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