ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 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 "power.h"
  33 #include "relational.h"
  34 #include "pseries.h"
  35 #include "symbol.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // absolute value
  42 //////////
  43
  44 static ex abs_evalf(const ex & arg)
  45 {
  46         BEGIN_TYPECHECK
  47                 TYPECHECK(arg,numeric)
  48         END_TYPECHECK(abs(arg))
  49
  50         return abs(ex_to<numeric>(arg));
  51 }
  52
  53 static ex abs_eval(const ex & arg)
  54 {
  55         if (is_ex_exactly_of_type(arg, numeric))
  56                 return abs(ex_to<numeric>(arg));
  57         else
  58                 return abs(arg).hold();
  59 }
  60
  61 REGISTER_FUNCTION(abs, eval_func(abs_eval).
  62                        evalf_func(abs_evalf));
  63
  64
  65 //////////
  66 // Complex sign
  67 //////////
  68
  69 static ex csgn_evalf(const ex & arg)
  70 {
  71         BEGIN_TYPECHECK
  72                 TYPECHECK(arg,numeric)
  73         END_TYPECHECK(csgn(arg))
  74
  75         return csgn(ex_to<numeric>(arg));
  76 }
  77
  78 static ex csgn_eval(const ex & arg)
  79 {
  80         if (is_ex_exactly_of_type(arg, numeric))
  81                 return csgn(ex_to<numeric>(arg));
  82
  83         else if (is_ex_of_type(arg, mul) &&
  84                  is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
  85                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
  86                 if (oc.is_real()) {
  87                         if (oc > 0)
  88                                 // csgn(42*x) -> csgn(x)
  89                                 return csgn(arg/oc).hold();
  90                         else
  91                                 // csgn(-42*x) -> -csgn(x)
  92                                 return -csgn(arg/oc).hold();
  93                 }
  94                 if (oc.real().is_zero()) {
  95                         if (oc.imag() > 0)
  96                                 // csgn(42*I*x) -> csgn(I*x)
  97                                 return csgn(I*arg/oc).hold();
  98                         else
  99                                 // csgn(-42*I*x) -> -csgn(I*x)
 100                                 return -csgn(I*arg/oc).hold();
 101                 }
 102         }
 103
 104         return csgn(arg).hold();
 105 }
 106
 107 static ex csgn_series(const ex & arg,
 108                       const relational & rel,
 109                       int order,
 110                       unsigned options)
 111 {
 112         const ex arg_pt = arg.subs(rel);
 113         if (arg_pt.info(info_flags::numeric)
 114             && ex_to<numeric>(arg_pt).real().is_zero()
 115             && !(options & series_options::suppress_branchcut))
 116                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 117
 118         epvector seq;
 119         seq.push_back(expair(csgn(arg_pt), _ex0()));
 120         return pseries(rel,seq);
 121 }
 122
 123 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 124                         evalf_func(csgn_evalf).
 125                         series_func(csgn_series));
 126
 127
 128 //////////
 129 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 130 //////////
 131
 132 static ex eta_evalf(const ex &x, const ex &y)
 133 {
 134         // It seems like we basically have to replicate the eval function here,
 135         // since the expression might not be fully evaluated yet.
 136         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 137                 return _ex0();
 138
 139         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 140                 const numeric nx = ex_to<numeric>(x);
 141                 const numeric ny = ex_to<numeric>(y);
 142                 const numeric nxy = ex_to<numeric>(x*y);
 143                 int cut = 0;
 144                 if (nx.is_real() && nx.is_negative())
 145                         cut -= 4;
 146                 if (ny.is_real() && ny.is_negative())
 147                         cut -= 4;
 148                 if (nxy.is_real() && nxy.is_negative())
 149                         cut += 4;
 150                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 151                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 152         }
 153
 154         return eta(x,y).hold();
 155 }
 156
 157 static ex eta_eval(const ex &x, const ex &y)
 158 {
 159         // trivial:  eta(x,c) -> 0  if c is real and positive
 160         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 161                 return _ex0();
 162
 163         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 164                 // don't call eta_evalf here because it would call Pi.evalf()!
