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 "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 namespace GiNaC {
  41
  42 //////////
  43 // absolute value
  44 //////////
  45
  46 static ex abs_evalf(const ex & arg)
  47 {
  48         BEGIN_TYPECHECK
  49                 TYPECHECK(arg,numeric)
  50         END_TYPECHECK(abs(arg))
  51
  52         return abs(ex_to_numeric(arg));
  53 }
  54
  55 static ex abs_eval(const ex & arg)
  56 {
  57         if (is_ex_exactly_of_type(arg, numeric))
  58                 return abs(ex_to_numeric(arg));
  59         else
  60                 return abs(arg).hold();
  61 }
  62
  63 REGISTER_FUNCTION(abs, eval_func(abs_eval).
  64                        evalf_func(abs_evalf));
  65
  66
  67 //////////
  68 // Complex sign
  69 //////////
  70
  71 static ex csgn_evalf(const ex & arg)
  72 {
  73         BEGIN_TYPECHECK
  74                 TYPECHECK(arg,numeric)
  75         END_TYPECHECK(csgn(arg))
  76
  77         return csgn(ex_to_numeric(arg));
  78 }
  79
  80 static ex csgn_eval(const ex & arg)
  81 {
  82         if (is_ex_exactly_of_type(arg, numeric))
  83                 return csgn(ex_to_numeric(arg));
  84
  85         else if (is_ex_exactly_of_type(arg, mul)) {
  86                 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
  87                 if (oc.is_real()) {
  88                         if (oc > 0)
  89                                 // csgn(42*x) -> csgn(x)
  90                                 return csgn(arg/oc).hold();
  91                         else
  92                                 // csgn(-42*x) -> -csgn(x)
  93                                 return -csgn(arg/oc).hold();
  94                 }
  95                 if (oc.real().is_zero()) {
  96                         if (oc.imag() > 0)
  97                                 // csgn(42*I*x) -> csgn(I*x)
  98                                 return csgn(I*arg/oc).hold();
  99                         else
 100                                 // csgn(-42*I*x) -> -csgn(I*x)
 101                                 return -csgn(I*arg/oc).hold();
 102                 }
 103         }
 104
 105         return csgn(arg).hold();
 106 }
 107
 108 static ex csgn_series(const ex & arg,
 109                       const relational & rel,
 110                       int order,
 111                       unsigned options)
 112 {
 113         const ex arg_pt = arg.subs(rel);
 114         if (arg_pt.info(info_flags::numeric)
 115             && ex_to_numeric(arg_pt).real().is_zero()
 116             && !(options & series_options::suppress_branchcut))
 117                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 118
 119         epvector seq;
 120         seq.push_back(expair(csgn(arg_pt), _ex0()));
 121         return pseries(rel,seq);
 122 }
 123
 124 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 125                         evalf_func(csgn_evalf).
 126                         series_func(csgn_series));
 127
 128
 129 //////////
 130 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
 131 //////////
 132
 133 static ex eta_evalf(const ex & x, const ex & y)
 134 {
 135         BEGIN_TYPECHECK
 136                 TYPECHECK(x,numeric)
 137                 TYPECHECK(y,numeric)
 138         END_TYPECHECK(eta(x,y))
 139
 140         numeric xim = imag(ex_to_numeric(x));
 141         numeric yim = imag(ex_to_numeric(y));
 142         numeric xyim = imag(ex_to_numeric(x*y));
 143         return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 144 }
 145
 146 static ex eta_eval(const ex & x, const ex & y)
 147 {
 148         if (is_ex_exactly_of_type(x, numeric) &&
 149                 is_ex_exactly_of_type(y, numeric)) {
 150                 // don't call eta_evalf here because it would call Pi.evalf()!
 151                 numeric xim = imag(ex_to_numeric(x));
 152                 numeric yim = imag(ex_to_numeric(y));
 153                 numeric xyim = imag(ex_to_numeric(x*y));
 154                 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 155         }
 156
 157         return eta(x,y).hold();
 158 }
 159
 160 static ex eta_series(const ex & arg1,
 161                      const ex & arg2,
 162                      const relational & rel,
 163                      int order,
 164                      unsigned options)
 165 {
 166         const ex arg1_pt = arg1.subs(rel);
 167         const ex arg2_pt = arg2.subs(rel);
 168         if (ex_to_numeric(arg1_pt).imag().is_zero() ||
 169                 ex_to_numeric(arg2_pt).imag().is_zero() ||
 170                 ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
 171                 throw (std::domain_error("eta_series(): on discontinuity"));
 172         }
 173         epvector seq;
 174         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
 175         return pseries(rel,seq);
 176 }
 177
 178 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 179                        evalf_func(eta_evalf).
 180                        series_func(eta_series).
