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 & x, const relational & rel, int order)
112 const ex x_pt = x.subs(rel);
113 if (x_pt.info(info_flags::numeric)) {
114 if (ex_to_numeric(x_pt).real().is_zero())
115 throw (std::domain_error("csgn_series(): on imaginary axis"));
117 seq.push_back(expair(csgn(x_pt), _ex0()));
118 return pseries(rel,seq);
121 seq.push_back(expair(csgn(x_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));
133 static ex Li2_eval(const ex & x)
139 if (x.is_equal(_ex1()))
140 return power(Pi,_ex2())/_ex6();
141 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
142 if (x.is_equal(_ex1_2()))
143 return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
144 // Li2(-1) -> -Pi^2/12
145 if (x.is_equal(_ex_1()))
146 return -power(Pi,_ex2())/_ex12();
147 // Li2(I) -> -Pi^2/48+Catalan*I
149 return power(Pi,_ex2())/_ex_48() + Catalan*I;
150 // Li2(-I) -> -Pi^2/48-Catalan*I
152 return power(Pi,_ex2())/_ex_48() - Catalan*I;
153 return Li2(x).hold();
156 static ex Li2_deriv(const ex & x, unsigned deriv_param)
158 GINAC_ASSERT(deriv_param==0);
160 // d/dx Li2(x) -> -log(1-x)/x
164 static ex Li2_series(const ex &x, const relational &rel, int order)
166 const ex x_pt = x.subs(rel);
167 if (!x_pt.is_zero() && !x_pt.is_equal(_ex1()))
168 throw do_taylor(); // caught by function::series()
169 // First case: x==0 (derivatives have poles)
170 if (x_pt.is_zero()) {
172 // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
173 // simply substitute x==0. The limit, however, exists: it is 1. We
174 // also know all higher derivatives' limits: (d/dx)^n Li2(x) == n!/n^2.
175 // So the primitive series expansion is Li2(x==0) == x + x^2/4 + x^3/9
177 // We first construct such a primitive series expansion manually in
178 // a dummy symbol s and then insert the argument's series expansion
179 // for s. Reexpanding the resulting series returns the desired result.
182 // construct manually the primitive expansion
183 for (int i=1; i<order; ++i)
184 ser += pow(s,i)/pow(numeric(i),numeric(2));
185 // substitute the argument's series expansion
186 ser = ser.subs(s==x.series(rel,order));
187 // maybe that was terminanting, so add a proper order term
189 nseq.push_back(expair(Order(_ex1()), numeric(order)));
190 ser += pseries(rel, nseq);
191 // reexpand will collapse the series again
192 ser = ser.series(rel,order);
194 // NOTE: Of course, this still does not allow us to compute anything
195 // like sin(Li2(x)).series(x==0,2), since then this code here is not
196 // reached and the derivative of sin(Li2(x)) doesn't allow the
197 // substitution x==0. Probably limits *are* needed for the general
200 // second problematic case: x real, >=1 (branch cut)
202 // TODO: Li2_series should do something around branch point?
203 // Careful: may involve logs!
206 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
207 derivative_func(Li2_deriv).
208 series_func(Li2_series));
214 static ex Li3_eval(const ex & x)
218 return Li3(x).hold();
221 REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
227 static ex factorial_evalf(const ex & x)
229 return factorial(x).hold();
232 static ex factorial_eval(const ex & x)
234 if (is_ex_exactly_of_type(x, numeric))
235 return factorial(ex_to_numeric(x));
237 return factorial(x).hold();
240 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
241 evalf_func(factorial_evalf));
247 static ex binomial_evalf(const ex & x, const ex & y)
249 return binomial(x, y).hold();
252 static ex binomial_eval(const ex & x, const ex &y)
254 if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
255 return binomial(ex_to_numeric(x), ex_to_numeric(y));
257 return binomial(x, y).hold();
260 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
261 evalf_func(binomial_evalf));
264 // Order term function (for truncated power series)
267 static ex Order_eval(const ex & x)
269 if (is_ex_exactly_of_type(x, numeric)) {
272 return Order(_ex1()).hold();
274 } else if (is_ex_exactly_of_type(x, mul)) {
276 mul *m = static_cast<mul *>(x.bp);
277 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric)) {
280 return Order(x / m->op(m->nops() - 1)).hold();
283 return Order(x).hold();
286 static ex Order_series(const ex & x, const relational & r, int order)
288 // Just wrap the function into a pseries object
290 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
291 const symbol *s = static_cast<symbol *>(r.lhs().bp);
292 new_seq.push_back(expair(Order(_ex1()), numeric(min(x.ldegree(*s), order))));
293 return pseries(r, new_seq);
296 // Differentiation is handled in function::derivative because of its special requirements
