpanding it will collapse the series again
284 return ser.series(rel, order);
285 // NB: Of course, this still does not allow us to compute anything
286 // like sin(Li2(x)).series(x==0,2), since then this code here is
287 // not reached and the derivative of sin(Li2(x)) doesn't allow the
288 // substitution x==0. Probably limits *are* needed for the general
289 // cases. In case L'Hospital's rule is implemented for limits and
290 // basic::series() takes care of this, this whole block is probably
293 // second special case: x==1 (branch point)
294 if (x_pt == _ex1()) {
296 // construct series manually in a dummy symbol s
298 ex ser = zeta(_ex2());
299 // manually construct the primitive expansion
300 for (int i=1; i<order; ++i)
301 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
302 // substitute the argument's series expansion
303 ser = ser.subs(s==x.series(rel, order));
304 // maybe that was terminating, so add a proper order term
306 nseq.push_back(expair(Order(_ex1()), order));
307 ser += pseries(rel, nseq);
308 // reexpanding it will collapse the series again
309 return ser.series(rel, order);
311 // third special case: x real, >=1 (branch cut)
312 if (!(options & series_options::suppress_branchcut) &&
313 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
315 // This is the branch cut: assemble the primitive series manually
316 // and then add the corresponding complex step function.
317 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
318 const ex point = rel.rhs();
321 // zeroth order term:
322 seq.push_back(expair(Li2(x_pt), _ex0()));
323 // compute the intermediate terms:
324 ex replarg = series(Li2(x), *s==foo, order);
325 for (unsigned i=1; i<replarg.nops()-1; ++i)
326 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
327 // append an order term:
328 seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
329 return pseries(rel, seq);
332 // all other cases should be safe, by now:
333 throw do_taylor(); // caught by function::series()
336 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
337 evalf_func(Li2_evalf).
338 derivative_func(Li2_deriv).
339 series_func(Li2_series).
340 latex_name("\\mbox{Li}_2"));
346 static ex Li3_eval(const ex & x)
350 return Li3(x).hold();
353 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
354 latex_name("\\mbox{Li}_3"));
360 static ex factorial_evalf(const ex & x)
362 return factorial(x).hold();
365 static ex factorial_eval(const ex & x)
367 if (is_ex_exactly_of_type(x, numeric))
368 return factorial(ex_to<numeric>(x));
370 return factorial(x).hold();
373 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
374 evalf_func(factorial_evalf));
380 static ex binomial_evalf(const ex & x, const ex & y)
382 return binomial(x, y).hold();
385 static ex binomial_eval(const ex & x, const ex &y)
387 if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
388 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
390 return binomial(x, y).hold();
393 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
394 evalf_func(binomial_evalf));
397 // Order term function (for truncated power series)
400 static ex Order_eval(const ex & x)
402 if (is_ex_exactly_of_type(x, numeric)) {
405 return Order(_ex1()).hold();
408 } else if (is_ex_exactly_of_type(x, mul)) {
409 mul *m = static_cast<mul *>(x.bp);
410 // O(c*expr) -> O(expr)
411 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
412 return Order(x / m->op(m->nops() - 1)).hold();
414 return Order(x).hold();
417 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
419 // Just wrap the function into a pseries object
421 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
422 const symbol *s = static_cast<symbol *>(r.lhs().bp);
423 new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
424 return pseries(r, new_seq);
427 // Differentiation is handled in function::derivative because of its special requirements
429 REGISTER_FUNCTION(Order, eval_func(Order_eval).
430 series_func(Order_series).
431 latex_name("\\mathcal{O}"));
434 // Solve linear system
437 ex lsolve(const ex &eqns, const ex &symbols)
439 // solve a system of linear equations
440 if (eqns.info(info_flags::relation_equal)) {
441 if (!symbols.info(info_flags::symbol))
442 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
443 const ex sol = lsolve(lst(eqns),lst(symbols));
445 GINAC_ASSERT(sol.nops()==1);
446 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
448 return sol.op(0).op(1); // return rhs of first solution
452 if (!eqns.info(info_flags::list)) {
453 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
455 for (unsigned i=0; i<eqns.nops(); i++) {
456 if (!eqns.op(i).info(info_flags::relation_equal)) {
457 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
460 if (!symbols.info(info_flags::list)) {
461 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
463 for (unsigned i=0; i<symbols.nops(); i++) {
464 if (!symbols.op(i).info(info_flags::symbol)) {
465 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
469 // build matrix from equation system
470 matrix sys(eqns.nops(),symbols.nops());
471 matrix rhs(eqns.nops(),1);
472 matrix vars(symbols.nops(),1);
474 for (unsigned r=0; r<eqns.nops(); r++) {
475 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
477 for (unsigned c=0; c<symbols.nops(); c++) {
478 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
479 linpart -= co*symbols.op(c);
482 linpart = linpart.expand();
486 // test if system is linear and fill vars matrix
487 for (unsigned i=0; i<symbols.nops(); i++) {
488 vars(i,0) = symbols.op(i);
489 if (sys.has(symbols.op(i)))
490 throw(std::logic_error("lsolve: system is not linear"));
491 if (rhs.has(symbols.op(i)))
492 throw(std::logic_error("lsolve: system is not linear"));
497 solution = sys.solve(vars,rhs);
498 } catch (const std::runtime_error & e) {
499 // Probably singular matrix or otherwise overdetermined system:
500 // It is consistent to return an empty list
503 GINAC_ASSERT(solution.cols()==1);
504 GINAC_ASSERT(solution.rows()==symbols.nops());
506 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
508 for (unsigned i=0; i<symbols.nops(); i++)
509 sollist.append(symbols.op(i)==solution(i,0));
514 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
515 * for static lib (so ginsh will see them). */
516 unsigned force_include_tgamma = function_index_tgamma;
517 unsigned force_include_zeta1 = function_index_zeta1;