 165                 const numeric nx = ex_to<numeric>(x);
 166                 const numeric ny = ex_to<numeric>(y);
 167                 const numeric nxy = ex_to<numeric>(x*y);
 168                 int cut = 0;
 169                 if (nx.is_real() && nx.is_negative())
 170                         cut -= 4;
 171                 if (ny.is_real() && ny.is_negative())
 172                         cut -= 4;
 173                 if (nxy.is_real() && nxy.is_negative())
 174                         cut += 4;
 175                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 176                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 177         }
 178
 179         return eta(x,y).hold();
 180 }
 181
 182 static ex eta_series(const ex & x, const ex & y,
 183                      const relational & rel,
 184                      int order,
 185                      unsigned options)
 186 {
 187         const ex x_pt = x.subs(rel);
 188         const ex y_pt = y.subs(rel);
 189         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 190             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 191             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 192                         throw (std::domain_error("eta_series(): on discontinuity"));
 193         epvector seq;
 194         seq.push_back(expair(eta(x_pt,y_pt), _ex0()));
 195         return pseries(rel,seq);
 196 }
 197
 198 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 199                        evalf_func(eta_evalf).
 200                                        series_func(eta_series).
 201                        latex_name("\\eta"));
 202
 203
 204 //////////
 205 // dilogarithm
 206 //////////
 207
 208 static ex Li2_evalf(const ex & x)
 209 {
 210         BEGIN_TYPECHECK
 211                 TYPECHECK(x,numeric)
 212         END_TYPECHECK(Li2(x))
 213
 214         return Li2(ex_to<numeric>(x));  // -> numeric Li2(numeric)
 215 }
 216
 217 static ex Li2_eval(const ex & x)
 218 {
 219         if (x.info(info_flags::numeric)) {
 220                 // Li2(0) -> 0
 221                 if (x.is_zero())
 222                         return _ex0();
 223                 // Li2(1) -> Pi^2/6
 224                 if (x.is_equal(_ex1()))
 225                         return power(Pi,_ex2())/_ex6();
 226                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 227                 if (x.is_equal(_ex1_2()))
 228                         return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
 229                 // Li2(-1) -> -Pi^2/12
 230                 if (x.is_equal(_ex_1()))
 231                         return -power(Pi,_ex2())/_ex12();
 232                 // Li2(I) -> -Pi^2/48+Catalan*I
 233                 if (x.is_equal(I))
 234                         return power(Pi,_ex2())/_ex_48() + Catalan*I;
 235                 // Li2(-I) -> -Pi^2/48-Catalan*I
 236                 if (x.is_equal(-I))
 237                         return power(Pi,_ex2())/_ex_48() - Catalan*I;
 238                 // Li2(float)
 239                 if (!x.info(info_flags::crational))
 240                         return Li2_evalf(x);
 241         }
 242
 243         return Li2(x).hold();
 244 }
 245
 246 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 247 {
 248         GINAC_ASSERT(deriv_param==0);
 249
 250         // d/dx Li2(x) -> -log(1-x)/x
 251         return -log(1-x)/x;
 252 }
 253
 254 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 255 {
 256         const ex x_pt = x.subs(rel);
 257         if (x_pt.info(info_flags::numeric)) {
 258                 // First special case: x==0 (derivatives have poles)
 259                 if (x_pt.is_zero()) {
 260                         // method:
 261                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 262                         // simply substitute x==0.  The limit, however, exists: it is 1.
 263                         // We also know all higher derivatives' limits:
 264                         // (d/dx)^n Li2(x) == n!/n^2.
 265                         // So the primitive series expansion is
 266                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 267                         // and so on.
 268                         // We first construct such a primitive series expansion manually in
 269                         // a dummy symbol s and then insert the argument's series expansion
 270                         // for s.  Reexpanding the resulting series returns the desired
 271                         // result.
