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_of_type(arg, mul) &&
  86                  is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
  87                 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
  88                 if (oc.is_real()) {
  89                         if (oc > 0)
  90                                 // csgn(42*x) -> csgn(x)
  91                                 return csgn(arg/oc).hold();
  92                         else
  93                                 // csgn(-42*x) -> -csgn(x)
  94                                 return -csgn(arg/oc).hold();
  95                 }
  96                 if (oc.real().is_zero()) {
  97                         if (oc.imag() > 0)
  98                                 // csgn(42*I*x) -> csgn(I*x)
  99                                 return csgn(I*arg/oc).hold();
 100                         else
 101                                 // csgn(-42*I*x) -> -csgn(I*x)
 102                                 return -csgn(I*arg/oc).hold();
 103                 }
 104         }
 105
 106         return csgn(arg).hold();
 107 }
 108
 109 static ex csgn_series(const ex & arg,
 110                       const relational & rel,
 111                       int order,
 112                       unsigned options)
 113 {
 114         const ex arg_pt = arg.subs(rel);
 115         if (arg_pt.info(info_flags::numeric)
 116             && ex_to_numeric(arg_pt).real().is_zero()
 117             && !(options & series_options::suppress_branchcut))
 118                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 119
 120         epvector seq;
 121         seq.push_back(expair(csgn(arg_pt), _ex0()));
 122         return pseries(rel,seq);
 123 }
 124
 125 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 126                         evalf_func(csgn_evalf).
 127                         series_func(csgn_series));
 128
 129
 130 //////////
 131 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
 132 //////////
 133
 134 static ex eta_evalf(const ex & x, const ex & y)
 135 {
 136         BEGIN_TYPECHECK
 137                 TYPECHECK(x,numeric)
 138                 TYPECHECK(y,numeric)
 139         END_TYPECHECK(eta(x,y))
 140
 141         numeric xim = imag(ex_to_numeric(x));
 142         numeric yim = imag(ex_to_numeric(y));
 143         numeric xyim = imag(ex_to_numeric(x*y));
 144         return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 145 }
 146
 147 static ex eta_eval(const ex & x, const ex & y)
 148 {
 149         if (is_ex_exactly_of_type(x, numeric) &&
 150                 is_ex_exactly_of_type(y, numeric)) {
 151                 // don't call eta_evalf here because it would call Pi.evalf()!
 152                 numeric xim = imag(ex_to_numeric(x));
 153                 numeric yim = imag(ex_to_numeric(y));
 154                 numeric xyim = imag(ex_to_numeric(x*y));
 155                 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 156         }
 157
 158         return eta(x,y).hold();
 159 }
 160
 161 static ex eta_series(const ex & arg1,
 162                      const ex & arg2,
 163                      const relational & rel,
 164                      int order,
 165                      unsigned options)
 166 {
 167         const ex arg1_pt = arg1.subs(rel);
 168         const ex arg2_pt = arg2.subs(rel);
 169         if (ex_to_numeric(arg1_pt).imag().is_zero() ||
 170                 ex_to_numeric(arg2_pt).imag().is_zero() ||
 171                 ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
 172                 throw (std::domain_error("eta_series(): on discontinuity"));
 173         }
 174         epvector seq;
 175         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
 176         return pseries(rel,seq);
 177 }
 178
 179 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 180                        evalf_func(eta_evalf).
 181                        series_func(eta_series).
 182                        latex_name("\\eta"));
 183
 184
 185 //////////
 186 // dilogarithm
 187 //////////
 188
 189 static ex Li2_evalf(const ex & x)
 190 {
 191         BEGIN_TYPECHECK
 192                 TYPECHECK(x,numeric)
 193         END_TYPECHECK(Li2(x))
 194
 195         return Li2(ex_to_numeric(x));  // -> numeric Li2(numeric)
 196 }
 197
 198 static ex Li2_eval(const ex & x)
 199 {
 200         if (x.info(info_flags::numeric)) {
 201                 // Li2(0) -> 0
 202                 if (x.is_zero())
 203                         return _ex0();
 204                 // Li2(1) -> Pi^2/6
 205                 if (x.is_equal(_ex1()))
 206                         return power(Pi,_ex2())/_ex6();
 207                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 208                 if (x.is_equal(_ex1_2()))
 209                         return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
 210                 // Li2(-1) -> -Pi^2/12
 211                 if (x.is_equal(_ex_1()))
 212                         return -power(Pi,_ex2())/_ex12();
 213                 // Li2(I) -> -Pi^2/48+Catalan*I
 214                 if (x.is_equal(I))
 215                         return power(Pi,_ex2())/_ex_48() + Catalan*I;
 216                 // Li2(-I) -> -Pi^2/48-Catalan*I
 217                 if (x.is_equal(-I))
 218                         return power(Pi,_ex2())/_ex_48() - Catalan*I;
 219                 // Li2(float)
 220                 if (!x.info(info_flags::crational))
 221                         return Li2_evalf(x);
 222         }
 223
 224         return Li2(x).hold();
 225 }
 226
 227 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 228 {
 229         GINAC_ASSERT(deriv_param==0);
 230
 231         // d/dx Li2(x) -> -log(1-x)/x
 232         return -log(1-x)/x;
 233 }
 234
 235 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 236 {
 237         const ex x_pt = x.subs(rel);
 238         if (x_pt.info(info_flags::numeric)) {
 239                 // First special case: x==0 (derivatives have poles)
 240                 if (x_pt.is_zero()) {
 241                         // method:
 242                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 243                         // simply substitute x==0.  The limit, however, exists: it is 1.
