ginac/inifcns_gamma.cpp

   1 /** @file inifcns_gamma.cpp
   2  *
   3  *  Implementation of Gamma-function, Beta-function, Polygamma-functions, and
   4  *  some related stuff. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  22  */
  23
  24 #include <vector>
  25 #include <stdexcept>
  26
  27 #include "inifcns.h"
  28 #include "constant.h"
  29 #include "pseries.h"
  30 #include "numeric.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "operators.h"
  34 #include "symbol.h"
  35 #include "symmetry.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // Logarithm of Gamma function
  42 //////////
  43
  44 static ex lgamma_evalf(const ex & x)
  45 {
  46         if (is_exactly_a<numeric>(x)) {
  47                 try {
  48                         return lgamma(ex_to<numeric>(x));
  49                 } catch (const dunno &e) { }
  50         }
  51
  52         return lgamma(x).hold();
  53 }
  54
  55
  56 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
  57  *  Knows about integer arguments and that's it.  Somebody ought to provide
  58  *  some good numerical evaluation some day...
  59  *
  60  *  @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
  61 static ex lgamma_eval(const ex & x)
  62 {
  63         if (x.info(info_flags::numeric)) {
  64                 // trap integer arguments:
  65                 if (x.info(info_flags::integer)) {
  66                         // lgamma(n) -> log((n-1)!) for postitive n
  67                         if (x.info(info_flags::posint))
  68                                 return log(factorial(x + _ex_1));
  69                         else
  70                                 throw (pole_error("lgamma_eval(): logarithmic pole",0));
  71                 }
  72                 //  lgamma_evalf should be called here once it becomes available
  73         }
  74
  75         return lgamma(x).hold();
  76 }
  77
  78
  79 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
  80 {
  81         GINAC_ASSERT(deriv_param==0);
  82
  83         // d/dx  lgamma(x) -> psi(x)
  84         return psi(x);
  85 }
  86
  87
  88 static ex lgamma_series(const ex & arg,
  89                         const relational & rel,
  90                         int order,
  91                         unsigned options)
  92 {
  93         // method:
  94         // Taylor series where there is no pole falls back to psi function
  95         // evaluation.
  96         // On a pole at -m we could use the recurrence relation
  97         //   lgamma(x) == lgamma(x+1)-log(x)
  98         // from which follows
  99         //   series(lgamma(x),x==-m,order) ==
 100         //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
 101         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 102         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 103                 throw do_taylor();  // caught by function::series()
 104         // if we got here we have to care for a simple pole of tgamma(-m):
 105         numeric m = -ex_to<numeric>(arg_pt);
 106         ex recur;
 107         for (numeric p = 0; p<=m; ++p)
 108                 recur += log(arg+p);
 109         return (lgamma(arg+m+_ex1)-recur).series(rel, order, options);
 110 }
 111
 112
 113 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 114                           evalf_func(lgamma_evalf).
 115                           derivative_func(lgamma_deriv).
 116                           series_func(lgamma_series).
 117                           latex_name("\\log \\Gamma"));
 118
 119
 120 //////////
 121 // true Gamma function
 122 //////////
 123
 124 static ex tgamma_evalf(const ex & x)
 125 {
 126         if (is_exactly_a<numeric>(x)) {
 127                 try {
 128                         return tgamma(ex_to<numeric>(x));
 129                 } catch (const dunno &e) { }
 130         }
 131
 132         return tgamma(x).hold();
 133 }
 134
 135
 136 /** Evaluation of tgamma(x), the true Gamma function.  Knows about integer
 137  *  arguments, half-integer arguments and that's it. Somebody ought to provide
 138  *  some good numerical evaluation some day...
 139  *
 140  *  @exception pole_error("tgamma_eval(): simple pole",0) */
 141 static ex tgamma_eval(const ex & x)
 142 {
 143         if (x.info(info_flags::numeric)) {
 144                 // trap integer arguments:
 145                 const numeric two_x = _num2*ex_to<numeric>(x);
 146                 if (two_x.is_even()) {
 147                         // tgamma(n) -> (n-1)! for postitive n
 148                         if (two_x.is_positive()) {
 149                                 return factorial(ex_to<numeric>(x).sub(_num1));
 150                         } else {
 151                                 throw (pole_error("tgamma_eval(): simple pole",1));
 152                         }
 153                 }
 154                 // trap half integer arguments:
 155                 if (two_x.is_integer()) {
 156                         // trap positive x==(n+1/2)
 157                         // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
 158                         if (two_x.is_positive()) {
 159                                 const numeric n = ex_to<numeric>(x).sub(_num1_2);
 160                                 return (doublefactorial(n.mul(_num2).sub(_num1)).div(pow(_num2,n))) * sqrt(Pi);
 161                         } else {
 162                                 // trap negative x==(-n+1/2)
 163                                 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
 164                                 const numeric n = abs(ex_to<numeric>(x).sub(_num1_2));
 165                                 return (pow(_num_2, n).div(doublefactorial(n.mul(_num2).sub(_num1))))*sqrt(Pi);
 166                         }
 167                 }
 168                 //  tgamma_evalf should be called here once it becomes available
 169         }
 170
 171         return tgamma(x).hold();
 172 }
 173
 174
 175 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 176 {
 177         GINAC_ASSERT(deriv_param==0);
 178
 179         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 180         return psi(x)*tgamma(x);
 181 }
 182
 183
 184 static ex tgamma_series(const ex & arg,
 185                         const relational & rel,
 186                         int order,
 187                         unsigned options)
 188 {
 189         // method:
 190         // Taylor series where there is no pole falls back to psi function
 191         // evaluation.
