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
35 #include "relational.h"
46 static ex abs_evalf(const ex & arg)
49 TYPECHECK(arg,numeric)
50 END_TYPECHECK(abs(arg))
52 return abs(ex_to_numeric(arg));
55 static ex abs_eval(const ex & arg)
57 if (is_ex_exactly_of_type(arg, numeric))
58 return abs(ex_to_numeric(arg));
60 return abs(arg).hold();
63 REGISTER_FUNCTION(abs, eval_func(abs_eval).
64 evalf_func(abs_evalf));
71 static ex csgn_evalf(const ex & arg)
74 TYPECHECK(arg,numeric)
75 END_TYPECHECK(csgn(arg))
77 return csgn(ex_to_numeric(arg));
80 static ex csgn_eval(const ex & arg)
82 if (is_ex_exactly_of_type(arg, numeric))
83 return csgn(ex_to_numeric(arg));
85 else if (is_ex_of_type(arg, mul) &&
86 is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
87 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
90 // csgn(42*x) -> csgn(x)
91 return csgn(arg/oc).hold();
93 // csgn(-42*x) -> -csgn(x)
94 return -csgn(arg/oc).hold();
96 if (oc.real().is_zero()) {
98 // csgn(42*I*x) -> csgn(I*x)
99 return csgn(I*arg/oc).hold();
101 // csgn(-42*I*x) -> -csgn(I*x)
102 return -csgn(I*arg/oc).hold();
106 return csgn(arg).hold();
109 static ex csgn_series(const ex & arg,
110 const relational & rel,
114 const ex arg_pt = arg.subs(rel);
115 if (arg_pt.info(info_flags::numeric)
116 && ex_to_numeric(arg_pt).real().is_zero()
117 && !(options & series_options::suppress_branchcut))
118 throw (std::domain_error("csgn_series(): on imaginary axis"));
121 seq.push_back(expair(csgn(arg_pt), _ex0()));
122 return pseries(rel,seq);
125 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
126 evalf_func(csgn_evalf).
127 series_func(csgn_series));
131 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
134 static ex eta_evalf(const ex & x, const ex & y)
139 END_TYPECHECK(eta(x,y))
141 numeric xim = imag(ex_to_numeric(x));
142 numeric yim = imag(ex_to_numeric(y));
143 numeric xyim = imag(ex_to_numeric(x*y));
144 return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
147 static ex eta_eval(const ex & x, const ex & y)
149 if (is_ex_exactly_of_type(x, numeric) &&
150 is_ex_exactly_of_type(y, numeric)) {
151 // don't call eta_evalf here because it would call Pi.evalf()!
152 numeric xim = imag(ex_to_numeric(x));
153 numeric yim = imag(ex_to_numeric(y));
154 numeric xyim = imag(ex_to_numeric(x*y));
155 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
158 return eta(x,y).hold();
161 static ex eta_series(const ex & arg1,
163 const relational & rel,
167 const ex arg1_pt = arg1.subs(rel);
168 const ex arg2_pt = arg2.subs(rel);
169 if (ex_to_numeric(arg1_pt).imag().is_zero() ||
170 ex_to_numeric(arg2_pt).imag().is_zero() ||
171 ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
172 throw (std::domain_error("eta_series(): on discontinuity"));
175 seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
176 return pseries(rel,seq);
179 REGISTER_FUNCTION(eta, eval_func(eta_eval).
180 evalf_func(eta_evalf).
181 series_func(eta_series).
182 latex_name("\\eta"));
189 static ex Li2_evalf(const ex & x)
193 END_TYPECHECK(Li2(x))
195 return Li2(ex_to_numeric(x)); // -> numeric Li2(numeric)
198 static ex Li2_eval(const ex & x)
200 if (x.info(info_flags::numeric)) {
205 if (x.is_equal(_ex1()))
206 return power(Pi,_ex2())/_ex6();
207 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
208 if (x.is_equal(_ex1_2()))
209 return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
210 // Li2(-1) -> -Pi^2/12
211 if (x.is_equal(_ex_1()))
212 return -power(Pi,_ex2())/_ex12();
213 // Li2(I) -> -Pi^2/48+Catalan*I
215 return power(Pi,_ex2())/_ex_48() + Catalan*I;
216 // Li2(-I) -> -Pi^2/48-Catalan*I
218 return power(Pi,_ex2())/_ex_48() - Catalan*I;
220 if (!x.info(info_flags::crational))
224 return Li2(x).hold();
227 static ex Li2_deriv(const ex & x, unsigned deriv_param)
229 GINAC_ASSERT(deriv_param==0);
231 // d/dx Li2(x) -> -log(1-x)/x
235 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
237 const ex x_pt = x.subs(rel);
238 if (x_pt.info(info_flags::numeric)) {
239 // First special case: x==0 (derivatives have poles)
240 if (x_pt.is_zero()) {
242 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
243 // simply substitute x==0. The limit, however, exists: it is 1.
