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_exactly_of_type(arg, mul)) {
86 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
89 // csgn(42*x) -> csgn(x)
90 return csgn(arg/oc).hold();
92 // csgn(-42*x) -> -csgn(x)
93 return -csgn(arg/oc).hold();
95 if (oc.real().is_zero()) {
97 // csgn(42*I*x) -> csgn(I*x)
98 return csgn(I*arg/oc).hold();
100 // csgn(-42*I*x) -> -csgn(I*x)
101 return -csgn(I*arg/oc).hold();
105 return csgn(arg).hold();
108 static ex csgn_series(const ex & arg,
109 const relational & rel,
113 const ex arg_pt = arg.subs(rel);
114 if (arg_pt.info(info_flags::numeric)
115 && ex_to_numeric(arg_pt).real().is_zero()
116 && !(options & series_options::suppress_branchcut))
117 throw (std::domain_error("csgn_series(): on imaginary axis"));
120 seq.push_back(expair(csgn(arg_pt), _ex0()));
121 return pseries(rel,seq);
124 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
125 evalf_func(csgn_evalf).
126 series_func(csgn_series));
130 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
133 static ex eta_evalf(const ex & x, const ex & y)
138 END_TYPECHECK(eta(x,y))
140 numeric xim = imag(ex_to_numeric(x));
141 numeric yim = imag(ex_to_numeric(y));
142 numeric xyim = imag(ex_to_numeric(x*y));
143 return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
146 static ex eta_eval(const ex & x, const ex & y)
148 if (is_ex_exactly_of_type(x, numeric) &&
149 is_ex_exactly_of_type(y, numeric)) {
150 // don't call eta_evalf here because it would call Pi.evalf()!
151 numeric xim = imag(ex_to_numeric(x));
152 numeric yim = imag(ex_to_numeric(y));
153 numeric xyim = imag(ex_to_numeric(x*y));
154 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
157 return eta(x,y).hold();
160 static ex eta_series(const ex & arg1,
162 const relational & rel,
166 const ex arg1_pt = arg1.subs(rel);
167 const ex arg2_pt = arg2.subs(rel);
168 if (ex_to_numeric(arg1_pt).imag().is_zero() ||
169 ex_to_numeric(arg2_pt).imag().is_zero() ||
170 ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
171 throw (std::domain_error("eta_series(): on discontinuity"));
174 seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
175 return pseries(rel,seq);
178 REGISTER_FUNCTION(eta, eval_func(eta_eval).
179 evalf_func(eta_evalf).
180 series_func(eta_series).
181 latex_name("\\eta"));
188 static ex Li2_evalf(const ex & x)
192 END_TYPECHECK(Li2(x))
194 return Li2(ex_to_numeric(x)); // -> numeric Li2(numeric)
197 static ex Li2_eval(const ex & x)
199 if (x.info(info_flags::numeric)) {
204 if (x.is_equal(_ex1()))
205 return power(Pi,_ex2())/_ex6();
206 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
207 if (x.is_equal(_ex1_2()))
208 return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
209 // Li2(-1) -> -Pi^2/12
210 if (x.is_equal(_ex_1()))
211 return -power(Pi,_ex2())/_ex12();
212 // Li2(I) -> -Pi^2/48+Catalan*I
214 return power(Pi,_ex2())/_ex_48() + Catalan*I;
215 // Li2(-I) -> -Pi^2/48-Catalan*I
217 return power(Pi,_ex2())/_ex_48() - Catalan*I;
219 if (!x.info(info_flags::crational))
223 return Li2(x).hold();
226 static ex Li2_deriv(const ex & x, unsigned deriv_param)
228 GINAC_ASSERT(deriv_param==0);
230 // d/dx Li2(x) -> -log(1-x)/x
234 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
236 const ex x_pt = x.subs(rel);
237 if (x_pt.info(info_flags::numeric)) {
238 // First special case: x==0 (derivatives have poles)
239 if (x_pt.is_zero()) {
241 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
242 // simply substitute x==0. The limit, however, exists: it is 1.
243 // We also know all higher derivatives' limits:
244 // (d/dx)^n Li2(x) == n!/n^2.
245 // So the primitive series expansion is
246 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
248 // We first construct such a primitive series expansion manually in
249 // a dummy symbol s and then insert the argument's series expansion
250 // for s. Reexpanding the resulting series returns the desired
254 // manually construct the primitive expansion
255 for (int i=1; i<order; ++i)
256 ser += pow(s,i) / pow(numeric(i), _num2());
257 // substitute the argument's series expansion
258 ser = ser.subs(s==x.series(rel, order));
259 // maybe that was terminating, so add a proper order term
261 nseq.push_back(expair(Order(_ex1()), order));
262 ser += pseries(rel, nseq);
263 // reexpanding it will collapse the series again
264 return ser.series(rel, order);
265 // NB: Of course, this still does not allow us to compute anything
266 // like sin(Li2(x)).series(x==0,2), since then this code here is
267 // not reached and the derivative of sin(Li2(x)) doesn't allow the
268 // substitution x==0. Probably limits *are* needed for the general
269 // cases. In case L'Hospital's rule is implemented for limits and
270 // basic::series() takes care of this, this whole block is probably
273 // second special case: x==1 (branch point)
274 if (x_pt == _ex1()) {
276 // construct series manually in a dummy symbol s
279 // manually construct the primitive expansion
280 for (int i=1; i<order; ++i)
281 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
282 // substitute the argument's series expansion
283 ser = ser.subs(s==x.series(rel, order));
284 // maybe that was terminating, so add a proper order term
286 nseq.push_back(expair(Order(_ex1()), order));
287 ser += pseries(rel, nseq);
288 // reexpanding it will collapse the series again
289 return ser.series(rel, order);
291 // third special case: x real, >=1 (branch cut)
292 if (!(options & series_options::suppress_branchcut) &&
293 ex_to_numeric(x_pt).is_real() && ex_to_numeric(x_pt)>1) {
295 // This is the branch cut: assemble the primitive series manually
296 // and then add the corresponding complex step function.
