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"
45 static ex abs_evalf(const ex & arg)
47 if (is_exactly_a<numeric>(arg))
48 return abs(ex_to<numeric>(arg));
50 return abs(arg).hold();
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)
71 if (is_exactly_a<numeric>(arg))
72 return csgn(ex_to<numeric>(arg));
74 return csgn(arg).hold();
77 static ex csgn_eval(const ex & arg)
79 if (is_ex_exactly_of_type(arg, numeric))
80 return csgn(ex_to<numeric>(arg));
82 else if (is_ex_of_type(arg, mul) &&
83 is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
84 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
87 // csgn(42*x) -> csgn(x)
88 return csgn(arg/oc).hold();
90 // csgn(-42*x) -> -csgn(x)
91 return -csgn(arg/oc).hold();
93 if (oc.real().is_zero()) {
95 // csgn(42*I*x) -> csgn(I*x)
96 return csgn(I*arg/oc).hold();
98 // csgn(-42*I*x) -> -csgn(I*x)
99 return -csgn(I*arg/oc).hold();
103 return csgn(arg).hold();
106 static ex csgn_series(const ex & arg,
107 const relational & rel,
111 const ex arg_pt = arg.subs(rel);
112 if (arg_pt.info(info_flags::numeric)
113 && ex_to<numeric>(arg_pt).real().is_zero()
114 && !(options & series_options::suppress_branchcut))
115 throw (std::domain_error("csgn_series(): on imaginary axis"));
118 seq.push_back(expair(csgn(arg_pt), _ex0()));
119 return pseries(rel,seq);
122 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
123 evalf_func(csgn_evalf).
124 series_func(csgn_series));
128 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
131 static ex eta_evalf(const ex &x, const ex &y)
133 // It seems like we basically have to replicate the eval function here,
134 // since the expression might not be fully evaluated yet.
135 if (x.info(info_flags::positive) || y.info(info_flags::positive))
138 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
139 const numeric nx = ex_to<numeric>(x);
140 const numeric ny = ex_to<numeric>(y);
141 const numeric nxy = ex_to<numeric>(x*y);
143 if (nx.is_real() && nx.is_negative())
145 if (ny.is_real() && ny.is_negative())
147 if (nxy.is_real() && nxy.is_negative())
149 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
150 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
153 return eta(x,y).hold();
156 static ex eta_eval(const ex &x, const ex &y)
158 // trivial: eta(x,c) -> 0 if c is real and positive
159 if (x.info(info_flags::positive) || y.info(info_flags::positive))
162 if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
163 // don't call eta_evalf here because it would call Pi.evalf()!
164 const numeric nx = ex_to<numeric>(x);
165 const numeric ny = ex_to<numeric>(y);
166 const numeric nxy = ex_to<numeric>(x*y);
168 if (nx.is_real() && nx.is_negative())
170 if (ny.is_real() && ny.is_negative())
172 if (nxy.is_real() && nxy.is_negative())
174 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
175 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
178 return eta(x,y).hold();
181 static ex eta_series(const ex & x, const ex & y,
182 const relational & rel,
186 const ex x_pt = x.subs(rel);
187 const ex y_pt = y.subs(rel);
188 if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
189 (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
190 ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
191 throw (std::domain_error("eta_series(): on discontinuity"));
193 seq.push_back(expair(eta(x_pt,y_pt), _ex0()));
194 return pseries(rel,seq);
197 REGISTER_FUNCTION(eta, eval_func(eta_eval).
198 evalf_func(eta_evalf).
199 series_func(eta_series).