 181                        latex_name("\\eta"));
 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                        latex_name("\\mbox{Li}_2"));
 321
 322 //////////
 323 // trilogarithm
 324 //////////
 325
 326 static ex Li3_eval(const ex & x)
 327 {
 328         if (x.is_zero())
 329                 return x;
 330         return Li3(x).hold();
 331 }
 332
 333 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 334                        latex_name("\\mbox{Li}_3"));
 335
 336 //////////
 337 // factorial
 338 //////////
 339
 340 static ex factorial_evalf(const ex & x)
 341 {
 342         return factorial(x).hold();
 343 }
 344
 345 static ex factorial_eval(const ex & x)
 346 {
 347         if (is_ex_exactly_of_type(x, numeric))
 348                 return factorial(ex_to_numeric(x));
 349         else
 350                 return factorial(x).hold();
 351 }
 352
 353 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 354                              evalf_func(factorial_evalf));
 355
 356 //////////
 357 // binomial
 358 //////////
 359
 360 static ex binomial_evalf(const ex & x, const ex & y)
 361 {
 362         return binomial(x, y).hold();
 363 }
 364
 365 static ex binomial_eval(const ex & x, const ex &y)
 366 {
 367         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
 368                 return binomial(ex_to_numeric(x), ex_to_numeric(y));
 369         else
 370                 return binomial(x, y).hold();
 371 }
 372
 373 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 374                             evalf_func(binomial_evalf));
 375
 376 //////////
 377 // Order term function (for truncated power series)
 378 //////////
 379
 380 static ex Order_eval(const ex & x)
 381 {
 382         if (is_ex_exactly_of_type(x, numeric)) {
 383                 // O(c) -> O(1) or 0
 384                 if (!x.is_zero())
 385                         return Order(_ex1()).hold();
 386                 else
 387                         return _ex0();
 388         } else if (is_ex_exactly_of_type(x, mul)) {
 389                 mul *m = static_cast<mul *>(x.bp);
 390                 // O(c*expr) -> O(expr)
 391                 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
 392                         return Order(x / m->op(m->nops() - 1)).hold();
 393         }
 394         return Order(x).hold();
 395 }
 396
 397 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 398 {
 399         // Just wrap the function into a pseries object
 400         epvector new_seq;
 401         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 402         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 403         new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
 404         return pseries(r, new_seq);
 405 }
 406
 407 // Differentiation is handled in function::derivative because of its special requirements
 408
 409 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 410                          series_func(Order_series).
 411                          latex_name("\\mathcal{O}"));
 412
 413 //////////
 414 // Inert partial differentiation operator
 415 //////////
 416
 417 static ex Derivative_eval(const ex & f, const ex & l)
 418 {
 419         if (!is_ex_exactly_of_type(f, function)) {
 420                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
 421         }
 422         if (!is_ex_exactly_of_type(l, lst)) {
 423                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
 424         }
 425         return Derivative(f, l).hold();
 426 }
 427
 428 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 429
 430 //////////
 431 // Solve linear system
 432 //////////
 433
 434 ex lsolve(const ex &eqns, const ex &symbols)
 435 {
 436         // solve a system of linear equations
 437         if (eqns.info(info_flags::relation_equal)) {
 438                 if (!symbols.info(info_flags::symbol))
 439                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 440                 ex sol=lsolve(lst(eqns),lst(symbols));
 441
 442                 GINAC_ASSERT(sol.nops()==1);
 443                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
 444
 445                 return sol.op(0).op(1); // return rhs of first solution
 446         }
 447
 448         // syntax checks
 449         if (!eqns.info(info_flags::list)) {
 450                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 451         }
 452         for (unsigned i=0; i<eqns.nops(); i++) {
 453                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 454                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 455                 }
 456         }
 457         if (!symbols.info(info_flags::list)) {
 458                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 459         }
 460         for (unsigned i=0; i<symbols.nops(); i++) {
 461                 if (!symbols.op(i).info(info_flags::symbol)) {
 462                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 463                 }
 464         }
 465
 466         // build matrix from equation system
 467         matrix sys(eqns.nops(),symbols.nops());
 468         matrix rhs(eqns.nops(),1);
 469         matrix vars(symbols.nops(),1);
 470
 471         for (unsigned r=0; r<eqns.nops(); r++) {
 472                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 473                 ex linpart = eq;
 474                 for (unsigned c=0; c<symbols.nops(); c++) {
 475                         ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
 476                         linpart -= co*symbols.op(c);
 477                         sys.set(r,c,co);
 478                 }
 479                 linpart = linpart.expand();
 480                 rhs.set(r,0,-linpart);
 481         }
 482
 483         // test if system is linear and fill vars matrix
 484         for (unsigned i=0; i<symbols.nops(); i++) {
 485                 vars.set(i,0,symbols.op(i));
 486                 if (sys.has(symbols.op(i)))
 487                         throw(std::logic_error("lsolve: system is not linear"));
 488                 if (rhs.has(symbols.op(i)))
 489                         throw(std::logic_error("lsolve: system is not linear"));
 490         }
 491
 492         matrix solution;
 493         try {
 494                 solution = sys.solve(vars,rhs);
 495         } catch (const std::runtime_error & e) {
 496                 // Probably singular matrix or otherwise overdetermined system:
 497                 // It is consistent to return an empty list
 498                 return lst();
 499         }
 500         GINAC_ASSERT(solution.cols()==1);
 501         GINAC_ASSERT(solution.rows()==symbols.nops());
 502
 503         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 504         lst sollist;
 505         for (unsigned i=0; i<symbols.nops(); i++)
 506                 sollist.append(symbols.op(i)==solution(i,0));
 507
 508         return sollist;
 509 }
 510
 511 /** non-commutative power. */
 512 ex ncpower(const ex &basis, unsigned exponent)
 513 {
 514         if (exponent==0) {
 515                 return _ex1();
 516         }
 517
 518         exvector v;
 519         v.reserve(exponent);
 520         for (unsigned i=0; i<exponent; ++i) {
 521                 v.push_back(basis);
 522         }
 523
 524         return ncmul(v,1);
 525 }
 526
 527 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
 528  *  for static lib (so ginsh will see them). */
 529 unsigned force_include_tgamma = function_index_tgamma;
 530 unsigned force_include_zeta1 = function_index_zeta1;
 531
 532 } // namespace GiNaC