244 // We also know all higher derivatives' limits:
245 // (d/dx)^n Li2(x) == n!/n^2.
246 // So the primitive series expansion is
247 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
249 // We first construct such a primitive series expansion manually in
250 // a dummy symbol s and then insert the argument's series expansion
251 // for s. Reexpanding the resulting series returns the desired
255 // manually construct the primitive expansion
256 for (int i=1; i<order; ++i)
257 ser += pow(s,i) / pow(numeric(i), _num2());
258 // substitute the argument's series expansion
259 ser = ser.subs(s==x.series(rel, order));
260 // maybe that was terminating, so add a proper order term
262 nseq.push_back(expair(Order(_ex1()), order));
263 ser += pseries(rel, nseq);
264 // reexpanding it will collapse the series again
265 return ser.series(rel, order);
266 // NB: Of course, this still does not allow us to compute anything
267 // like sin(Li2(x)).series(x==0,2), since then this code here is
268 // not reached and the derivative of sin(Li2(x)) doesn't allow the
269 // substitution x==0. Probably limits *are* needed for the general
270 // cases. In case L'Hospital's rule is implemented for limits and
271 // basic::series() takes care of this, this whole block is probably
274 // second special case: x==1 (branch point)
275 if (x_pt == _ex1()) {
277 // construct series manually in a dummy symbol s
280 // manually construct the primitive expansion
281 for (int i=1; i<order; ++i)
282 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
283 // substitute the argument's series expansion
284 ser = ser.subs(s==x.series(rel, order));
285 // maybe that was terminating, so add a proper order term
287 nseq.push_back(expair(Order(_ex1()), order));
288 ser += pseries(rel, nseq);
289 // reexpanding it will collapse the series again
290 return ser.series(rel, order);
292 // third special case: x real, >=1 (branch cut)
293 if (!(options & series_options::suppress_branchcut) &&
294 ex_to_numeric(x_pt).is_real() && ex_to_numeric(x_pt)>1) {
296 // This is the branch cut: assemble the primitive series manually
297 // and then add the corresponding complex step function.
298 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
299 const ex point = rel.rhs();
302 // zeroth order term:
303 seq.push_back(expair(Li2(x_pt), _ex0()));
304 // compute the intermediate terms:
305 ex replarg = series(Li2(x), *s==foo, order);
306 for (unsigned i=1; i<replarg.nops()-1; ++i)
307 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
308 // append an order term:
309 seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
310 return pseries(rel, seq);
313 // all other cases should be safe, by now:
314 throw do_taylor(); // caught by function::series()
317 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
318 evalf_func(Li2_evalf).
319 derivative_func(Li2_deriv).
320 series_func(Li2_series).
321 latex_name("\\mbox{Li}_2"));
327 static ex Li3_eval(const ex & x)
331 return Li3(x).hold();
334 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
335 latex_name("\\mbox{Li}_3"));
341 static ex factorial_evalf(const ex & x)
343 return factorial(x).hold();
346 static ex factorial_eval(const ex & x)
348 if (is_ex_exactly_of_type(x, numeric))
349 return factorial(ex_to_numeric(x));
351 return factorial(x).hold();
354 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
355 evalf_func(factorial_evalf));
361 static ex binomial_evalf(const ex & x, const ex & y)
363 return binomial(x, y).hold();
366 static ex binomial_eval(const ex & x, const ex &y)
368 if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
369 return binomial(ex_to_numeric(x), ex_to_numeric(y));
371 return binomial(x, y).hold();
374 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
375 evalf_func(binomial_evalf));
378 // Order term function (for truncated power series)
381 static ex Order_eval(const ex & x)
383 if (is_ex_exactly_of_type(x, numeric)) {
386 return Order(_ex1()).hold();
389 } else if (is_ex_exactly_of_type(x, mul)) {
390 mul *m = static_cast<mul *>(x.bp);
391 // O(c*expr) -> O(expr)
392 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
393 return Order(x / m->op(m->nops() - 1)).hold();
395 return Order(x).hold();
398 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
400 // Just wrap the function into a pseries object
402 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
403 const symbol *s = static_cast<symbol *>(r.lhs().bp);
404 new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
405 return pseries(r, new_seq);
408 // Differentiation is handled in function::derivative because of its special requirements
410 REGISTER_FUNCTION(Order, eval_func(Order_eval).
411 series_func(Order_series).
