3 * Implementation of GiNaC's initially known functions. */
6 * GiNaC Copyright (C) 1999-2000 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"
40 #ifndef NO_NAMESPACE_GINAC
42 #endif // ndef NO_NAMESPACE_GINAC
48 static ex abs_evalf(const ex & x)
54 return abs(ex_to_numeric(x));
57 static ex abs_eval(const ex & x)
59 if (is_ex_exactly_of_type(x, numeric))
60 return abs(ex_to_numeric(x));
65 REGISTER_FUNCTION(abs, eval_func(abs_eval).
66 evalf_func(abs_evalf));
73 static ex csgn_evalf(const ex & x)
77 END_TYPECHECK(csgn(x))
79 return csgn(ex_to_numeric(x));
82 static ex csgn_eval(const ex & x)
84 if (is_ex_exactly_of_type(x, numeric))
85 return csgn(ex_to_numeric(x));
87 else if (is_ex_exactly_of_type(x, mul)) {
88 numeric oc = ex_to_numeric(x.op(x.nops()-1));
91 // csgn(42*x) -> csgn(x)
92 return csgn(x/oc).hold();
94 // csgn(-42*x) -> -csgn(x)
95 return -csgn(x/oc).hold();
97 if (oc.real().is_zero()) {
99 // csgn(42*I*x) -> csgn(I*x)
100 return csgn(I*x/oc).hold();
102 // csgn(-42*I*x) -> -csgn(I*x)
103 return -csgn(I*x/oc).hold();
107 return csgn(x).hold();
110 static ex csgn_series(const ex & arg,
111 const relational & rel,
115 const ex arg_pt = arg.subs(rel);
116 if (arg_pt.info(info_flags::numeric)) {
117 if (ex_to_numeric(arg_pt).real().is_zero())
118 throw (std::domain_error("csgn_series(): on imaginary axis"));
120 seq.push_back(expair(csgn(arg_pt), _ex0()));
121 return pseries(rel,seq);
124 seq.push_back(expair(csgn(arg_pt), _ex0()));
125 return pseries(rel,seq);
128 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
129 evalf_func(csgn_evalf).
130 series_func(csgn_series));
136 static ex Li2_evalf(const ex & x)
140 END_TYPECHECK(Li2(x))
142 return Li2(ex_to_numeric(x)); // -> numeric Li2(numeric)
145 static ex Li2_eval(const ex & x)
147 if (x.info(info_flags::numeric)) {
152 if (x.is_equal(_ex1()))
153 return power(Pi,_ex2())/_ex6();
154 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
155 if (x.is_equal(_ex1_2()))
156 return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
157 // Li2(-1) -> -Pi^2/12
158 if (x.is_equal(_ex_1()))
159 return -power(Pi,_ex2())/_ex12();
160 // Li2(I) -> -Pi^2/48+Catalan*I
162 return power(Pi,_ex2())/_ex_48() + Catalan*I;
163 // Li2(-I) -> -Pi^2/48-Catalan*I
165 return power(Pi,_ex2())/_ex_48() - Catalan*I;
167 if (!x.info(info_flags::crational))
171 return Li2(x).hold();
174 static ex Li2_deriv(const ex & x, unsigned deriv_param)
176 GINAC_ASSERT(deriv_param==0);
178 // d/dx Li2(x) -> -log(1-x)/x
182 static ex Li2_series(const ex &x, const relational &rel, int order, bool branchcut)
184 const ex x_pt = x.subs(rel);
185 if (x_pt.info(info_flags::numeric)) {
186 // First special case: x==0 (derivatives have poles)
187 if (x_pt.is_zero()) {
189 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
190 // simply substitute x==0. The limit, however, exists: it is 1.
191 // We also know all higher derivatives' limits:
192 // (d/dx)^n Li2(x) == n!/n^2.
193 // So the primitive series expansion is
194 // Li2(x==0) == x + x^2/4 + x^3/9 + ...
