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 "utils.h"
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // absolute value
  42 //////////
  43
  44 static ex abs_evalf(const ex & arg)
  45 {
  46         BEGIN_TYPECHECK
  47                 TYPECHECK(arg,numeric)
  48         END_TYPECHECK(abs(arg))
  49
  50         return abs(ex_to<numeric>(arg));
  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         BEGIN_TYPECHECK
  72                 TYPECHECK(arg,numeric)
  73         END_TYPECHECK(csgn(arg))
  74
  75         return csgn(ex_to<numeric>(arg));
  76 }
  77
  78 static ex csgn_eval(const ex & arg)
  79 {
  80         if (is_ex_exactly_of_type(arg, numeric))
  81                 return csgn(ex_to<numeric>(arg));
  82
  83         else if (is_ex_of_type(arg, mul) &&
  84                  is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
  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: log(x*y) == log(x) + log(y) + eta(x,y).
 130 //////////
 131
 132 static ex eta_evalf(const ex & x, const ex & y)
 133 {
 134         BEGIN_TYPECHECK
 135                 TYPECHECK(x,numeric)
 136                 TYPECHECK(y,numeric)
 137         END_TYPECHECK(eta(x,y))
 138
 139         numeric xim = imag(ex_to<numeric>(x));
 140         numeric yim = imag(ex_to<numeric>(y));
 141         numeric xyim = imag(ex_to<numeric>(x*y));
 142         return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 143 }
 144
 145 static ex eta_eval(const ex & x, const ex & y)
 146 {
 147         if (is_ex_exactly_of_type(x, numeric) &&
 148                 is_ex_exactly_of_type(y, numeric)) {
 149                 // don't call eta_evalf here because it would call Pi.evalf()!
 150                 numeric xim = imag(ex_to<numeric>(x));
 151                 numeric yim = imag(ex_to<numeric>(y));
 152                 numeric xyim = imag(ex_to<numeric>(x*y));
 153                 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 154         }
 155
 156         return eta(x,y).hold();
 157 }
 158
 159 static ex eta_series(const ex & arg1,
 160                      const ex & arg2,
 161                      const relational & rel,
 162                      int order,
 163                      unsigned options)
 164 {
 165         const ex arg1_pt = arg1.subs(rel);
 166         const ex arg2_pt = arg2.subs(rel);
 167         if (ex_to<numeric>(arg1_pt).imag().is_zero() ||
 168                 ex_to<numeric>(arg2_pt).imag().is_zero() ||
 169                 ex_to<numeric>(arg1_pt*arg2_pt).imag().is_zero()) {
 170                 throw (std::domain_error("eta_series(): on discontinuity"));
 171         }
 172         epvector seq;
 173         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
 174         return pseries(rel,seq);
 175 }
 176
 177 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 178                        evalf_func(eta_evalf).
 179                        series_func(eta_series).
 180                        latex_name("\\eta"));
 181
 182
 183 //////////
 184 // dilogarithm
 185 //////////
 186
 187 static ex Li2_evalf(const ex & x)
 188 {
 189         BEGIN_TYPECHECK
 190                 TYPECHECK(x,numeric)
 191         END_TYPECHECK(Li2(x))
 192
 193         return Li2(ex_to<numeric>(x));  // -> numeric Li2(numeric)
 194 }
 195
 196 static ex Li2_eval(const ex & x)
 197 {
 198         if (x.info(info_flags::numeric)) {
 199                 // Li2(0) -> 0
 200                 if (x.is_zero())
 201                         return _ex0();
 202                 // Li2(1) -> Pi^2/6
 203                 if (x.is_equal(_ex1()))
 204                         return power(Pi,_ex2())/_ex6();
 205                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 206                 if (x.is_equal(_ex1_2()))
 207                         return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
 208                 // Li2(-1) -> -Pi^2/12
 209                 if (x.is_equal(_ex_1()))
 210                         return -power(Pi,_ex2())/_ex12();
 211                 // Li2(I) -> -Pi^2/48+Catalan*I
 212                 if (x.is_equal(I))
 213                         return power(Pi,_ex2())/_ex_48() + Catalan*I;
 214                 // Li2(-I) -> -Pi^2/48-Catalan*I
 215                 if (x.is_equal(-I))
 216                         return power(Pi,_ex2())/_ex_48() - Catalan*I;
 217                 // Li2(float)
 218                 if (!x.info(info_flags::crational))
 219                         return Li2_evalf(x);
 220         }
 221
 222         return Li2(x).hold();
 223 }
 224
 225 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 226 {
 227         GINAC_ASSERT(deriv_param==0);
 228
 229         // d/dx Li2(x) -> -log(1-x)/x
 230         return -log(1-x)/x;
 231 }
 232
 233 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 234 {
 235         const ex x_pt = x.subs(rel);
 236         if (x_pt.info(info_flags::numeric)) {
 237                 // First special case: x==0 (derivatives have poles)
 238                 if (x_pt.is_zero()) {
 239                         // method:
 240                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 241                         // simply substitute x==0.  The limit, however, exists: it is 1.
