ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 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_exactly_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 // This function is closely related to the unwinding number K, sometimes found
 130 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 131 //////////
 132
 133 static ex eta_evalf(const ex &x, const ex &y)
 134 {
 135         // It seems like we basically have to replicate the eval function here,
 136         // since the expression might not be fully evaluated yet.
 137         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 138                 return _ex0;
 139
 140         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 141                 const numeric nx = ex_to<numeric>(x);
 142                 const numeric ny = ex_to<numeric>(y);
 143                 const numeric nxy = ex_to<numeric>(x*y);
 144                 int cut = 0;
 145                 if (nx.is_real() && nx.is_negative())
 146                         cut -= 4;
 147                 if (ny.is_real() && ny.is_negative())
 148                         cut -= 4;
 149                 if (nxy.is_real() && nxy.is_negative())
 150                         cut += 4;
 151                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 152                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 153         }
 154
 155         return eta(x,y).hold();
 156 }
 157
 158 static ex eta_eval(const ex &x, const ex &y)
 159 {
 160         // trivial:  eta(x,c) -> 0  if c is real and positive
 161         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 162                 return _ex0;
 163
 164         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 165                 // don't call eta_evalf here because it would call Pi.evalf()!
 166                 const numeric nx = ex_to<numeric>(x);
 167                 const numeric ny = ex_to<numeric>(y);
 168                 const numeric nxy = ex_to<numeric>(x*y);
 169                 int cut = 0;
 170                 if (nx.is_real() && nx.is_negative())
 171                         cut -= 4;
 172                 if (ny.is_real() && ny.is_negative())
 173                         cut -= 4;
 174                 if (nxy.is_real() && nxy.is_negative())
 175                         cut += 4;
 176                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 177                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 178         }
 179
 180         return eta(x,y).hold();
 181 }
 182
 183 static ex eta_series(const ex & x, const ex & y,
 184                      const relational & rel,
 185                      int order,
 186                      unsigned options)
 187 {
 188         const ex x_pt = x.subs(rel);
 189         const ex y_pt = y.subs(rel);
 190         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 191             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 192             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 193                         throw (std::domain_error("eta_series(): on discontinuity"));
 194         epvector seq;
 195         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 196         return pseries(rel,seq);
 197 }
 198
 199 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 200                        evalf_func(eta_evalf).
 201                        series_func(eta_series).
 202                        latex_name("\\eta").
 203                        set_symmetry(sy_symm(0, 1)));
 204
 205
 206 //////////
 207 // dilogarithm
 208 //////////
 209
 210 static ex Li2_evalf(const ex & x)
 211 {
 212         if (is_exactly_a<numeric>(x))
 213                 return Li2(ex_to<numeric>(x));
 214
 215         return Li2(x).hold();
 216 }
 217
 218 static ex Li2_eval(const ex & x)
 219 {
 220         if (x.info(info_flags::numeric)) {
 221                 // Li2(0) -> 0
 222                 if (x.is_zero())
 223                         return _ex0;
 224                 // Li2(1) -> Pi^2/6
 225                 if (x.is_equal(_ex1))
 226                         return power(Pi,_ex2)/_ex6;
 227                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 228                 if (x.is_equal(_ex1_2))
 229                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 230                 // Li2(-1) -> -Pi^2/12
 231                 if (x.is_equal(_ex_1))
 232                         return -power(Pi,_ex2)/_ex12;
 233                 // Li2(I) -> -Pi^2/48+Catalan*I
 234                 if (x.is_equal(I))
 235                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 236                 // Li2(-I) -> -Pi^2/48-Catalan*I
 237                 if (x.is_equal(-I))
 238                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 239                 // Li2(float)
 240                 if (!x.info(info_flags::crational))
 241                         return Li2(ex_to<numeric>(x));
 242         }
 243
 244         return Li2(x).hold();
 245 }
 246
 247 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 248 {
 249         GINAC_ASSERT(deriv_param==0);
 250
 251         // d/dx Li2(x) -> -log(1-x)/x
 252         return -log(_ex1-x)/x;
 253 }
 254
 255 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 256 {
 257         const ex x_pt = x.subs(rel);
 258         if (x_pt.info(info_flags::numeric)) {
 259                 // First special case: x==0 (derivatives have poles)
 260                 if (x_pt.is_zero()) {
 261                         // method:
 262                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 263                         // simply substitute x==0.  The limit, however, exists: it is 1.