196 // We first construct such a primitive series expansion manually in
197 // a dummy symbol s and then insert the argument's series expansion
198 // for s. Reexpanding the resulting series returns the desired
202 // manually construct the primitive expansion
203 for (int i=1; i<order; ++i)
204 ser += pow(s,i) / pow(numeric(i), _num2());
205 // substitute the argument's series expansion
206 ser = ser.subs(s==x.series(rel, order));
207 // maybe that was terminating, so add a proper order term
209 nseq.push_back(expair(Order(_ex1()), order));
210 ser += pseries(rel, nseq);
211 // reexpanding it will collapse the series again
212 return ser.series(rel, order);
213 // NB: Of course, this still does not allow us to compute anything
214 // like sin(Li2(x)).series(x==0,2), since then this code here is
215 // not reached and the derivative of sin(Li2(x)) doesn't allow the
216 // substitution x==0. Probably limits *are* needed for the general
217 // cases. In case L'Hospital's rule is implemented for limits and
218 // basic::series() takes care of this, this whole block is probably
221 // second special case: x==1 (branch point)
222 if (x_pt == _ex1()) {
224 // construct series manually in a dummy symbol s
227 // manually construct the primitive expansion
228 for (int i=1; i<order; ++i)
229 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
230 // substitute the argument's series expansion
231 ser = ser.subs(s==x.series(rel, order));
232 // maybe that was terminating, so add a proper order term
234 nseq.push_back(expair(Order(_ex1()), order));
235 ser += pseries(rel, nseq);
236 // reexpanding it will collapse the series again
237 return ser.series(rel, order);
239 // third special case: x real, >=1 (branch cut)
240 if (ex_to_numeric(x_pt).is_real() && ex_to_numeric(x_pt)>1) {
242 // This is the branch cut: assemble the primitive series manually
243 // and then add the corresponding complex step function.
244 const symbol *s = static_cast<symbol *>(rel.lhs().bp);
245 const ex point = rel.rhs();
248 // zeroth order term:
249 seq.push_back(expair(Li2(x_pt), _ex0()));
250 // compute the intermediate terms:
251 ex replarg = series(Li2(x), *s==foo, order);
252 for (unsigned i=1; i<replarg.nops()-1; ++i)
253 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,branchcut).op(0).subs(foo==*s),i));
254 // append an order term:
255 seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
256 return pseries(rel, seq);
259 // all other cases should be safe, by now:
260 throw do_taylor(); // caught by function::series()
263 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
264 evalf_func(Li2_evalf).
265 derivative_func(Li2_deriv).
266 series_func(Li2_series));
272 static ex Li3_eval(const ex & x)
276 return Li3(x).hold();
279 REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
285 static ex factorial_evalf(const ex & x)
287 return factorial(x).hold();
290 static ex factorial_eval(const ex & x)
292 if (is_ex_exactly_of_type(x, numeric))
293 return factorial(ex_to_numeric(x));
295 return factorial(x).hold();
298 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
299 evalf_func(factorial_evalf));
305 static ex binomial_evalf(const ex & x, const ex & y)
307 return binomial(x, y).hold();
310 static ex binomial_eval(const ex & x, const ex &y)
312 if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
313 return binomial(ex_to_numeric(x), ex_to_numeric(y));
315 return binomial(x, y).hold();
318 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
319 evalf_func(binomial_evalf));
322 // Order term function (for truncated power series)
325 static ex Order_eval(const ex & x)
327 if (is_ex_exactly_of_type(x, numeric)) {
330 return Order(_ex1()).hold();
332 } else if (is_ex_exactly_of_type(x, mul)) {
334 mul *m = static_cast<mul *>(x.bp);
335 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric)) {
338 return Order(x / m->op(m->nops() - 1)).hold();
341 return Order(x).hold();
344 static ex Order_series(const ex & x, const relational & r, int order, bool branchcut)
346 // Just wrap the function into a pseries object
348 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
349 const symbol *s = static_cast<symbol *>(r.lhs().bp);
350 new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
351 return pseries(r, new_seq);
354 // Differentiation is handled in function::derivative because of its special requirements