 242                         // We also know all higher derivatives' limits:
 243                         // (d/dx)^n Li2(x) == n!/n^2.
 244                         // So the primitive series expansion is
 245                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 246                         // and so on.
 247                         // We first construct such a primitive series expansion manually in
 248                         // a dummy symbol s and then insert the argument's series expansion
 249                         // for s.  Reexpanding the resulting series returns the desired
 250                         // result.
 251                         const symbol s;
 252                         ex ser;
 253                         // manually construct the primitive expansion
 254                         for (int i=1; i<order; ++i)
 255                                 ser += pow(s,i) / pow(numeric(i), _num2());
 256                         // substitute the argument's series expansion
 257                         ser = ser.subs(s==x.series(rel, order));
 258                         // maybe that was terminating, so add a proper order term
 259                         epvector nseq;
 260                         nseq.push_back(expair(Order(_ex1()), order));
 261                         ser += pseries(rel, nseq);
 262                         // reexpanding it will collapse the series again
 263                         return ser.series(rel, order);
 264                         // NB: Of course, this still does not allow us to compute anything
 265                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 266                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 267                         // substitution x==0.  Probably limits *are* needed for the general
 268                         // cases.  In case L'Hospital's rule is implemented for limits and
 269                         // basic::series() takes care of this, this whole block is probably
 270                         // obsolete!
 271                 }
 272                 // second special case: x==1 (branch point)
 273                 if (x_pt == _ex1()) {
 274                         // method:
 275                         // construct series manually in a dummy symbol s
 276                         const symbol s;
 277                         ex ser = zeta(2);
 278                         // manually construct the primitive expansion
 279                         for (int i=1; i<order; ++i)
 280                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 281                         // substitute the argument's series expansion
 282                         ser = ser.subs(s==x.series(rel, order));
 283                         // maybe that was terminating, so add a proper order term
 284                         epvector nseq;
 285                         nseq.push_back(expair(Order(_ex1()), order));
 286                         ser += pseries(rel, nseq);
 287                         // reexpanding it will collapse the series again
 288                         return ser.series(rel, order);
 289                 }
 290                 // third special case: x real, >=1 (branch cut)
 291                 if (!(options & series_options::suppress_branchcut) &&
 292                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 293                         // method:
 294                         // This is the branch cut: assemble the primitive series manually
 295                         // and then add the corresponding complex step function.
 296                         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 297                         const ex point = rel.rhs();
 298                         const symbol foo;
 299                         epvector seq;
 300                         // zeroth order term:
 301                         seq.push_back(expair(Li2(x_pt), _ex0()));
 302                         // compute the intermediate terms:
 303                         ex replarg = series(Li2(x), *s==foo, order);
 304                         for (unsigned i=1; i<replarg.nops()-1; ++i)
 305                                 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
 306                         // append an order term:
 307                         seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
 308                         return pseries(rel, seq);
 309                 }
 310         }
 311         // all other cases should be safe, by now:
 312         throw do_taylor();  // caught by function::series()
 313 }
 314
 315 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 316                        evalf_func(Li2_evalf).
 317                        derivative_func(Li2_deriv).
 318                        series_func(Li2_series).