 264                         // We also know all higher derivatives' limits:
 265                         // (d/dx)^n Li2(x) == n!/n^2.
 266                         // So the primitive series expansion is
 267                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 268                         // and so on.
 269                         // We first construct such a primitive series expansion manually in
 270                         // a dummy symbol s and then insert the argument's series expansion
 271                         // for s.  Reexpanding the resulting series returns the desired
 272                         // result.
 273                         const symbol s;
 274                         ex ser;
 275                         // manually construct the primitive expansion
 276                         for (int i=1; i<order; ++i)
 277                                 ser += pow(s,i) / pow(numeric(i), _num2);
 278                         // substitute the argument's series expansion
 279                         ser = ser.subs(s==x.series(rel, order));
 280                         // maybe that was terminating, so add a proper order term
 281                         epvector nseq;
 282                         nseq.push_back(expair(Order(_ex1), order));
 283                         ser += pseries(rel, nseq);
 284                         // reexpanding it will collapse the series again
 285                         return ser.series(rel, order);
 286                         // NB: Of course, this still does not allow us to compute anything
 287                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 288                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 289                         // substitution x==0.  Probably limits *are* needed for the general
 290                         // cases.  In case L'Hospital's rule is implemented for limits and
 291                         // basic::series() takes care of this, this whole block is probably
 292                         // obsolete!
 293                 }
 294                 // second special case: x==1 (branch point)
 295                 if (x_pt.is_equal(_ex1)) {
 296                         // method:
 297                         // construct series manually in a dummy symbol s
 298                         const symbol s;
 299                         ex ser = zeta(_ex2);
 300                         // manually construct the primitive expansion
 301                         for (int i=1; i<order; ++i)
 302                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 303                         // substitute the argument's series expansion
 304                         ser = ser.subs(s==x.series(rel, order));
 305                         // maybe that was terminating, so add a proper order term
 306                         epvector nseq;
 307                         nseq.push_back(expair(Order(_ex1), order));
 308                         ser += pseries(rel, nseq);
 309                         // reexpanding it will collapse the series again
 310                         return ser.series(rel, order);
 311                 }
 312                 // third special case: x real, >=1 (branch cut)
 313                 if (!(options & series_options::suppress_branchcut) &&
 314                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 315                         // method:
 316                         // This is the branch cut: assemble the primitive series manually
 317                         // and then add the corresponding complex step function.
 318                         const symbol &s = ex_to<symbol>(rel.lhs());
 319                         const ex point = rel.rhs();
 320                         const symbol foo;
 321                         epvector seq;
 322                         // zeroth order term:
 323                         seq.push_back(expair(Li2(x_pt), _ex0));
 324                         // compute the intermediate terms:
 325                         ex replarg = series(Li2(x), s==foo, order);
 326                         for (unsigned i=1; i<replarg.nops()-1; ++i)
 327                                 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s),i));
 328                         // append an order term:
 329                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 330                         return pseries(rel, seq);
 331                 }
 332         }
 333         // all other cases should be safe, by now:
 334         throw do_taylor();  // caught by function::series()
 335 }
 336
 337 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 338                        evalf_func(Li2_evalf).
 339                        derivative_func(Li2_deriv).
 340                        series_func(Li2_series).