 272                         const symbol s;
 273                         ex ser;
 274                         // manually construct the primitive expansion
 275                         for (int i=1; i<order; ++i)
 276                                 ser += pow(s,i) / pow(numeric(i), _num2());
 277                         // substitute the argument's series expansion
 278                         ser = ser.subs(s==x.series(rel, order));
 279                         // maybe that was terminating, so add a proper order term
 280                         epvector nseq;
 281                         nseq.push_back(expair(Order(_ex1()), order));
 282                         ser += pseries(rel, nseq);
 283                         // reexpanding it will collapse the series again
 284                         return ser.series(rel, order);
 285                         // NB: Of course, this still does not allow us to compute anything
 286                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 287                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 288                         // substitution x==0.  Probably limits *are* needed for the general
 289                         // cases.  In case L'Hospital's rule is implemented for limits and
 290                         // basic::series() takes care of this, this whole block is probably
 291                         // obsolete!
 292                 }
 293                 // second special case: x==1 (branch point)
 294                 if (x_pt == _ex1()) {
 295                         // method:
 296                         // construct series manually in a dummy symbol s
 297                         const symbol s;
 298                         ex ser = zeta(_ex2());
 299                         // manually construct the primitive expansion
 300                         for (int i=1; i<order; ++i)
 301                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 302                         // substitute the argument's series expansion
 303                         ser = ser.subs(s==x.series(rel, order));
 304                         // maybe that was terminating, so add a proper order term
 305                         epvector nseq;
 306                         nseq.push_back(expair(Order(_ex1()), order));
 307                         ser += pseries(rel, nseq);
 308                         // reexpanding it will collapse the series again
 309                         return ser.series(rel, order);
 310                 }
 311                 // third special case: x real, >=1 (branch cut)
 312                 if (!(options & series_options::suppress_branchcut) &&
 313                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 314                         // method:
 315                         // This is the branch cut: assemble the primitive series manually
 316                         // and then add the corresponding complex step function.
 317                         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 318                         const ex point = rel.rhs();
 319                         const symbol foo;
 320                         epvector seq;
 321                         // zeroth order term:
 322                         seq.push_back(expair(Li2(x_pt), _ex0()));
 323                         // compute the intermediate terms:
 324                         ex replarg = series(Li2(x), *s==foo, order);
 325                         for (unsigned i=1; i<replarg.nops()-1; ++i)
 326                                 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
 327                         // append an order term:
 328                         seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
 329                         return pseries(rel, seq);
 330                 }
 331         }
 332         // all other cases should be safe, by now:
 333         throw do_taylor();  // caught by function::series()
 334 }
 335
 336 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 337                        evalf_func(Li2_evalf).
 338                        derivative_func(Li2_deriv).
 339                        series_func(Li2_series).
 340                        latex_name("\\mbox{Li}_2"));
 341
 342 //////////
 343 // trilogarithm
 344 //////////
 345
 346 static ex Li3_eval(const ex & x)
 347 {
 348         if (x.is_zero())
 349                 return x;
 350         return Li3(x).hold();
 351 }
 352
 353 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 354                        latex_name("\\mbox{Li}_3"));
 355
 356 //////////
 357 // factorial
 358 //////////
 359
 360 static ex factorial_evalf(const ex & x)
 361 {
 362         return factorial(x).hold();
 363 }
 364
 365 static ex factorial_eval(const ex & x)
 366 {
 367         if (is_ex_exactly_of_type(x, numeric))
 368                 return factorial(ex_to<numeric>(x));
 369         else
 370                 return factorial(x).hold();
 371 }
 372
 373 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 374                              evalf_func(factorial_evalf));
 375
 376 //////////
 377 // binomial
 378 //////////
 379
 380 static ex binomial_evalf(const ex & x, const ex & y)
 381 {
 382         return binomial(x, y).hold();
 383 }
 384
 385 static ex binomial_eval(const ex & x, const ex &y)
 386 {
 387         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
 388                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 389         else
 390                 return binomial(x, y).hold();
 391 }
 392
 393 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 394                             evalf_func(binomial_evalf));
 395
 396 //////////
 397 // Order term function (for truncated power series)
 398 //////////
 399
 400 static ex Order_eval(const ex & x)
 401 {
 402         if (is_ex_exactly_of_type(x, numeric)) {
 403                 // O(c) -> O(1) or 0
 404                 if (!x.is_zero())
 405                         return Order(_ex1()).hold();
 406                 else
 407                         return _ex0();
 408         } else if (is_ex_exactly_of_type(x, mul)) {
 409                 mul *m = static_cast<mul *>(x.bp);
 410                 // O(c*expr) -> O(expr)
 411                 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
 412                         return Order(x / m->op(m->nops() - 1)).hold();
 413         }
 414         return Order(x).hold();
 415 }
 416
 417 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 418 {
 419         // Just wrap the function into a pseries object
 420         epvector new_seq;
 421         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 422         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 423         new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
 424         return pseries(r, new_seq);
 425 }
 426
 427 // Differentiation is handled in function::derivative because of its special requirements
 428
 429 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 430                          series_func(Order_series).