 192         // On a pole at -m use the recurrence relation
 193         //   tgamma(x) == tgamma(x+1) / x
 194         // from which follows
 195         //   series(tgamma(x),x==-m,order) ==
 196         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
 197         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 198         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 199                 throw do_taylor();  // caught by function::series()
 200         // if we got here we have to care for a simple pole at -m:
 201         const numeric m = -ex_to<numeric>(arg_pt);
 202         ex ser_denom = _ex1;
 203         for (numeric p; p<=m; ++p)
 204                 ser_denom *= arg+p;
 205         return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order+1, options);
 206 }
 207
 208
 209 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 210                           evalf_func(tgamma_evalf).
 211                           derivative_func(tgamma_deriv).
 212                           series_func(tgamma_series).
 213                           latex_name("\\Gamma"));
 214
 215
 216 //////////
 217 // beta-function
 218 //////////
 219
 220 static ex beta_evalf(const ex & x, const ex & y)
 221 {
 222         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
 223                 try {
 224                         return tgamma(ex_to<numeric>(x))*tgamma(ex_to<numeric>(y))/tgamma(ex_to<numeric>(x+y));
 225                 } catch (const dunno &e) { }
 226         }
 227
 228         return beta(x,y).hold();
 229 }
 230
 231
 232 static ex beta_eval(const ex & x, const ex & y)
 233 {
 234         if (x.is_equal(_ex1))
 235                 return 1/y;
 236         if (y.is_equal(_ex1))
 237                 return 1/x;
 238         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 239                 // treat all problematic x and y that may not be passed into tgamma,
 240                 // because they would throw there although beta(x,y) is well-defined
 241                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 242                 const numeric &nx = ex_to<numeric>(x);
 243                 const numeric &ny = ex_to<numeric>(y);
 244                 if (nx.is_real() && nx.is_integer() &&
 245                         ny.is_real() && ny.is_integer()) {
 246                         if (nx.is_negative()) {
 247                                 if (nx<=-ny)
 248                                         return pow(_num_1, ny)*beta(1-x-y, y);
 249                                 else
 250                                         throw (pole_error("beta_eval(): simple pole",1));
 251                         }
 252                         if (ny.is_negative()) {
 253                                 if (ny<=-nx)
 254                                         return pow(_num_1, nx)*beta(1-y-x, x);
 255                                 else
 256                                         throw (pole_error("beta_eval(): simple pole",1));
 257                         }
 258                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 259                 }
 260                 // no problem in numerator, but denominator has pole:
 261                 if ((nx+ny).is_real() &&
 262                     (nx+ny).is_integer() &&
 263                    !(nx+ny).is_positive())
 264                          return _ex0;
 265                 // beta_evalf should be called here once it becomes available
 266         }
 267
 268         return beta(x,y).hold();
 269 }
 270
 271
 272 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
 273 {
 274         GINAC_ASSERT(deriv_param<2);
 275         ex retval;
 276
 277         // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
 278         if (deriv_param==0)
 279                 retval = (psi(x)-psi(x+y))*beta(x,y);
 280         // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
 281         if (deriv_param==1)
 282                 retval = (psi(y)-psi(x+y))*beta(x,y);
 283         return retval;
 284 }
 285
 286
 287 static ex beta_series(const ex & arg1,
 288                       const ex & arg2,
 289                       const relational & rel,
 290                       int order,
 291                       unsigned options)
 292 {
 293         // method:
 294         // Taylor series where there is no pole of one of the tgamma functions
 295         // falls back to beta function evaluation.  Otherwise, fall back to
 296         // tgamma series directly.