 341                        latex_name("\\mbox{Li}_2"));
 342
 343 //////////
 344 // trilogarithm
 345 //////////
 346
 347 static ex Li3_eval(const ex & x)
 348 {
 349         if (x.is_zero())
 350                 return x;
 351         return Li3(x).hold();
 352 }
 353
 354 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 355                        latex_name("\\mbox{Li}_3"));
 356
 357 //////////
 358 // factorial
 359 //////////
 360
 361 static ex factorial_evalf(const ex & x)
 362 {
 363         return factorial(x).hold();
 364 }
 365
 366 static ex factorial_eval(const ex & x)
 367 {
 368         if (is_ex_exactly_of_type(x, numeric))
 369                 return factorial(ex_to<numeric>(x));
 370         else
 371                 return factorial(x).hold();
 372 }
 373
 374 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 375                              evalf_func(factorial_evalf));
 376
 377 //////////
 378 // binomial
 379 //////////
 380
 381 static ex binomial_evalf(const ex & x, const ex & y)
 382 {
 383         return binomial(x, y).hold();
 384 }
 385
 386 static ex binomial_eval(const ex & x, const ex &y)
 387 {
 388         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
 389                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 390         else
 391                 return binomial(x, y).hold();
 392 }
 393
 394 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 395                             evalf_func(binomial_evalf));
 396
 397 //////////
 398 // Order term function (for truncated power series)
 399 //////////
 400
 401 static ex Order_eval(const ex & x)
 402 {
 403         if (is_ex_exactly_of_type(x, numeric)) {
 404                 // O(c) -> O(1) or 0
 405                 if (!x.is_zero())
 406                         return Order(_ex1).hold();
 407                 else
 408                         return _ex0;
 409         } else if (is_ex_exactly_of_type(x, mul)) {
 410                 const mul &m = ex_to<mul>(x);
 411                 // O(c*expr) -> O(expr)
 412                 if (is_ex_exactly_of_type(m.op(m.nops() - 1), numeric))
 413                         return Order(x / m.op(m.nops() - 1)).hold();
 414         }
 415         return Order(x).hold();
 416 }
 417
 418 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 419 {
 420         // Just wrap the function into a pseries object
 421         epvector new_seq;
 422         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 423         const symbol &s = ex_to<symbol>(r.lhs());
 424         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 425         return pseries(r, new_seq);
 426 }
 427
 428 // Differentiation is handled in function::derivative because of its special requirements
 429
 430 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 431                          series_func(Order_series).
 432                          latex_name("\\mathcal{O}"));
 433
 434 //////////
 435 // Solve linear system
 436 //////////
 437
 438 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 439 {
 440         // solve a system of linear equations
 441         if (eqns.info(info_flags::relation_equal)) {
 442                 if (!symbols.info(info_flags::symbol))
 443                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 444                 const ex sol = lsolve(lst(eqns),lst(symbols));
 445
 446                 GINAC_ASSERT(sol.nops()==1);
 447                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 448
 449                 return sol.op(0).op(1); // return rhs of first solution
 450         }
 451
 452         // syntax checks
 453         if (!eqns.info(info_flags::list)) {
 454                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 455         }
 456         for (unsigned i=0; i<eqns.nops(); i++) {
 457                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 458                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 459                 }
 460         }
 461         if (!symbols.info(info_flags::list)) {
 462                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 463         }
 464         for (unsigned i=0; i<symbols.nops(); i++) {
 465                 if (!symbols.op(i).info(info_flags::symbol)) {
 466                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 467                 }
 468         }
 469
 470         // build matrix from equation system
 471         matrix sys(eqns.nops(),symbols.nops());
 472         matrix rhs(eqns.nops(),1);
 473         matrix vars(symbols.nops(),1);
 474
 475         for (unsigned r=0; r<eqns.nops(); r++) {
 476                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 477                 ex linpart = eq;
 478                 for (unsigned c=0; c<symbols.nops(); c++) {
 479                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 480                         linpart -= co*symbols.op(c);
 481                         sys(r,c) = co;
 482                 }
 483                 linpart = linpart.expand();
 484                 rhs(r,0) = -linpart;
 485         }
 486
 487         // test if system is linear and fill vars matrix
 488         for (unsigned i=0; i<symbols.nops(); i++) {
 489                 vars(i,0) = symbols.op(i);
 490                 if (sys.has(symbols.op(i)))
 491                         throw(std::logic_error("lsolve: system is not linear"));
 492                 if (rhs.has(symbols.op(i)))
 493                         throw(std::logic_error("lsolve: system is not linear"));
 494         }
 495
 496         matrix solution;
 497         try {
 498                 solution = sys.solve(vars,rhs,options);
 499         } catch (const std::runtime_error & e) {
 500                 // Probably singular matrix or otherwise overdetermined system:
 501                 // It is consistent to return an empty list
 502                 return lst();
 503         }
 504         GINAC_ASSERT(solution.cols()==1);
 505         GINAC_ASSERT(solution.rows()==symbols.nops());
 506
 507         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 508         lst sollist;
 509         for (unsigned i=0; i<symbols.nops(); i++)
 510                 sollist.append(symbols.op(i)==solution(i,0));
 511
 512         return sollist;
 513 }
 514
 515 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 516  * for static lib (so ginsh will see them). */
 517 unsigned force_include_tgamma = function_index_tgamma;
 518 unsigned force_include_zeta1 = function_index_zeta1;
 519
 520 } // namespace GiNaC