ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 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 "operators.h"
  34 #include "relational.h"
  35 #include "pseries.h"
  36 #include "symbol.h"
  37 #include "symmetry.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 //////////
  43 // absolute value
  44 //////////
  45
  46 static ex abs_evalf(const ex & arg)
  47 {
  48         if (is_exactly_a<numeric>(arg))
  49                 return abs(ex_to<numeric>(arg));
  50
  51         return abs(arg).hold();
  52 }
  53
  54 static ex abs_eval(const ex & arg)
  55 {
  56         if (is_exactly_a<numeric>(arg))
  57                 return abs(ex_to<numeric>(arg));
  58         else
  59                 return abs(arg).hold();
  60 }
  61
  62 REGISTER_FUNCTION(abs, eval_func(abs_eval).
  63                        evalf_func(abs_evalf));
  64
  65
  66 //////////
  67 // Complex sign
  68 //////////
  69
  70 static ex csgn_evalf(const ex & arg)
  71 {
  72         if (is_exactly_a<numeric>(arg))
  73                 return csgn(ex_to<numeric>(arg));
  74
  75         return csgn(arg).hold();
  76 }
  77
  78 static ex csgn_eval(const ex & arg)
  79 {
  80         if (is_exactly_a<numeric>(arg))
  81                 return csgn(ex_to<numeric>(arg));
  82
  83         else if (is_exactly_a<mul>(arg) &&
  84                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
  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 // This function is closely related to the unwinding number K, sometimes found
 131 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 132 //////////
 133
 134 static ex eta_evalf(const ex &x, const ex &y)
 135 {
 136         // It seems like we basically have to replicate the eval function here,
 137         // since the expression might not be fully evaluated yet.
 138         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 139                 return _ex0;
 140
 141         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 142                 const numeric nx = ex_to<numeric>(x);
 143                 const numeric ny = ex_to<numeric>(y);
 144                 const numeric nxy = ex_to<numeric>(x*y);
 145                 int cut = 0;
 146                 if (nx.is_real() && nx.is_negative())
 147                         cut -= 4;
 148                 if (ny.is_real() && ny.is_negative())
 149                         cut -= 4;
 150                 if (nxy.is_real() && nxy.is_negative())
 151                         cut += 4;
 152                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 153                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 154         }
 155
 156         return eta(x,y).hold();
 157 }
 158
 159 static ex eta_eval(const ex &x, const ex &y)
 160 {
 161         // trivial:  eta(x,c) -> 0  if c is real and positive
 162         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 163                 return _ex0;
 164
 165         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 166                 // don't call eta_evalf here because it would call Pi.evalf()!
 167                 const numeric nx = ex_to<numeric>(x);
 168                 const numeric ny = ex_to<numeric>(y);
 169                 const numeric nxy = ex_to<numeric>(x*y);
 170                 int cut = 0;
 171                 if (nx.is_real() && nx.is_negative())
 172                         cut -= 4;
 173                 if (ny.is_real() && ny.is_negative())
 174                         cut -= 4;
 175                 if (nxy.is_real() && nxy.is_negative())
 176                         cut += 4;
 177                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 178                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 179         }
 180
 181         return eta(x,y).hold();
 182 }
 183
 184 static ex eta_series(const ex & x, const ex & y,
 185                      const relational & rel,
 186                      int order,
 187                      unsigned options)
 188 {
 189         const ex x_pt = x.subs(rel);
 190         const ex y_pt = y.subs(rel);
 191         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 192             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 193             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 194                         throw (std::domain_error("eta_series(): on discontinuity"));
 195         epvector seq;
 196         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 197         return pseries(rel,seq);
 198 }
 199
 200 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 201                        evalf_func(eta_evalf).
 202                        series_func(eta_series).
 203                        latex_name("\\eta").