297 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
298 const ex point = rel.rhs();
301 // zeroth order term:
302 seq.push_back(expair(Li2(x_pt), _ex0()));
303 // compute the intermediate terms:
304 ex replarg = series(Li2(x), *s==foo, order);
305 for (unsigned i=1; i<replarg.nops()-1; ++i)
306 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
307 // append an order term:
308 seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
309 return pseries(rel, seq);
312 // all other cases should be safe, by now:
313 throw do_taylor(); // caught by function::series()
316 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
317 evalf_func(Li2_evalf).
318 derivative_func(Li2_deriv).
319 series_func(Li2_series).
320 latex_name("\\mbox{Li}_2"));
326 static ex Li3_eval(const ex & x)
330 return Li3(x).hold();
333 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
334 latex_name("\\mbox{Li}_3"));
340 static ex factorial_evalf(const ex & x)
342 return factorial(x).hold();
345 static ex factorial_eval(const ex & x)
347 if (is_ex_exactly_of_type(x, numeric))
348 return factorial(ex_to_numeric(x));
350 return factorial(x).hold();
353 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
354 evalf_func(factorial_evalf));
360 static ex binomial_evalf(const ex & x, const ex & y)
362 return binomial(x, y).hold();
365 static ex binomial_eval(const ex & x, const ex &y)
367 if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
368 return binomial(ex_to_numeric(x), ex_to_numeric(y));
370 return binomial(x, y).hold();
373 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
374 evalf_func(binomial_evalf));
377 // Order term function (for truncated power series)
380 static ex Order_eval(const ex & x)
382 if (is_ex_exactly_of_type(x, numeric)) {
385 return Order(_ex1()).hold();
388 } else if (is_ex_exactly_of_type(x, mul)) {
389 mul *m = static_cast<mul *>(x.bp);
390 // O(c*expr) -> O(expr)
391 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
392 return Order(x / m->op(m->nops() - 1)).hold();
394 return Order(x).hold();
397 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
399 // Just wrap the function into a pseries object
401 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
402 const symbol *s = static_cast<symbol *>(r.lhs().bp);
403 new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
404 return pseries(r, new_seq);
407 // Differentiation is handled in function::derivative because of its special requirements
409 REGISTER_FUNCTION(Order, eval_func(Order_eval).
410 series_func(Order_series).
411 latex_name("\\mathcal{O}"));
414 // Inert partial differentiation operator
417 static ex Derivative_eval(const ex & f, const ex & l)
419 if (!is_ex_exactly_of_type(f, function)) {
420 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
422 if (!is_ex_exactly_of_type(l, lst)) {
423 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
425 return Derivative(f, l).hold();
428 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
431 // Solve linear system
434 ex lsolve(const ex &eqns, const ex &symbols)
436 // solve a system of linear equations
437 if (eqns.info(info_flags::relation_equal)) {
438 if (!symbols.info(info_flags::symbol))
439 throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
440 ex sol=lsolve(lst(eqns),lst(symbols));
442 GINAC_ASSERT(sol.nops()==1);
443 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
445 return sol.op(0).op(1); // return rhs of first solution
449 if (!eqns.info(info_flags::list)) {
450 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
452 for (unsigned i=0; i<eqns.nops(); i++) {
453 if (!eqns.op(i).info(info_flags::relation_equal)) {
454 throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
457 if (!symbols.info(info_flags::list)) {
458 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
460 for (unsigned i=0; i<symbols.nops(); i++) {
461 if (!symbols.op(i).info(info_flags::symbol)) {
462 throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
466 // build matrix from equation system
467 matrix sys(eqns.nops(),symbols.nops());
468 matrix rhs(eqns.nops(),1);
469 matrix vars(symbols.nops(),1);
471 for (unsigned r=0; r<eqns.nops(); r++) {
472 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
474 for (unsigned c=0; c<symbols.nops(); c++) {
475 ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
476 linpart -= co*symbols.op(c);
479 linpart = linpart.expand();
480 rhs.set(r,0,-linpart);
483 // test if system is linear and fill vars matrix
484 for (unsigned i=0; i<symbols.nops(); i++) {
485 vars.set(i,0,symbols.op(i));
486 if (sys.has(symbols.op(i)))
487 throw(std::logic_error("lsolve: system is not linear"));
488 if (rhs.has(symbols.op(i)))
489 throw(std::logic_error("lsolve: system is not linear"));
494 solution = sys.solve(vars,rhs);
495 } catch (const std::runtime_error & e) {
496 // Probably singular matrix or otherwise overdetermined system:
497 // It is consistent to return an empty list
500 GINAC_ASSERT(solution.cols()==1);
501 GINAC_ASSERT(solution.rows()==symbols.nops());
503 // return list of equations of the form lst(var1==sol1,var2==sol2,...)
505 for (unsigned i=0; i<symbols.nops(); i++)
506 sollist.append(symbols.op(i)==solution(i,0));
511 /** non-commutative power. */
512 ex ncpower(const ex &basis, unsigned exponent)
520 for (unsigned i=0; i<exponent; ++i) {
527 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
528 * for static lib (so ginsh will see them). */
529 unsigned force_include_tgamma = function_index_tgamma;
530 unsigned force_include_zeta1 = function_index_zeta1;