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 #ifndef NO_NAMESPACE_GINAC
  41 namespace GiNaC {
  42 #endif // ndef NO_NAMESPACE_GINAC
  43
  44 //////////
  45 // absolute value
  46 //////////
  47
  48 static ex abs_evalf(const ex & arg)
  49 {
  50         BEGIN_TYPECHECK
  51                 TYPECHECK(arg,numeric)
  52         END_TYPECHECK(abs(arg))
  53
  54         return abs(ex_to_numeric(arg));
  55 }
  56
  57 static ex abs_eval(const ex & arg)
  58 {
  59         if (is_ex_exactly_of_type(arg, numeric))
  60                 return abs(ex_to_numeric(arg));
  61         else
  62                 return abs(arg).hold();
  63 }
  64
  65 REGISTER_FUNCTION(abs, eval_func(abs_eval).
  66                        evalf_func(abs_evalf));
  67
  68
  69 //////////
  70 // Complex sign
  71 //////////
  72
  73 static ex csgn_evalf(const ex & arg)
  74 {
  75         BEGIN_TYPECHECK
  76                 TYPECHECK(arg,numeric)
  77         END_TYPECHECK(csgn(arg))
  78
  79         return csgn(ex_to_numeric(arg));
  80 }
  81
  82 static ex csgn_eval(const ex & arg)
  83 {
  84         if (is_ex_exactly_of_type(arg, numeric))
  85                 return csgn(ex_to_numeric(arg));
  86
  87         else if (is_ex_exactly_of_type(arg, mul)) {
  88                 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
  89                 if (oc.is_real()) {
  90                         if (oc > 0)
  91                                 // csgn(42*x) -> csgn(x)
  92                                 return csgn(arg/oc).hold();
  93                         else
  94                                 // csgn(-42*x) -> -csgn(x)
  95                                 return -csgn(arg/oc).hold();
  96                 }
  97                 if (oc.real().is_zero()) {
  98                         if (oc.imag() > 0)
  99                                 // csgn(42*I*x) -> csgn(I*x)
 100                                 return csgn(I*arg/oc).hold();
 101                         else
 102                                 // csgn(-42*I*x) -> -csgn(I*x)
 103                                 return -csgn(I*arg/oc).hold();
 104                 }
 105         }
 106
 107         return csgn(arg).hold();
 108 }
 109
 110 static ex csgn_series(const ex & arg,
 111                       const relational & rel,
 112                       int order,
 113                       unsigned options)
 114 {
 115         const ex arg_pt = arg.subs(rel);
 116         if (arg_pt.info(info_flags::numeric)
 117             && ex_to_numeric(arg_pt).real().is_zero()
 118             && !(options & series_options::suppress_branchcut))
 119                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 120
 121         epvector seq;
 122         seq.push_back(expair(csgn(arg_pt), _ex0()));
 123         return pseries(rel,seq);
 124 }
 125
 126 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 127                         evalf_func(csgn_evalf).
 128                         series_func(csgn_series));
 129
 130
 131 //////////
 132 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
 133 //////////
 134
 135 static ex eta_evalf(const ex & x, const ex & y)
 136 {
 137         BEGIN_TYPECHECK
 138                 TYPECHECK(x,numeric)
 139                 TYPECHECK(y,numeric)
 140         END_TYPECHECK(eta(x,y))
 141
 142         numeric xim = imag(ex_to_numeric(x));
 143         numeric yim = imag(ex_to_numeric(y));
 144         numeric xyim = imag(ex_to_numeric(x*y));
 145         return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 146 }
 147
 148 static ex eta_eval(const ex & x, const ex & y)
 149 {
 150         if (is_ex_exactly_of_type(x, numeric) &&
 151                 is_ex_exactly_of_type(y, numeric)) {
 152                 // don't call eta_evalf here because it would call Pi.evalf()!
 153                 numeric xim = imag(ex_to_numeric(x));
 154                 numeric yim = imag(ex_to_numeric(y));
 155                 numeric xyim = imag(ex_to_numeric(x*y));
 156                 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
 157         }
 158
 159         return eta(x,y).hold();
 160 }
 161
 162 static ex eta_series(const ex & arg1,
 163                      const ex & arg2,
 164                      const relational & rel,
 165                      int order,
 166                      unsigned options)
 167 {
 168         const ex arg1_pt = arg1.subs(rel);
 169         const ex arg2_pt = arg2.subs(rel);
 170         if (ex_to_numeric(arg1_pt).imag().is_zero() ||
 171                 ex_to_numeric(arg2_pt).imag().is_zero() ||
 172                 ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
 173                 throw (std::domain_error("eta_series(): on discontinuity"));
 174         }
 175         epvector seq;
 176         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
 177         return pseries(rel,seq);
 178 }
 179
 180 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 181                        evalf_func(eta_evalf).