 244                         // We also know all higher derivatives' limits:
 245                         // (d/dx)^n Li2(x) == n!/n^2.
 246                         // So the primitive series expansion is
 247                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 248                         // and so on.
 249                         // We first construct such a primitive series expansion manually in
 250                         // a dummy symbol s and then insert the argument's series expansion
 251                         // for s.  Reexpanding the resulting series returns the desired
 252                         // result.
 253                         const symbol s;
 254                         ex ser;
 255                         // manually construct the primitive expansion
 256                         for (int i=1; i<order; ++i)
 257                                 ser += pow(s,i) / pow(numeric(i), _num2());
 258                         // substitute the argument's series expansion
 259                         ser = ser.subs(s==x.series(rel, order));
 260                         // maybe that was terminating, so add a proper order term
 261                         epvector nseq;
 262                         nseq.push_back(expair(Order(_ex1()), order));
 263                         ser += pseries(rel, nseq);
 264                         // reexpanding it will collapse the series again
 265                         return ser.series(rel, order);
 266                         // NB: Of course, this still does not allow us to compute anything
 267                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 268                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 269                         // substitution x==0.  Probably limits *are* needed for the general
 270                         // cases.  In case L'Hospital's rule is implemented for limits and
 271                         // basic::series() takes care of this, this whole block is probably
 272                         // obsolete!
 273                 }
 274                 // second special case: x==1 (branch point)
 275                 if (x_pt == _ex1()) {
 276                         // method:
 277                         // construct series manually in a dummy symbol s
 278                         const symbol s;
 279                         ex ser = zeta(2);
 280                         // manually construct the primitive expansion
 281                         for (int i=1; i<order; ++i)
 282                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 283                         // substitute the argument's series expansion
 284                         ser = ser.subs(s==x.series(rel, order));
 285                         // maybe that was terminating, so add a proper order term
 286                         epvector nseq;
 287                         nseq.push_back(expair(Order(_ex1()), order));
 288                         ser += pseries(rel, nseq);
 289                         // reexpanding it will collapse the series again
 290                         return ser.series(rel, order);
 291                 }
 292                 // third special case: x real, >=1 (branch cut)
 293                 if (!(options & series_options::suppress_branchcut) &&
 294                         ex_to_numeric(x_pt).is_real() && ex_to_numeric(x_pt)>1) {
 295                         // method:
 296                         // This is the branch cut: assemble the primitive series manually
 297                         // and then add the corresponding complex step function.
 298                         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 299                         const ex point = rel.rhs();
 300                         const symbol foo;
 301                         epvector seq;
 302                         // zeroth order term:
 303                         seq.push_back(expair(Li2(x_pt), _ex0()));
 304                         // compute the intermediate terms:
 305                         ex replarg = series(Li2(x), *s==foo, order);
 306                         for (unsigned i=1; i<replarg.nops()-1; ++i)
 307                                 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
 308                         // append an order term:
 309                         seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
 310                         return pseries(rel, seq);
 311                 }
 312         }
 313         // all other cases should be safe, by now:
 314         throw do_taylor();  // caught by function::series()
 315 }
 316
 317 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 318                        evalf_func(Li2_evalf).
 319                        derivative_func(Li2_deriv).
 320                        series_func(Li2_series).
 321                        latex_name("\\mbox{Li}_2"));
 322
 323 //////////
 324 // trilogarithm
 325 //////////
 326
 327 static ex Li3_eval(const ex & x)
 328 {
 329         if (x.is_zero())
 330                 return x;
 331         return Li3(x).hold();
 332 }
 333
 334 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 335                        latex_name("\\mbox{Li}_3"));
 336
 337 //////////
 338 // factorial
 339 //////////
 340
 341 static ex factorial_evalf(const ex & x)
 342 {
 343         return factorial(x).hold();
 344 }
 345
 346 static ex factorial_eval(const ex & x)
 347 {
 348         if (is_ex_exactly_of_type(x, numeric))
 349                 return factorial(ex_to_numeric(x));
 350         else
 351                 return factorial(x).hold();
 352 }
 353
 354 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 355                              evalf_func(factorial_evalf));
 356
 357 //////////
 358 // binomial
 359 //////////
 360
 361 static ex binomial_evalf(const ex & x, const ex & y)
 362 {
 363         return binomial(x, y).hold();
 364 }
 365
 366 static ex binomial_eval(const ex & x, const ex &y)
 367 {
 368         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
 369                 return binomial(ex_to_numeric(x), ex_to_numeric(y));
 370         else
 371                 return binomial(x, y).hold();
 372 }
 373
 374 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 375                             evalf_func(binomial_evalf));
 376
 377 //////////
 378 // Order term function (for truncated power series)
 379 //////////
 380
 381 static ex Order_eval(const ex & x)
 382 {
 383         if (is_ex_exactly_of_type(x, numeric)) {
 384                 // O(c) -> O(1) or 0
 385                 if (!x.is_zero())
 386                         return Order(_ex1()).hold();
 387                 else
 388                         return _ex0();
 389         } else if (is_ex_exactly_of_type(x, mul)) {
 390                 mul *m = static_cast<mul *>(x.bp);
 391                 // O(c*expr) -> O(expr)
 392                 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
 393                         return Order(x / m->op(m->nops() - 1)).hold();
 394         }
 395         return Order(x).hold();
 396 }
 397
 398 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 399 {
 400         // Just wrap the function into a pseries object
 401         epvector new_seq;
 402         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 403         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 404         new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
 405         return pseries(r, new_seq);
 406 }
 407
 408 // Differentiation is handled in function::derivative because of its special requirements
 409
 410 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 411                          series_func(Order_series).