 431                          latex_name("\\mathcal{O}"));
 432
 433 //////////
 434 // Solve linear system
 435 //////////
 436
 437 ex lsolve(const ex &eqns, const ex &symbols)
 438 {
 439         // solve a system of linear equations
 440         if (eqns.info(info_flags::relation_equal)) {
 441                 if (!symbols.info(info_flags::symbol))
 442                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 443                 const ex sol = lsolve(lst(eqns),lst(symbols));
 444
 445                 GINAC_ASSERT(sol.nops()==1);
 446                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
 447
 448                 return sol.op(0).op(1); // return rhs of first solution
 449         }
 450
 451         // syntax checks
 452         if (!eqns.info(info_flags::list)) {
 453                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 454         }
 455         for (unsigned i=0; i<eqns.nops(); i++) {
 456                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 457                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 458                 }
 459         }
 460         if (!symbols.info(info_flags::list)) {
 461                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 462         }
 463         for (unsigned i=0; i<symbols.nops(); i++) {
 464                 if (!symbols.op(i).info(info_flags::symbol)) {
 465                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 466                 }
 467         }
 468
 469         // build matrix from equation system
 470         matrix sys(eqns.nops(),symbols.nops());
 471         matrix rhs(eqns.nops(),1);
 472         matrix vars(symbols.nops(),1);
 473
 474         for (unsigned r=0; r<eqns.nops(); r++) {
 475                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 476                 ex linpart = eq;
 477                 for (unsigned c=0; c<symbols.nops(); c++) {
 478                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 479                         linpart -= co*symbols.op(c);
 480                         sys(r,c) = co;
 481                 }
 482                 linpart = linpart.expand();
 483                 rhs(r,0) = -linpart;
 484         }
 485
 486         // test if system is linear and fill vars matrix
 487         for (unsigned i=0; i<symbols.nops(); i++) {
 488                 vars(i,0) = symbols.op(i);
 489                 if (sys.has(symbols.op(i)))
 490                         throw(std::logic_error("lsolve: system is not linear"));
 491                 if (rhs.has(symbols.op(i)))
 492                         throw(std::logic_error("lsolve: system is not linear"));
 493         }
 494
 495         matrix solution;
 496         try {
 497                 solution = sys.solve(vars,rhs);
 498         } catch (const std::runtime_error & e) {
 499                 // Probably singular matrix or otherwise overdetermined system:
 500                 // It is consistent to return an empty list
 501                 return lst();
 502         }
 503         GINAC_ASSERT(solution.cols()==1);
 504         GINAC_ASSERT(solution.rows()==symbols.nops());
 505
 506         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 507         lst sollist;
 508         for (unsigned i=0; i<symbols.nops(); i++)
 509                 sollist.append(symbols.op(i)==solution(i,0));
 510
 511         return sollist;
 512 }
 513
 514 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 515  * for static lib (so ginsh will see them). */
 516 unsigned force_include_tgamma = function_index_tgamma;
 517 unsigned force_include_zeta1 = function_index_zeta1;
 518
 519 } // namespace GiNaC