 182                        series_func(eta_series));
 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
 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
 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
 411 //////////
 412 // Inert partial differentiation operator
 413 //////////
 414
 415 static ex Derivative_eval(const ex & f, const ex & l)
 416 {
 417         if (!is_ex_exactly_of_type(f, function)) {
 418                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
 419         }
 420         if (!is_ex_exactly_of_type(l, lst)) {
 421                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
 422         }
 423         return Derivative(f, l).hold();
 424 }
 425
 426 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 427
 428 //////////
 429 // Solve linear system
 430 //////////
 431
 432 ex lsolve(const ex &eqns, const ex &symbols)
 433 {
 434         // solve a system of linear equations
 435         if (eqns.info(info_flags::relation_equal)) {
 436                 if (!symbols.info(info_flags::symbol))
 437                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 438                 ex sol=lsolve(lst(eqns),lst(symbols));
 439
 440                 GINAC_ASSERT(sol.nops()==1);
 441                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
 442
 443                 return sol.op(0).op(1); // return rhs of first solution
 444         }
 445
 446         // syntax checks
 447         if (!eqns.info(info_flags::list)) {
 448                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 449         }
 450         for (unsigned i=0; i<eqns.nops(); i++) {
 451                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 452                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 453                 }
 454         }
 455         if (!symbols.info(info_flags::list)) {
 456                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 457         }
 458         for (unsigned i=0; i<symbols.nops(); i++) {
 459                 if (!symbols.op(i).info(info_flags::symbol)) {
 460                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 461                 }
 462         }
 463
 464         // build matrix from equation system
 465         matrix sys(eqns.nops(),symbols.nops());
 466         matrix rhs(eqns.nops(),1);
 467         matrix vars(symbols.nops(),1);
 468
 469         for (unsigned r=0; r<eqns.nops(); r++) {
 470                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 471                 ex linpart = eq;
 472                 for (unsigned c=0; c<symbols.nops(); c++) {
 473                         ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
 474                         linpart -= co*symbols.op(c);
 475                         sys.set(r,c,co);
 476                 }
 477                 linpart = linpart.expand();
 478                 rhs.set(r,0,-linpart);
 479         }
 480
 481         // test if system is linear and fill vars matrix
 482         for (unsigned i=0; i<symbols.nops(); i++) {
 483                 vars.set(i,0,symbols.op(i));
 484                 if (sys.has(symbols.op(i)))
 485                         throw(std::logic_error("lsolve: system is not linear"));
 486                 if (rhs.has(symbols.op(i)))
 487                         throw(std::logic_error("lsolve: system is not linear"));
 488         }
 489
 490         matrix solution;
 491         try {
 492                 solution = sys.solve(vars,rhs);
 493         } catch (const std::runtime_error & e) {
 494                 // Probably singular matrix or otherwise overdetermined system:
 495                 // It is consistent to return an empty list
 496                 return lst();
 497         }
 498         GINAC_ASSERT(solution.cols()==1);
 499         GINAC_ASSERT(solution.rows()==symbols.nops());
 500
 501         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 502         lst sollist;
 503         for (unsigned i=0; i<symbols.nops(); i++)
 504                 sollist.append(symbols.op(i)==solution(i,0));
 505
 506         return sollist;
 507 }
 508
 509 /** non-commutative power. */
 510 ex ncpower(const ex &basis, unsigned exponent)
 511 {
 512         if (exponent==0) {
 513                 return _ex1();
 514         }
 515
 516         exvector v;
 517         v.reserve(exponent);
 518         for (unsigned i=0; i<exponent; ++i) {
 519                 v.push_back(basis);
 520         }
 521
 522         return ncmul(v,1);
 523 }
 524
 525 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
 526  *  for static lib (so ginsh will see them). */
 527 unsigned force_include_tgamma = function_index_tgamma;
 528 unsigned force_include_zeta1 = function_index_zeta1;
 529
 530 #ifndef NO_NAMESPACE_GINAC
 531 } // namespace GiNaC
 532 #endif // ndef NO_NAMESPACE_GINAC