298 REGISTER_FUNCTION(Order, eval_func(Order_eval).
299 series_func(Order_series));
302 // Inert partial differentiation operator
305 static ex Derivative_eval(const ex & f, const ex & l)
307 if (!is_ex_exactly_of_type(f, function)) {
308 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
310 if (!is_ex_exactly_of_type(l, lst)) {
311 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
313 return Derivative(f, l).hold();
316 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
319 // Solve linear system
322 ex lsolve(const ex &eqns, const ex &symbols)
324 // solve a system of linear equations
325 if (eqns.info(info_flags::relation_equal)) {
326 if (!symbols.info(info_flags::symbol)) {
327 throw(std::invalid_argument("lsolve: 2nd argument must be a symbol"));
329 ex sol=lsolve(lst(eqns),lst(symbols));
331 GINAC_ASSERT(sol.nops()==1);
332 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
334 return sol.op(0).op(1); // return rhs of first solution
338 if (!eqns.info(info_flags::list)) {
339 throw(std::invalid_argument("lsolve: 1st argument must be a list"));
341 for (unsigned i=0; i<eqns.nops(); i++) {
342 if (!eqns.op(i).info(info_flags::relation_equal)) {
343 throw(std::invalid_argument("lsolve: 1st argument must be a list of equations"));
346 if (!symbols.info(info_flags::list)) {
347 throw(std::invalid_argument("lsolve: 2nd argument must be a list"));
349 for (unsigned i=0; i<symbols.nops(); i++) {
350 if (!symbols.op(i).info(info_flags::symbol)) {
351 throw(std::invalid_argument("lsolve: 2nd argument must be a list of symbols"));
355 // build matrix from equation system
356 matrix sys(eqns.nops(),symbols.nops());
357 matrix rhs(eqns.nops(),1);
358 matrix vars(symbols.nops(),1);
360 for (unsigned r=0; r<eqns.nops(); r++) {
361 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
363 for (unsigned c=0; c<symbols.nops(); c++) {
364 ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
365 linpart -= co*symbols.op(c);
368 linpart=linpart.expand();
369 rhs.set(r,0,-linpart);
372 // test if system is linear and fill vars matrix
373 for (unsigned i=0; i<symbols.nops(); i++) {
374 vars.set(i,0,symbols.op(i));
375 if (sys.has(symbols.op(i)))
376 throw(std::logic_error("lsolve: system is not linear"));
377 if (rhs.has(symbols.op(i)))
378 throw(std::logic_error("lsolve: system is not linear"));
381 //matrix solution=sys.solve(rhs);
384 solution = sys.fraction_free_elim(vars,rhs);
385 } catch (const runtime_error & e) {
386 // probably singular matrix (or other error)
387 // return empty solution list
388 // cerr << e.what() << endl;
392 // return a list of equations
393 if (solution.cols()!=1) {
394 throw(std::runtime_error("lsolve: strange number of columns returned from matrix::solve"));
396 if (solution.rows()!=symbols.nops()) {
397 cout << "symbols.nops()=" << symbols.nops() << endl;
398 cout << "solution.rows()=" << solution.rows() << endl;
399 throw(std::runtime_error("lsolve: strange number of rows returned from matrix::solve"));
402 // return list of the form lst(var1==sol1,var2==sol2,...)
404 for (unsigned i=0; i<symbols.nops(); i++) {
405 sollist.append(symbols.op(i)==solution(i,0));
411 /** non-commutative power. */
412 ex ncpower(const ex &basis, unsigned exponent)
420 for (unsigned i=0; i<exponent; ++i) {
427 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
428 * for static lib (so ginsh will see them). */
429 unsigned force_include_tgamma = function_index_tgamma;
430 unsigned force_include_zeta1 = function_index_zeta1;
432 #ifndef NO_NAMESPACE_GINAC
434 #endif // ndef NO_NAMESPACE_GINAC