ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
   7  *
   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.
  12  *
  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.
  17  *
  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
  21  */
  22
  23 #include <vector>
  24 #include <stdexcept>
  25
  26 #include "inifcns.h"
  27 #include "ex.h"
  28 #include "constant.h"
  29 #include "lst.h"
  30 #include "matrix.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "power.h"
  35 #include "relational.h"
  36 #include "pseries.h"
  37 #include "symbol.h"
  38 #include "utils.h"
  39
  40 #ifndef NO_NAMESPACE_GINAC
  41 namespace GiNaC {
  42 #endif // ndef NO_NAMESPACE_GINAC
  43
  44 //////////
  45 // absolute value
  46 //////////
  47
  48 static ex abs_evalf(const ex & arg)
  49 {
  50         BEGIN_TYPECHECK
  51                 TYPECHECK(arg,numeric)
  52         END_TYPECHECK(abs(arg))
  53
  54         return abs(ex_to_numeric(arg));
  55 }
  56
  57 static ex abs_eval(const ex & arg)
  58 {
  59         if (is_ex_exactly_of_type(arg, numeric))
  60                 return abs(ex_to_numeric(arg));
  61         else
  62                 return abs(arg).hold();
  63 }
  64
  65 REGISTER_FUNCTION(abs, eval_func(abs_eval).
  66                        evalf_func(abs_evalf));
  67
  68
  69 //////////
  70 // Complex sign
  71 //////////
  72
  73 static ex csgn_evalf(const ex & arg)
  74 {
  75         BEGIN_TYPECHECK
  76                 TYPECHECK(arg,numeric)
  77         END_TYPECHECK(csgn(arg))
  78
  79         return csgn(ex_to_numeric(arg));
  80 }
  81
  82 static ex csgn_eval(const ex & arg)
  83 {
  84         if (is_ex_exactly_of_type(arg, numeric))
  85                 return csgn(ex_to_numeric(arg));
  86
  87         else if (is_ex_exactly_of_type(arg, mul)) {
  88                 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
  89                 if (oc.is_real()) {
  90                         if (oc > 0)
  91                                 // csgn(42*x) -> csgn(x)
  92                                 return csgn(arg/oc).hold();
  93                         else
  94                                 // csgn(-42*x) -> -csgn(x)
  95                                 return -csgn(arg/oc).hold();
  96                 }
  97                 if (oc.real().is_zero()) {
  98                         if (oc.imag() > 0)
  99                                 // csgn(42*I*x) -> csgn(I*x)
 100                                 return csgn(I*arg/oc).hold();
 101                         else
 102                                 // csgn(-42*I*x) -> -csgn(I*x)
 103                                 return -csgn(I*arg/oc).hold();
 104                 }
 105         }
 106
 107         return csgn(arg).hold();
 108 }
 109
 110 static ex csgn_series(const ex & arg,
 111                       const relational & rel,
 112                       int order,
 113                       unsigned options)
 114 {
 115         const ex arg_pt = arg.subs(rel);
 116         if (arg_pt.info(info_flags::numeric)
 117          && ex_to_numeric(arg_pt).real().is_zero())
 118                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 119
 120         epvector seq;
 121         seq.push_back(expair(csgn(arg_pt), _ex0()));
 122         return pseries(rel,seq);
 123 }
 124
 125 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 126                         evalf_func(csgn_evalf).
 127                         series_func(csgn_series));
 128
 129
 130 //////////
 131 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
 132 //////////
 133
 134 static ex eta_evalf(const ex & x, const ex & y)
 135 {
 136         BEGIN_TYPECHECK
 137                 TYPECHECK(x,numeric)
 138                 TYPECHECK(y,numeric)
 139         END_TYPECHECK(eta(x,y))
 140
 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));
 145 }
 146
 147 static ex eta_eval(const ex & x, const ex & y)
 148 {
 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));
 156         }
 157
 158         return eta(x,y).hold();
 159 }
 160
 161 static ex eta_series(const ex & arg1,
 162                      const ex & arg2,
 163                      const relational & rel,
 164                      int order,
 165                      unsigned options)
 166 {
 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"));
 173         }
 174         epvector seq;
 175         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
 176         return pseries(rel,seq);
 177 }
 178
 179 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 180                        evalf_func(eta_evalf).