 297         const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern);
 298         const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
 299         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 300         const symbol &s = ex_to<symbol>(rel.lhs());
 301         ex arg1_ser, arg2_ser, arg1arg2_ser;
 302         if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
 303             (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
 304                 throw do_taylor();  // caught by function::series()
 305         // trap the case where arg1 is on a pole:
 306         if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
 307                 arg1_ser = tgamma(arg1+s).series(rel, order, options);
 308         else
 309                 arg1_ser = tgamma(arg1).series(rel,order);
 310         // trap the case where arg2 is on a pole:
 311         if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
 312                 arg2_ser = tgamma(arg2+s).series(rel, order, options);
 313         else
 314                 arg2_ser = tgamma(arg2).series(rel,order);
 315         // trap the case where arg1+arg2 is on a pole:
 316         if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
 317                 arg1arg2_ser = tgamma(arg2+arg1+s).series(rel, order, options);
 318         else
 319                 arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
 320         // compose the result (expanding all the terms):
 321         return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 322 }
 323
 324
 325 REGISTER_FUNCTION(beta, eval_func(beta_eval).
 326                         evalf_func(beta_evalf).
 327                         derivative_func(beta_deriv).
 328                         series_func(beta_series).
 329                         latex_name("\\mbox{B}").
 330                                                 set_symmetry(sy_symm(0, 1)));
 331
 332
 333 //////////
 334 // Psi-function (aka digamma-function)
 335 //////////
 336
 337 static ex psi1_evalf(const ex & x)
 338 {
 339         if (is_exactly_a<numeric>(x)) {
 340                 try {
 341                         return psi(ex_to<numeric>(x));
 342                 } catch (const dunno &e) { }
 343         }
 344
 345         return psi(x).hold();
 346 }
 347
 348 /** Evaluation of digamma-function psi(x).
 349  *  Somebody ought to provide some good numerical evaluation some day... */
 350 static ex psi1_eval(const ex & x)
 351 {
 352         if (x.info(info_flags::numeric)) {
 353                 const numeric &nx = ex_to<numeric>(x);
 354                 if (nx.is_integer()) {
 355                         // integer case
 356                         if (nx.is_positive()) {
 357                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 358                                 numeric rat = 0;
 359                                 for (numeric i(nx+_num_1); i>0; --i)
 360                                         rat += i.inverse();
 361                                 return rat-Euler;
 362                         } else {
 363                                 // for non-positive integers there is a pole:
 364                                 throw (pole_error("psi_eval(): simple pole",1));
 365                         }
 366                 }
 367                 if ((_num2*nx).is_integer()) {
 368                         // half integer case
 369                         if (nx.is_positive()) {
 370                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 371                                 numeric rat = 0;
 372                                 for (numeric i = (nx+_num_1)*_num2; i>0; i-=_num2)
 373                                         rat += _num2*i.inverse();
 374                                 return rat-Euler-_ex2*log(_ex2);
 375                         } else {
 376                                 // use the recurrence relation
 377                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 378                                 // to relate psi(-m-1/2) to psi(1/2):
 379                                 //   psi(-m-1/2) == psi(1/2) + r
 380                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 381                                 numeric recur = 0;
 382                                 for (numeric p = nx; p<0; ++p)
 383                                         recur -= pow(p, _num_1);
 384                                 return recur+psi(_ex1_2);
 385                         }
 386                 }
 387                 //  psi1_evalf should be called here once it becomes available
 388         }
 389
 390         return psi(x).hold();
 391 }
 392
 393 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 394 {
 395         GINAC_ASSERT(deriv_param==0);
 396
 397         // d/dx psi(x) -> psi(1,x)
 398         return psi(_ex1, x);
 399 }
 400
 401 static ex psi1_series(const ex & arg,
 402                       const relational & rel,
 403                       int order,
 404                       unsigned options)
 405 {
 406         // method:
 407         // Taylor series where there is no pole falls back to polygamma function
 408         // evaluation.
 409         // On a pole at -m use the recurrence relation
 410         //   psi(x) == psi(x+1) - 1/z
 411         // from which follows
 412         //   series(psi(x),x==-m,order) ==
 413         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 414         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 415         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 416                 throw do_taylor();  // caught by function::series()
 417         // if we got here we have to care for a simple pole at -m:
 418         const numeric m = -ex_to<numeric>(arg_pt);
 419         ex recur;
 420         for (numeric p; p<=m; ++p)
 421                 recur += power(arg+p,_ex_1);
 422         return (psi(arg+m+_ex1)-recur).series(rel, order, options);
 423 }
 424
 425 unsigned psi1_SERIAL::serial =
 426         function::register_new(function_options("psi").
 427                                eval_func(psi1_eval).
 428                                evalf_func(psi1_evalf).
 429                                derivative_func(psi1_deriv).
 430                                series_func(psi1_series).
 431                                latex_name("\\psi").