201 set_symmetry(sy_symm(0, 1)));
208 static ex Li2_evalf(const ex & x)
210 if (is_exactly_a<numeric>(x))
211 return Li2(ex_to<numeric>(x));
213 return Li2(x).hold();
216 static ex Li2_eval(const ex & x)
218 if (x.info(info_flags::numeric)) {
223 if (x.is_equal(_ex1()))
224 return power(Pi,_ex2())/_ex6();
225 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
226 if (x.is_equal(_ex1_2()))
227 return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
228 // Li2(-1) -> -Pi^2/12
229 if (x.is_equal(_ex_1()))
230 return -power(Pi,_ex2())/_ex12();
231 // Li2(I) -> -Pi^2/48+Catalan*I
233 return power(Pi,_ex2())/_ex_48() + Catalan*I;
234 // Li2(-I) -> -Pi^2/48-Catalan*I
236 return power(Pi,_ex2())/_ex_48() - Catalan*I;
238 if (!x.info(info_flags::crational))
239 return Li2(ex_to<numeric>(x));
242 return Li2(x).hold();
245 static ex Li2_deriv(const ex & x, unsigned deriv_param)
247 GINAC_ASSERT(deriv_param==0);
249 // d/dx Li2(x) -> -log(1-x)/x
250 return -log(_ex1()-x)/x;
253 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
255 const ex x_pt = x.subs(rel);
256 if (x_pt.info(info_flags::numeric)) {
257 // First special case: x==0 (derivatives have poles)
258 if (x_pt.is_zero()) {
260 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
261 // simply substitute x==0. The limit, however, exists: it is 1.
262 // We also know all higher derivatives' limits:
263 // (d/dx)^n Li2(x) == n!/n^2.
264 // So the primitive series expansion is
265 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
267 // We first construct such a primitive series expansion manually in
268 // a dummy symbol s and then insert the argument's series expansion
269 // for s. Reexpanding the resulting series returns the desired
273 // manually construct the primitive expansion
274 for (int i=1; i<order; ++i)
275 ser += pow(s,i) / pow(numeric(i), _num2());
276 // substitute the argument's series expansion
277 ser = ser.subs(s==x.series(rel, order));
278 // maybe that was terminating, so add a proper order term
280 nseq.push_back(expair(Order(_ex1()), order));
281 ser += pseries(rel, nseq);
282 // reexpanding it will collapse the series again
283 return ser.series(rel, order);
284 // NB: Of course, this still does not allow us to compute anything
285 // like sin(Li2(x)).series(x==0,2), since then this code here is
286 // not reached and the derivative of sin(Li2(x)) doesn't allow the
287 // substitution x==0. Probably limits *are* needed for the general
288 // cases. In case L'Hospital's rule is implemented for limits and
289 // basic::series() takes care of this, this whole block is probably
292 // second special case: x==1 (branch point)
293 if (x_pt.is_equal(_ex1())) {
295 // construct series manually in a dummy symbol s
297 ex ser = zeta(_ex2());
298 // manually construct the primitive expansion
299 for (int i=1; i<order; ++i)
300 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
301 // substitute the argument's series expansion
302 ser = ser.subs(s==x.series(rel, order));
303 // maybe that was terminating, so add a proper order term
305 nseq.push_back(expair(Order(_ex1()), order));
306 ser += pseries(rel, nseq);
307 // reexpanding it will collapse the series again
308 return ser.series(rel, order);
310 // third special case: x real, >=1 (branch cut)
311 if (!(options & series_options::suppress_branchcut) &&
312 ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
314 // This is the branch cut: assemble the primitive series manually
315 // and then add the corresponding complex step function.
316 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
317 const ex point = rel.rhs();
320 // zeroth order term:
321 seq.push_back(expair(Li2(x_pt), _ex0()));
322 // compute the intermediate terms:
323 ex replarg = series(Li2(x), *s==foo, order);
324 for (unsigned i=1; i<replarg.nops()-1; ++i)
325 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
326 // append an order term:
327 seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
328 return pseries(rel, seq);
331 // all other cases should be safe, by now:
332 throw do_taylor(); // caught by function::series()
335 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
336 evalf_func(Li2_evalf).
337 derivative_func(Li2_deriv).
338 series_func(Li2_series).