 319                        latex_name("\\mbox{Li}_2"));
 320
 321 //////////
 322 // trilogarithm
 323 //////////
 324
 325 static ex Li3_eval(const ex & x)
 326 {
 327         if (x.is_zero())
 328                 return x;
 329         return Li3(x).hold();
 330 }
 331
 332 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 333                        latex_name("\\mbox{Li}_3"));
 334
 335 //////////
 336 // factorial
 337 //////////
 338
 339 static ex factorial_evalf(const ex & x)
 340 {
 341         return factorial(x).hold();
 342 }
 343
 344 static ex factorial_eval(const ex & x)
 345 {
 346         if (is_ex_exactly_of_type(x, numeric))
 347                 return factorial(ex_to<numeric>(x));
 348         else
 349                 return factorial(x).hold();
 350 }
 351
 352 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 353                              evalf_func(factorial_evalf));
 354
 355 //////////
 356 // binomial
 357 //////////
 358
 359 static ex binomial_evalf(const ex & x, const ex & y)
 360 {
 361         return binomial(x, y).hold();
 362 }
 363
 364 static ex binomial_eval(const ex & x, const ex &y)
 365 {
 366         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
 367                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 368         else
 369                 return binomial(x, y).hold();
 370 }
 371
 372 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 373                             evalf_func(binomial_evalf));
 374
 375 //////////
 376 // Order term function (for truncated power series)
 377 //////////
 378
 379 static ex Order_eval(const ex & x)
 380 {
 381         if (is_ex_exactly_of_type(x, numeric)) {
 382                 // O(c) -> O(1) or 0
 383                 if (!x.is_zero())
 384                         return Order(_ex1()).hold();
 385                 else
 386                         return _ex0();
 387         } else if (is_ex_exactly_of_type(x, mul)) {
 388                 mul *m = static_cast<mul *>(x.bp);
 389                 // O(c*expr) -> O(expr)
 390                 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
 391                         return Order(x / m->op(m->nops() - 1)).hold();
 392         }
 393         return Order(x).hold();
 394 }
 395
 396 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 397 {
 398         // Just wrap the function into a pseries object
 399         epvector new_seq;
 400         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 401         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 402         new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
 403         return pseries(r, new_seq);
 404 }
 405
 406 // Differentiation is handled in function::derivative because of its special requirements
 407
 408 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 409                          series_func(Order_series).
 410                          latex_name("\\mathcal{O}"));
 411
 412 //////////
 413 // Inert partial differentiation operator
 414 //////////
 415
 416 static ex Derivative_eval(const ex & f, const ex & l)
 417 {
 418         if (!is_ex_exactly_of_type(f, function)) {
 419                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
 420         }
 421         if (!is_ex_exactly_of_type(l, lst)) {
 422                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
 423         }
 424         return Derivative(f, l).hold();
 425 }
 426
 427 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 428
 429 //////////
 430 // Solve linear system
 431 //////////
 432
 433 ex lsolve(const ex &eqns, const ex &symbols)
 434 {
 435         // solve a system of linear equations
 436         if (eqns.info(info_flags::relation_equal)) {
 437                 if (!symbols.info(info_flags::symbol))
 438                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 439                 ex sol=lsolve(lst(eqns),lst(symbols));
 440
 441                 GINAC_ASSERT(sol.nops()==1);
 442                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
 443
 444                 return sol.op(0).op(1); // return rhs of first solution
 445         }
 446
 447         // syntax checks
 448         if (!eqns.info(info_flags::list)) {
 449                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 450         }
 451         for (unsigned i=0; i<eqns.nops(); i++) {
 452                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 453                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 454                 }
 455         }
 456         if (!symbols.info(info_flags::list)) {
 457                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 458         }
 459         for (unsigned i=0; i<symbols.nops(); i++) {
 460                 if (!symbols.op(i).info(info_flags::symbol)) {
 461                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 462                 }
 463         }
 464
 465         // build matrix from equation system
 466         matrix sys(eqns.nops(),symbols.nops());
 467         matrix rhs(eqns.nops(),1);
 468         matrix vars(symbols.nops(),1);
 469
 470         for (unsigned r=0; r<eqns.nops(); r++) {
 471                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 472                 ex linpart = eq;
 473                 for (unsigned c=0; c<symbols.nops(); c++) {
 474                         ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 475                         linpart -= co*symbols.op(c);
 476                         sys(r,c) = co;
 477                 }
 478                 linpart = linpart.expand();
 479                 rhs(r,0) = -linpart;
 480         }
 481
 482         // test if system is linear and fill vars matrix
 483         for (unsigned i=0; i<symbols.nops(); i++) {
 484                 vars(i,0) = symbols.op(i);
 485                 if (sys.has(symbols.op(i)))
 486                         throw(std::logic_error("lsolve: system is not linear"));
 487                 if (rhs.has(symbols.op(i)))
 488                         throw(std::logic_error("lsolve: system is not linear"));
 489         }
 490
 491         matrix solution;
 492         try {
 493                 solution = sys.solve(vars,rhs);
 494         } catch (const std::runtime_error & e) {
 495                 // Probably singular matrix or otherwise overdetermined system:
 496                 // It is consistent to return an empty list
 497                 return lst();
 498         }
 499         GINAC_ASSERT(solution.cols()==1);
 500         GINAC_ASSERT(solution.rows()==symbols.nops());
 501
 502         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 503         lst sollist;
 504         for (unsigned i=0; i<symbols.nops(); i++)
 505                 sollist.append(symbols.op(i)==solution(i,0));
 506
 507         return sollist;
 508 }
 509
 510 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 511  * for static lib (so ginsh will see them). */
 512 unsigned force_include_tgamma = function_index_tgamma;
 513 unsigned force_include_zeta1 = function_index_zeta1;
 514
 515 } // namespace GiNaC