 181                        series_func(eta_series));
 182
 183
 184 //////////
 185 // dilogarithm
 186 //////////
 187
 188 static ex Li2_evalf(const ex & x)
 189 {
 190         BEGIN_TYPECHECK
 191                 TYPECHECK(x,numeric)
 192         END_TYPECHECK(Li2(x))
 193
 194         return Li2(ex_to_numeric(x));  // -> numeric Li2(numeric)
 195 }
 196
 197 static ex Li2_eval(const ex & x)
 198 {
 199         if (x.info(info_flags::numeric)) {
 200                 // Li2(0) -> 0
 201                 if (x.is_zero())
 202                         return _ex0();
 203                 // Li2(1) -> Pi^2/6
 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
 213                 if (x.is_equal(I))
 214                         return power(Pi,_ex2())/_ex_48() + Catalan*I;
 215                 // Li2(-I) -> -Pi^2/48-Catalan*I
 216                 if (x.is_equal(-I))
 217                         return power(Pi,_ex2())/_ex_48() - Catalan*I;
 218                 // Li2(float)
 219                 if (!x.info(info_flags::crational))
 220                         return Li2_evalf(x);
 221         }
 222
 223         return Li2(x).hold();
 224 }
 225
 226 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 227 {
 228         GINAC_ASSERT(deriv_param==0);
 229
 230         // d/dx Li2(x) -> -log(1-x)/x
 231         return -log(1-x)/x;
 232 }
 233
 234 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 235 {
 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()) {
 240                         // method:
 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 + ...
 247                         // and so on.
 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
 251                         // result.
 252                         const symbol s;
 253                         ex ser;
 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
 260                         epvector nseq;
 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
 271                         // obsolete!
 272                 }
 273                 // second special case: x==1 (branch point)
 274                 if (x_pt == _ex1()) {
 275                         // method:
 276                         // construct series manually in a dummy symbol s
 277                         const symbol s;
 278                         ex ser = zeta(2);
 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
 285                         epvector nseq;
 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);
 290                 }
 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) {
 294                         // method:
 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();
 299                         const symbol foo;
 300                         epvector seq;
 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);
 310                 }
 311         }
 312         // all other cases should be safe, by now:
 313         throw do_taylor();  // caught by function::series()
 314 }
 315
 316 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 317                        evalf_func(Li2_evalf).
 318                        derivative_func(Li2_deriv).
 319                        series_func(Li2_series));
 320
 321 //////////
 322 // trilogarithm
 323 //////////
 324
 325 static ex Li3_eval(const ex & x)
 326 {
 327         if (x.is_zero())
 328                 return x;
 329         return Li3(x).hold();
 330 }
 331
 332 REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
 333
 334 //////////
 335 // factorial
 336 //////////
 337
 338 static ex factorial_evalf(const ex & x)
 339 {
 340         return factorial(x).hold();
 341 }
 342
 343 static ex factorial_eval(const ex & x)
 344 {
 345         if (is_ex_exactly_of_type(x, numeric))
 346                 return factorial(ex_to_numeric(x));
 347         else
 348                 return factorial(x).hold();
 349 }
 350
 351 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 352                              evalf_func(factorial_evalf));
 353
 354 //////////
 355 // binomial
 356 //////////
 357
 358 static ex binomial_evalf(const ex & x, const ex & y)
 359 {
 360         return binomial(x, y).hold();
 361 }
 362
 363 static ex binomial_eval(const ex & x, const ex &y)
 364 {
 365         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
 366                 return binomial(ex_to_numeric(x), ex_to_numeric(y));
 367         else
 368                 return binomial(x, y).hold();
 369 }
 370
 371 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 372                             evalf_func(binomial_evalf));
 373
 374 //////////
 375 // Order term function (for truncated power series)
 376 //////////
 377
 378 static ex Order_eval(const ex & x)
 379 {
 380         if (is_ex_exactly_of_type(x, numeric)) {
 381                 // O(c) -> O(1) or 0
 382                 if (!x.is_zero())
 383                         return Order(_ex1()).hold();
 384                 else
 385                         return _ex0();
 386         } else if (is_ex_exactly_of_type(x, mul)) {
 387                 mul *m = static_cast<mul *>(x.bp);
 388                 // O(c*expr) -> O(expr)
 389                 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
 390                         return Order(x / m->op(m->nops() - 1)).hold();
 391         }