 204                        set_symmetry(sy_symm(0, 1)));
 205
 206
 207 //////////
 208 // dilogarithm
 209 //////////
 210
 211 static ex Li2_evalf(const ex & x)
 212 {
 213         if (is_exactly_a<numeric>(x))
 214                 return Li2(ex_to<numeric>(x));
 215
 216         return Li2(x).hold();
 217 }
 218
 219 static ex Li2_eval(const ex & x)
 220 {
 221         if (x.info(info_flags::numeric)) {
 222                 // Li2(0) -> 0
 223                 if (x.is_zero())
 224                         return _ex0;
 225                 // Li2(1) -> Pi^2/6
 226                 if (x.is_equal(_ex1))
 227                         return power(Pi,_ex2)/_ex6;
 228                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 229                 if (x.is_equal(_ex1_2))
 230                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 231                 // Li2(-1) -> -Pi^2/12
 232                 if (x.is_equal(_ex_1))
 233                         return -power(Pi,_ex2)/_ex12;
 234                 // Li2(I) -> -Pi^2/48+Catalan*I
 235                 if (x.is_equal(I))
 236                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 237                 // Li2(-I) -> -Pi^2/48-Catalan*I
 238                 if (x.is_equal(-I))
 239                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 240                 // Li2(float)
 241                 if (!x.info(info_flags::crational))
 242                         return Li2(ex_to<numeric>(x));
 243         }
 244
 245         return Li2(x).hold();
 246 }
 247
 248 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 249 {
 250         GINAC_ASSERT(deriv_param==0);
 251
 252         // d/dx Li2(x) -> -log(1-x)/x
 253         return -log(_ex1-x)/x;
 254 }
 255
 256 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 257 {
 258         const ex x_pt = x.subs(rel);
 259         if (x_pt.info(info_flags::numeric)) {
 260                 // First special case: x==0 (derivatives have poles)
 261                 if (x_pt.is_zero()) {
 262                         // method:
 263                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 264                         // simply substitute x==0.  The limit, however, exists: it is 1.
 265                         // We also know all higher derivatives' limits:
 266                         // (d/dx)^n Li2(x) == n!/n^2.
 267                         // So the primitive series expansion is
 268                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 269                         // and so on.
 270                         // We first construct such a primitive series expansion manually in
 271                         // a dummy symbol s and then insert the argument's series expansion
 272                         // for s.  Reexpanding the resulting series returns the desired
 273                         // result.
 274                         const symbol s;
 275                         ex ser;
 276                         // manually construct the primitive expansion
 277                         for (int i=1; i<order; ++i)
 278                                 ser += pow(s,i) / pow(numeric(i), _num2);
 279                         // substitute the argument's series expansion
 280                         ser = ser.subs(s==x.series(rel, order));
 281                         // maybe that was terminating, so add a proper order term
 282                         epvector nseq;
 283                         nseq.push_back(expair(Order(_ex1), order));
 284                         ser += pseries(rel, nseq);
 285                         // reexpanding it will collapse the series again
 286                         return ser.series(rel, order);
 287                         // NB: Of course, this still does not allow us to compute anything
 288                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 289                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 290                         // substitution x==0.  Probably limits *are* needed for the general
 291                         // cases.  In case L'Hospital's rule is implemented for limits and
 292                         // basic::series() takes care of this, this whole block is probably
 293                         // obsolete!
 294                 }
 295                 // second special case: x==1 (branch point)
 296                 if (x_pt.is_equal(_ex1)) {
 297                         // method:
 298                         // construct series manually in a dummy symbol s
 299                         const symbol s;
 300                         ex ser = zeta(_ex2);
 301                         // manually construct the primitive expansion
 302                         for (int i=1; i<order; ++i)
 303                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 304                         // substitute the argument's series expansion
 305                         ser = ser.subs(s==x.series(rel, order));
 306                         // maybe that was terminating, so add a proper order term
 307                         epvector nseq;
 308                         nseq.push_back(expair(Order(_ex1), order));
 309                         ser += pseries(rel, nseq);
 310                         // reexpanding it will collapse the series again
 311                         return ser.series(rel, order);
 312                 }
 313                 // third special case: x real, >=1 (branch cut)
 314                 if (!(options & series_options::suppress_branchcut) &&
 315                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 316                         // method:
 317                         // This is the branch cut: assemble the primitive series manually
 318                         // and then add the corresponding complex step function.
 319                         const symbol &s = ex_to<symbol>(rel.lhs());
 320                         const ex point = rel.rhs();
 321                         const symbol foo;
 322                         epvector seq;
 323                         // zeroth order term:
 324                         seq.push_back(expair(Li2(x_pt), _ex0));
 325                         // compute the intermediate terms:
 326                         ex replarg = series(Li2(x), s==foo, order);
 327                         for (size_t i=1; i<replarg.nops()-1; ++i)
 328                                 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s),i));
 329                         // append an order term:
 330                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 331                         return pseries(rel, seq);
 332                 }
 333         }
 334         // all other cases should be safe, by now:
 335         throw do_taylor();  // caught by function::series()
 336 }
 337
 338 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 339                        evalf_func(Li2_evalf).
 340                        derivative_func(Li2_deriv).
 341                        series_func(Li2_series).