 412                          latex_name("\\mathcal{O}"));
 413
 414 //////////
 415 // Inert partial differentiation operator
 416 //////////
 417
 418 static ex Derivative_eval(const ex & f, const ex & l)
 419 {
 420         if (!is_ex_exactly_of_type(f, function)) {
 421                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
 422         }
 423         if (!is_ex_exactly_of_type(l, lst)) {
 424                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
 425         }
 426         return Derivative(f, l).hold();
 427 }
 428
 429 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 430
 431 //////////
 432 // Solve linear system
 433 //////////
 434
 435 ex lsolve(const ex &eqns, const ex &symbols)
 436 {
 437         // solve a system of linear equations
 438         if (eqns.info(info_flags::relation_equal)) {
 439                 if (!symbols.info(info_flags::symbol))
 440                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 441                 ex sol=lsolve(lst(eqns),lst(symbols));
 442
 443                 GINAC_ASSERT(sol.nops()==1);
 444                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
 445
 446                 return sol.op(0).op(1); // return rhs of first solution
 447         }
 448
 449         // syntax checks
 450         if (!eqns.info(info_flags::list)) {
 451                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 452         }
 453         for (unsigned i=0; i<eqns.nops(); i++) {
 454                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 455                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 456                 }
 457         }
 458         if (!symbols.info(info_flags::list)) {
 459                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 460         }
 461         for (unsigned i=0; i<symbols.nops(); i++) {
 462                 if (!symbols.op(i).info(info_flags::symbol)) {
 463                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 464                 }
 465         }
 466
 467         // build matrix from equation system
 468         matrix sys(eqns.nops(),symbols.nops());
 469         matrix rhs(eqns.nops(),1);
 470         matrix vars(symbols.nops(),1);
 471
 472         for (unsigned r=0; r<eqns.nops(); r++) {
 473                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 474                 ex linpart = eq;
 475                 for (unsigned c=0; c<symbols.nops(); c++) {
 476                         ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
 477                         linpart -= co*symbols.op(c);
 478                         sys.set(r,c,co);
 479                 }
 480                 linpart = linpart.expand();
 481                 rhs.set(r,0,-linpart);
 482         }
 483
 484         // test if system is linear and fill vars matrix
 485         for (unsigned i=0; i<symbols.nops(); i++) {
 486                 vars.set(i,0,symbols.op(i));
 487                 if (sys.has(symbols.op(i)))
 488                         throw(std::logic_error("lsolve: system is not linear"));
 489                 if (rhs.has(symbols.op(i)))
 490                         throw(std::logic_error("lsolve: system is not linear"));
 491         }
 492
 493         matrix solution;
 494         try {
 495                 solution = sys.solve(vars,rhs);
 496         } catch (const std::runtime_error & e) {
 497                 // Probably singular matrix or otherwise overdetermined system:
 498                 // It is consistent to return an empty list
 499                 return lst();
 500         }
 501         GINAC_ASSERT(solution.cols()==1);
 502         GINAC_ASSERT(solution.rows()==symbols.nops());
 503
 504         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 505         lst sollist;
 506         for (unsigned i=0; i<symbols.nops(); i++)
 507                 sollist.append(symbols.op(i)==solution(i,0));
 508
 509         return sollist;
 510 }
 511
 512 /** non-commutative power. */
 513 ex ncpower(const ex &basis, unsigned exponent)
 514 {
 515         if (exponent==0) {
 516                 return _ex1();
 517         }
 518
 519         exvector v;
 520         v.reserve(exponent);
 521         for (unsigned i=0; i<exponent; ++i) {
 522                 v.push_back(basis);
 523         }
 524
 525         return ncmul(v,1);
 526 }
 527
 528 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
 529  *  for static lib (so ginsh will see them). */
 530 unsigned force_include_tgamma = function_index_tgamma;
 531 unsigned force_include_zeta1 = function_index_zeta1;
 532
 533 } // namespace GiNaC