 392         return Order(x).hold();
 393 }
 394
 395 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 396 {
 397         // Just wrap the function into a pseries object
 398         epvector new_seq;
 399         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 400         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 401         new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
 402         return pseries(r, new_seq);
 403 }
 404
 405 // Differentiation is handled in function::derivative because of its special requirements
 406
 407 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 408                          series_func(Order_series));
 409
 410 //////////
 411 // Inert partial differentiation operator
 412 //////////
 413
 414 static ex Derivative_eval(const ex & f, const ex & l)
 415 {
 416         if (!is_ex_exactly_of_type(f, function)) {
 417                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
 418         }
 419         if (!is_ex_exactly_of_type(l, lst)) {
 420                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
 421         }
 422         return Derivative(f, l).hold();
 423 }
 424
 425 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 426
 427 //////////
 428 // Solve linear system
 429 //////////
 430
 431 ex lsolve(const ex &eqns, const ex &symbols)
 432 {
 433         // solve a system of linear equations
 434         if (eqns.info(info_flags::relation_equal)) {
 435                 if (!symbols.info(info_flags::symbol))
 436                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 437                 ex sol=lsolve(lst(eqns),lst(symbols));
 438
 439                 GINAC_ASSERT(sol.nops()==1);
 440                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
 441
 442                 return sol.op(0).op(1); // return rhs of first solution
 443         }
 444
 445         // syntax checks
 446         if (!eqns.info(info_flags::list)) {
 447                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 448         }
 449         for (unsigned i=0; i<eqns.nops(); i++) {
 450                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 451                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 452                 }
 453         }
 454         if (!symbols.info(info_flags::list)) {
 455                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 456         }
 457         for (unsigned i=0; i<symbols.nops(); i++) {
 458                 if (!symbols.op(i).info(info_flags::symbol)) {
 459                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 460                 }
 461         }
 462
 463         // build matrix from equation system
 464         matrix sys(eqns.nops(),symbols.nops());
 465         matrix rhs(eqns.nops(),1);
 466         matrix vars(symbols.nops(),1);
 467
 468         for (unsigned r=0; r<eqns.nops(); r++) {
 469                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 470                 ex linpart = eq;
 471                 for (unsigned c=0; c<symbols.nops(); c++) {
 472                         ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
 473                         linpart -= co*symbols.op(c);
 474                         sys.set(r,c,co);
 475                 }
 476                 linpart = linpart.expand();
 477                 rhs.set(r,0,-linpart);
 478         }
 479
 480         // test if system is linear and fill vars matrix
 481         for (unsigned i=0; i<symbols.nops(); i++) {
 482                 vars.set(i,0,symbols.op(i));
 483                 if (sys.has(symbols.op(i)))
 484                         throw(std::logic_error("lsolve: system is not linear"));
 485                 if (rhs.has(symbols.op(i)))
 486                         throw(std::logic_error("lsolve: system is not linear"));
 487         }
 488
 489         matrix solution;
 490         try {
 491                 solution = sys.solve(vars,rhs);
 492         } catch (const runtime_error & e) {
 493                 // Probably singular matrix or otherwise overdetermined system:
 494                 // It is consistent to return an empty list
 495                 return lst();
 496         }
 497         GINAC_ASSERT(solution.cols()==1);
 498         GINAC_ASSERT(solution.rows()==symbols.nops());
 499
 500         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 501         lst sollist;
 502         for (unsigned i=0; i<symbols.nops(); i++)
 503                 sollist.append(symbols.op(i)==solution(i,0));
 504
 505         return sollist;
 506 }
 507
 508 /** non-commutative power. */
 509 ex ncpower(const ex &basis, unsigned exponent)
 510 {
 511         if (exponent==0) {
 512                 return _ex1();
 513         }
 514
 515         exvector v;
 516         v.reserve(exponent);
 517         for (unsigned i=0; i<exponent; ++i) {
 518                 v.push_back(basis);
 519         }
 520
 521         return ncmul(v,1);
 522 }
 523
 524 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
 525  *  for static lib (so ginsh will see them). */
 526 unsigned force_include_tgamma = function_index_tgamma;
 527 unsigned force_include_zeta1 = function_index_zeta1;
 528
 529 #ifndef NO_NAMESPACE_GINAC
 530 } // namespace GiNaC
 531 #endif // ndef NO_NAMESPACE_GINAC