 432                                overloaded(2));
 433
 434 //////////
 435 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 436 //////////
 437
 438 static ex psi2_evalf(const ex & n, const ex & x)
 439 {
 440         if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
 441                 try {
 442                         return psi(ex_to<numeric>(n),ex_to<numeric>(x));
 443                 } catch (const dunno &e) { }
 444         }
 445
 446         return psi(n,x).hold();
 447 }
 448
 449 /** Evaluation of polygamma-function psi(n,x).
 450  *  Somebody ought to provide some good numerical evaluation some day... */
 451 static ex psi2_eval(const ex & n, const ex & x)
 452 {
 453         // psi(0,x) -> psi(x)
 454         if (n.is_zero())
 455                 return psi(x);
 456         // psi(-1,x) -> log(tgamma(x))
 457         if (n.is_equal(_ex_1))
 458                 return log(tgamma(x));
 459         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 460                 x.info(info_flags::numeric)) {
 461                 const numeric &nn = ex_to<numeric>(n);
 462                 const numeric &nx = ex_to<numeric>(x);
 463                 if (nx.is_integer()) {
 464                         // integer case
 465                         if (nx.is_equal(_num1))
 466                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 467                                 return pow(_num_1,nn+_num1)*factorial(nn)*zeta(ex(nn+_num1));
 468                         if (nx.is_positive()) {
 469                                 // use the recurrence relation
 470                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 471                                 // to relate psi(n,m) to psi(n,1):
 472                                 //   psi(n,m) == psi(n,1) + r
 473                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 474                                 numeric recur = 0;
 475                                 for (numeric p = 1; p<nx; ++p)
 476                                         recur += pow(p, -nn+_num_1);
 477                                 recur *= factorial(nn)*pow(_num_1, nn);
 478                                 return recur+psi(n,_ex1);
 479                         } else {
 480                                 // for non-positive integers there is a pole:
 481                                 throw (pole_error("psi2_eval(): pole",1));
 482                         }
 483                 }
 484                 if ((_num2*nx).is_integer()) {
 485                         // half integer case
 486                         if (nx.is_equal(_num1_2))
 487                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 488                                 return pow(_num_1,nn+_num1)*factorial(nn)*(pow(_num2,nn+_num1) + _num_1)*zeta(ex(nn+_num1));
 489                         if (nx.is_positive()) {
 490                                 const numeric m = nx - _num1_2;
 491                                 // use the multiplication formula
 492                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 493                                 // to revert to positive integer case
 494                                 return psi(n,_num2*m)*pow(_num2,nn+_num1)-psi(n,m);
 495                         } else {
 496                                 // use the recurrence relation
 497                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 498                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 499                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 500                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 501                                 numeric recur = 0;
 502                                 for (numeric p = nx; p<0; ++p)
 503                                         recur += pow(p, -nn+_num_1);
 504                                 recur *= factorial(nn)*pow(_num_1, nn+_num_1);
 505                                 return recur+psi(n,_ex1_2);
 506                         }
 507                 }
 508                 //  psi2_evalf should be called here once it becomes available
 509         }
 510
 511         return psi(n, x).hold();
 512 }
 513
 514 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 515 {
 516         GINAC_ASSERT(deriv_param<2);
 517
 518         if (deriv_param==0) {
 519                 // d/dn psi(n,x)
 520                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 521         }
 522         // d/dx psi(n,x) -> psi(n+1,x)
 523         return psi(n+_ex1, x);
 524 }
 525
 526 static ex psi2_series(const ex & n,
 527                       const ex & arg,
 528                       const relational & rel,
 529                       int order,
 530                       unsigned options)
 531 {
 532         // method:
 533         // Taylor series where there is no pole falls back to polygamma function
 534         // evaluation.
 535         // On a pole at -m use the recurrence relation
 536         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 537         // from which follows
 538         //   series(psi(x),x==-m,order) ==
 539         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 540         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 541         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 542         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 543                 throw do_taylor();  // caught by function::series()
 544         // if we got here we have to care for a pole of order n+1 at -m:
 545         const numeric m = -ex_to<numeric>(arg_pt);
 546         ex recur;
 547         for (numeric p; p<=m; ++p)
 548                 recur += power(arg+p,-n+_ex_1);
 549         recur *= factorial(n)*power(_ex_1,n);
 550         return (psi(n, arg+m+_ex1)-recur).series(rel, order, options);
 551 }
 552
 553 unsigned psi2_SERIAL::serial =
 554         function::register_new(function_options("psi").
 555                                eval_func(psi2_eval).
 556                                evalf_func(psi2_evalf).
 557                                derivative_func(psi2_deriv).
 558                                series_func(psi2_series).
 559                                latex_name("\\psi").
 560                                overloaded(2));
 561
 562
 563 } // namespace GiNaC