412 latex_name("\\mathcal{O}"));
415 // Inert partial differentiation operator
418 static ex Derivative_eval(const ex & f, const ex & l)
420 if (!is_ex_exactly_of_type(f, function)) {
421 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
423 if (!is_ex_exactly_of_type(l, lst)) {
424 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
426 return Derivative(f, l).hold();
429 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
432 // Solve linear system
435 ex lsolve(const ex &eqns, const ex &symbols)
437 // solve a system of linear equations
438 if (eqns.info(info_flags::relation_equal)) {
439 if (!symbols.info(info_flags::symbol))
440 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
441 ex sol=lsolve(lst(eqns),lst(symbols));
443 GINAC_ASSERT(sol.nops()==1);
444 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
446 return sol.op(0).op(1); // return rhs of first solution
450 if (!eqns.info(info_flags::list)) {
451 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
453 for (unsigned i=0; i<eqns.nops(); i++) {
454 if (!eqns.op(i).info(info_flags::relation_equal)) {
455 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
458 if (!symbols.info(info_flags::list)) {
459 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
461 for (unsigned i=0; i<symbols.nops(); i++) {
462 if (!symbols.op(i).info(info_flags::symbol)) {
463 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
467 // build matrix from equation system
468 matrix sys(eqns.nops(),symbols.nops());
469 matrix rhs(eqns.nops(),1);
470 matrix vars(symbols.nops(),1);
472 for (unsigned r=0; r<eqns.nops(); r++) {
473 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
475 for (unsigned c=0; c<symbols.nops(); c++) {
476 ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
477 linpart -= co*symbols.op(c);
480 linpart = linpart.expand();
481 rhs.set(r,0,-linpart);
484 // test if system is linear and fill vars matrix
485 for (unsigned i=0; i<symbols.nops(); i++) {
486 vars.set(i,0,symbols.op(i));
487 if (sys.has(symbols.op(i)))
488 throw(std::logic_error("lsolve: system is not linear"));
489 if (rhs.has(symbols.op(i)))
490 throw(std::logic_error("lsolve: system is not linear"));
495 solution = sys.solve(vars,rhs);
496 } catch (const std::runtime_error & e) {
497 // Probably singular matrix or otherwise overdetermined system:
498 // It is consistent to return an empty list
501 GINAC_ASSERT(solution.cols()==1);
502 GINAC_ASSERT(solution.rows()==symbols.nops());
504 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
506 for (unsigned i=0; i<symbols.nops(); i++)
507 sollist.append(symbols.op(i)==solution(i,0));
512 /** non-commutative power. */
513 ex ncpow(const ex & basis, unsigned exponent)
520 for (unsigned i=0; i<exponent; ++i)
523 return ncmul(v, true);
526 // Symmetrize/antisymmetrize over a vector of objects
527 static ex symm(const ex & e, exvector::const_iterator first, exvector::const_iterator last, bool asymmetric)
529 // Need at least 2 objects for this operation
530 int num = last - first;
534 // Sort object vector, transform it into a list, and make a copy so we
535 // will know which objects get substituted for which
537 iv_lst.insert(iv_lst.begin(), first, last);
538 shaker_sort(iv_lst.begin(), iv_lst.end(), ex_is_less());
539 lst orig_lst(iv_lst);
541 // Loop over all permutations (the first permutation, which is the
542 // identity, is unrolled)
544 while (next_permutation(iv_lst.begin(), iv_lst.end(), ex_is_less())) {
545 ex term = e.subs(orig_lst, lst(iv_lst));
547 exlist test_lst = iv_lst;
548 term *= permutation_sign(test_lst.begin(), test_lst.end(), ex_is_less());
552 return sum / factorial(numeric(num));
555 ex symmetrize(const ex & e, exvector::const_iterator first, exvector::const_iterator last)
557 return symm(e, first, last, false);
560 ex antisymmetrize(const ex & e, exvector::const_iterator first, exvector::const_iterator last)
562 return symm(e, first, last, true);
565 /** Symmetrize expression over a list of objects (symbols, indices). */
566 ex ex::symmetrize(const lst & l) const
570 for (unsigned i=0; i<l.nops(); i++)
571 v.push_back(l.op(i));
572 return symm(*this, v.begin(), v.end(), false);
575 /** Antisymmetrize expression over a list of objects (symbols, indices). */
576 ex ex::antisymmetrize(const lst & l) const
580 for (unsigned i=0; i<l.nops(); i++)
581 v.push_back(l.op(i));
582 return symm(*this, v.begin(), v.end(), true);
585 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
586 * for static lib (so ginsh will see them). */
587 unsigned force_include_tgamma = function_index_tgamma;
588 unsigned force_include_zeta1 = function_index_zeta1;