339 latex_name("\\mbox{Li}_2"));
345 static ex Li3_eval(const ex & x)
349 return Li3(x).hold();
352 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
353 latex_name("\\mbox{Li}_3"));
359 static ex factorial_evalf(const ex & x)
361 return factorial(x).hold();
364 static ex factorial_eval(const ex & x)
366 if (is_ex_exactly_of_type(x, numeric))
367 return factorial(ex_to<numeric>(x));
369 return factorial(x).hold();
372 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
373 evalf_func(factorial_evalf));
379 static ex binomial_evalf(const ex & x, const ex & y)
381 return binomial(x, y).hold();
384 static ex binomial_eval(const ex & x, const ex &y)
386 if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
387 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
389 return binomial(x, y).hold();
392 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
393 evalf_func(binomial_evalf));
396 // Order term function (for truncated power series)
399 static ex Order_eval(const ex & x)
401 if (is_ex_exactly_of_type(x, numeric)) {
404 return Order(_ex1()).hold();
407 } else if (is_ex_exactly_of_type(x, mul)) {
408 mul *m = static_cast<mul *>(x.bp);
409 // O(c*expr) -> O(expr)
410 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
411 return Order(x / m->op(m->nops() - 1)).hold();
413 return Order(x).hold();
416 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
418 // Just wrap the function into a pseries object
420 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
421 const symbol *s = static_cast<symbol *>(r.lhs().bp);
422 new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
423 return pseries(r, new_seq);
426 // Differentiation is handled in function::derivative because of its special requirements
428 REGISTER_FUNCTION(Order, eval_func(Order_eval).
429 series_func(Order_series).
430 latex_name("\\mathcal{O}"));
433 // Solve linear system
436 ex lsolve(const ex &eqns, const ex &symbols)
438 // solve a system of linear equations
439 if (eqns.info(info_flags::relation_equal)) {
440 if (!symbols.info(info_flags::symbol))
441 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
442 const ex sol = lsolve(lst(eqns),lst(symbols));
444 GINAC_ASSERT(sol.nops()==1);
445 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
447 return sol.op(0).op(1); // return rhs of first solution
451 if (!eqns.info(info_flags::list)) {
452 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
454 for (unsigned i=0; i<eqns.nops(); i++) {
455 if (!eqns.op(i).info(info_flags::relation_equal)) {
456 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
459 if (!symbols.info(info_flags::list)) {
460 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
462 for (unsigned i=0; i<symbols.nops(); i++) {
463 if (!symbols.op(i).info(info_flags::symbol)) {
464 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
468 // build matrix from equation system
469 matrix sys(eqns.nops(),symbols.nops());
470 matrix rhs(eqns.nops(),1);
471 matrix vars(symbols.nops(),1);
473 for (unsigned r=0; r<eqns.nops(); r++) {
474 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
476 for (unsigned c=0; c<symbols.nops(); c++) {
477 const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
478 linpart -= co*symbols.op(c);
481 linpart = linpart.expand();
485 // test if system is linear and fill vars matrix
486 for (unsigned i=0; i<symbols.nops(); i++) {
487 vars(i,0) = symbols.op(i);
488 if (sys.has(symbols.op(i)))
489 throw(std::logic_error("lsolve: system is not linear"));
490 if (rhs.has(symbols.op(i)))
491 throw(std::logic_error("lsolve: system is not linear"));
496 solution = sys.solve(vars,rhs);
497 } catch (const std::runtime_error & e) {
498 // Probably singular matrix or otherwise overdetermined system:
499 // It is consistent to return an empty list
502 GINAC_ASSERT(solution.cols()==1);
503 GINAC_ASSERT(solution.rows()==symbols.nops());
505 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
507 for (unsigned i=0; i<symbols.nops(); i++)
508 sollist.append(symbols.op(i)==solution(i,0));
513 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
514 * for static lib (so ginsh will see them). */
515 unsigned force_include_tgamma = function_index_tgamma;
516 unsigned force_include_zeta1 = function_index_zeta1;