356 REGISTER_FUNCTION(Order, eval_func(Order_eval).
357 series_func(Order_series));
360 // Inert partial differentiation operator
363 static ex Derivative_eval(const ex & f, const ex & l)
365 if (!is_ex_exactly_of_type(f, function)) {
366 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
368 if (!is_ex_exactly_of_type(l, lst)) {
369 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
371 return Derivative(f, l).hold();
374 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
377 // Solve linear system
380 ex lsolve(const ex &eqns, const ex &symbols)
382 // solve a system of linear equations
383 if (eqns.info(info_flags::relation_equal)) {
384 if (!symbols.info(info_flags::symbol))
385 throw(std::invalid_argument("lsolve: 2nd argument must be a symbol"));
386 ex sol=lsolve(lst(eqns),lst(symbols));
388 GINAC_ASSERT(sol.nops()==1);
389 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
391 return sol.op(0).op(1); // return rhs of first solution
395 if (!eqns.info(info_flags::list)) {
396 throw(std::invalid_argument("lsolve: 1st argument must be a list"));
398 for (unsigned i=0; i<eqns.nops(); i++) {
399 if (!eqns.op(i).info(info_flags::relation_equal)) {
400 throw(std::invalid_argument("lsolve: 1st argument must be a list of equations"));
403 if (!symbols.info(info_flags::list)) {
404 throw(std::invalid_argument("lsolve: 2nd argument must be a list"));
406 for (unsigned i=0; i<symbols.nops(); i++) {
407 if (!symbols.op(i).info(info_flags::symbol)) {
408 throw(std::invalid_argument("lsolve: 2nd argument must be a list of symbols"));
412 // build matrix from equation system
413 matrix sys(eqns.nops(),symbols.nops());
414 matrix rhs(eqns.nops(),1);
415 matrix vars(symbols.nops(),1);
417 for (unsigned r=0; r<eqns.nops(); r++) {
418 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
420 for (unsigned c=0; c<symbols.nops(); c++) {
421 ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
422 linpart -= co*symbols.op(c);
425 linpart=linpart.expand();
426 rhs.set(r,0,-linpart);
429 // test if system is linear and fill vars matrix
430 for (unsigned i=0; i<symbols.nops(); i++) {
431 vars.set(i,0,symbols.op(i));
432 if (sys.has(symbols.op(i)))
433 throw(std::logic_error("lsolve: system is not linear"));
434 if (rhs.has(symbols.op(i)))
435 throw(std::logic_error("lsolve: system is not linear"));
438 //matrix solution=sys.solve(rhs);
441 solution = sys.fraction_free_elim(vars,rhs);
442 } catch (const runtime_error & e) {
443 // probably singular matrix (or other error)
444 // return empty solution list
445 // cerr << e.what() << endl;
449 // return a list of equations
450 if (solution.cols()!=1) {
451 throw(std::runtime_error("lsolve: strange number of columns returned from matrix::solve"));
453 if (solution.rows()!=symbols.nops()) {
454 cout << "symbols.nops()=" << symbols.nops() << endl;
455 cout << "solution.rows()=" << solution.rows() << endl;
456 throw(std::runtime_error("lsolve: strange number of rows returned from matrix::solve"));
459 // return list of the form lst(var1==sol1,var2==sol2,...)
461 for (unsigned i=0; i<symbols.nops(); i++) {
462 sollist.append(symbols.op(i)==solution(i,0));
468 /** non-commutative power. */
469 ex ncpower(const ex &basis, unsigned exponent)
477 for (unsigned i=0; i<exponent; ++i) {
484 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
485 * for static lib (so ginsh will see them). */
486 unsigned force_include_tgamma = function_index_tgamma;
487 unsigned force_include_zeta1 = function_index_zeta1;
489 #ifndef NO_NAMESPACE_GINAC
491 #endif // ndef NO_NAMESPACE_GINAC