 342                        latex_name("\\mbox{Li}_2"));
 343
 344 //////////
 345 // trilogarithm
 346 //////////
 347
 348 static ex Li3_eval(const ex & x)
 349 {
 350         if (x.is_zero())
 351                 return x;
 352         return Li3(x).hold();
 353 }
 354
 355 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 356                        latex_name("\\mbox{Li}_3"));
 357
 358 //////////
 359 // factorial
 360 //////////
 361
 362 static ex factorial_evalf(const ex & x)
 363 {
 364         return factorial(x).hold();
 365 }
 366
 367 static ex factorial_eval(const ex & x)
 368 {
 369         if (is_exactly_a<numeric>(x))
 370                 return factorial(ex_to<numeric>(x));
 371         else
 372                 return factorial(x).hold();
 373 }
 374
 375 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 376                              evalf_func(factorial_evalf));
 377
 378 //////////
 379 // binomial
 380 //////////
 381
 382 static ex binomial_evalf(const ex & x, const ex & y)
 383 {
 384         return binomial(x, y).hold();
 385 }
 386
 387 static ex binomial_eval(const ex & x, const ex &y)
 388 {
 389         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y))
 390                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 391         else
 392                 return binomial(x, y).hold();
 393 }
 394
 395 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 396                             evalf_func(binomial_evalf));
 397
 398 //////////
 399 // Order term function (for truncated power series)
 400 //////////
 401
 402 static ex Order_eval(const ex & x)
 403 {
 404         if (is_exactly_a<numeric>(x)) {
 405                 // O(c) -> O(1) or 0
 406                 if (!x.is_zero())
 407                         return Order(_ex1).hold();
 408                 else
 409                         return _ex0;
 410         } else if (is_exactly_a<mul>(x)) {
 411                 const mul &m = ex_to<mul>(x);
 412                 // O(c*expr) -> O(expr)
 413                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 414                         return Order(x / m.op(m.nops() - 1)).hold();
 415         }
 416         return Order(x).hold();
 417 }
 418
 419 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 420 {
 421         // Just wrap the function into a pseries object
 422         epvector new_seq;
 423         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 424         const symbol &s = ex_to<symbol>(r.lhs());
 425         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 426         return pseries(r, new_seq);
 427 }
 428
 429 // Differentiation is handled in function::derivative because of its special requirements
 430
 431 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 432                          series_func(Order_series).
 433                          latex_name("\\mathcal{O}"));
 434
 435 //////////
 436 // Solve linear system
 437 //////////
 438
 439 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 440 {
 441         // solve a system of linear equations
 442         if (eqns.info(info_flags::relation_equal)) {
 443                 if (!symbols.info(info_flags::symbol))
 444                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 445                 const ex sol = lsolve(lst(eqns),lst(symbols));
 446
 447                 GINAC_ASSERT(sol.nops()==1);
 448                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 449
 450                 return sol.op(0).op(1); // return rhs of first solution
 451         }
 452
 453         // syntax checks
 454         if (!eqns.info(info_flags::list)) {
 455                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 456         }
 457         for (size_t i=0; i<eqns.nops(); i++) {
 458                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 459                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 460                 }
 461         }
 462         if (!symbols.info(info_flags::list)) {
 463                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 464         }
 465         for (size_t i=0; i<symbols.nops(); i++) {
 466                 if (!symbols.op(i).info(info_flags::symbol)) {
 467                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 468                 }
 469         }
 470
 471         // build matrix from equation system
 472         matrix sys(eqns.nops(),symbols.nops());
 473         matrix rhs(eqns.nops(),1);
 474         matrix vars(symbols.nops(),1);
 475
 476         for (size_t r=0; r<eqns.nops(); r++) {
 477                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 478                 ex linpart = eq;
 479                 for (size_t c=0; c<symbols.nops(); c++) {
 480                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 481                         linpart -= co*symbols.op(c);
 482                         sys(r,c) = co;
 483                 }
 484                 linpart = linpart.expand();
 485                 rhs(r,0) = -linpart;
 486         }
 487
 488         // test if system is linear and fill vars matrix
 489         for (size_t i=0; i<symbols.nops(); i++) {
 490                 vars(i,0) = symbols.op(i);
 491                 if (sys.has(symbols.op(i)))
 492                         throw(std::logic_error("lsolve: system is not linear"));
 493                 if (rhs.has(symbols.op(i)))
 494                         throw(std::logic_error("lsolve: system is not linear"));
 495         }
 496
 497         matrix solution;
 498         try {
 499                 solution = sys.solve(vars,rhs,options);
 500         } catch (const std::runtime_error & e) {
 501                 // Probably singular matrix or otherwise overdetermined system:
 502                 // It is consistent to return an empty list
 503                 return lst();
 504         }
 505         GINAC_ASSERT(solution.cols()==1);
 506         GINAC_ASSERT(solution.rows()==symbols.nops());
 507
 508         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 509         lst sollist;
 510         for (size_t i=0; i<symbols.nops(); i++)
 511                 sollist.append(symbols.op(i)==solution(i,0));
 512
 513         return sollist;
 514 }
 515
 516 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 517  * for static lib (so ginsh will see them). */
 518 unsigned force_include_tgamma = tgamma_SERIAL::serial;
 519 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 520
 521 } // namespace GiNaC