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) 1999-2010 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include "inifcns.h"
  25 #include "constant.h"
  26 #include "pseries.h"
  27 #include "numeric.h"
  28 #include "power.h"
  29 #include "relational.h"
  30 #include "operators.h"
  31 #include "symbol.h"
  32 #include "symmetry.h"
  33 #include "utils.h"
  34
  35 #include <stdexcept>
  36 #include <vector>
  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  *  Handles integer arguments as a special case.
  58  *
  59  *  @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
  60 static ex lgamma_eval(const ex & x)
  61 {
  62         if (x.info(info_flags::numeric)) {
  63                 // trap integer arguments:
  64                 if (x.info(info_flags::integer)) {
  65                         // lgamma(n) -> log((n-1)!) for postitive n
  66                         if (x.info(info_flags::posint))
  67                                 return log(factorial(x + _ex_1));
  68                         else
  69                                 throw (pole_error("lgamma_eval(): logarithmic pole",0));
  70                 }
  71                 if (!ex_to<numeric>(x).is_rational())
  72                         return lgamma(ex_to<numeric>(x));
  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_p)*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_p));
 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_p);
 160                                 return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,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_p));
 165                                 return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi);
 166                         }
 167                 }
 168                 if (!ex_to<numeric>(x).is_rational())
 169                         return tgamma(ex_to<numeric>(x));
 170         }
 171
 172         return tgamma(x).hold();
 173 }
 174
 175
 176 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 177 {
 178         GINAC_ASSERT(deriv_param==0);
 179
 180         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 181         return psi(x)*tgamma(x);
 182 }
 183
 184
 185 static ex tgamma_series(const ex & arg,
 186                         const relational & rel,
 187                         int order,
 188                         unsigned options)
 189 {
 190         // method:
 191         // Taylor series where there is no pole falls back to psi function
 192         // evaluation.
 193         // On a pole at -m use the recurrence relation
 194         //   tgamma(x) == tgamma(x+1) / x
 195         // from which follows
 196         //   series(tgamma(x),x==-m,order) ==
 197         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order);
 198         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 199         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 200                 throw do_taylor();  // caught by function::series()
 201         // if we got here we have to care for a simple pole at -m:
 202         const numeric m = -ex_to<numeric>(arg_pt);
 203         ex ser_denom = _ex1;
 204         for (numeric p; p<=m; ++p)
 205                 ser_denom *= arg+p;
 206         return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order, options);
 207 }
 208
 209
 210 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 211                           evalf_func(tgamma_evalf).
 212                           derivative_func(tgamma_deriv).
 213                           series_func(tgamma_series).
 214                           latex_name("\\Gamma"));
 215
 216
 217 //////////
 218 // beta-function
 219 //////////
 220
 221 static ex beta_evalf(const ex & x, const ex & y)
 222 {
 223         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
 224                 try {
 225                         return exp(lgamma(ex_to<numeric>(x))+lgamma(ex_to<numeric>(y))-lgamma(ex_to<numeric>(x+y)));
 226                 } catch (const dunno &e) { }
 227         }
 228
 229         return beta(x,y).hold();
 230 }
 231
 232
 233 static ex beta_eval(const ex & x, const ex & y)
 234 {
 235         if (x.is_equal(_ex1))
 236                 return 1/y;
 237         if (y.is_equal(_ex1))
 238                 return 1/x;
 239         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 240                 // treat all problematic x and y that may not be passed into tgamma,
 241                 // because they would throw there although beta(x,y) is well-defined
 242                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 243                 const numeric &nx = ex_to<numeric>(x);
 244                 const numeric &ny = ex_to<numeric>(y);
 245                 if (nx.is_real() && nx.is_integer() &&
 246                         ny.is_real() && ny.is_integer()) {
 247                         if (nx.is_negative()) {
 248                                 if (nx<=-ny)
 249                                         return pow(*_num_1_p, ny)*beta(1-x-y, y);
 250                                 else
 251                                         throw (pole_error("beta_eval(): simple pole",1));
 252                         }
 253                         if (ny.is_negative()) {
 254                                 if (ny<=-nx)
 255                                         return pow(*_num_1_p, nx)*beta(1-y-x, x);
 256                                 else
 257                                         throw (pole_error("beta_eval(): simple pole",1));
 258                         }
 259                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 260                 }
 261                 // no problem in numerator, but denominator has pole:
 262                 if ((nx+ny).is_real() &&
 263                     (nx+ny).is_integer() &&
 264                    !(nx+ny).is_positive())
 265                          return _ex0;
 266                 if (!ex_to<numeric>(x).is_rational() || !ex_to<numeric>(x).is_rational())
 267                         return evalf(beta(x, y).hold());
 268         }
 269
 270         return beta(x,y).hold();
 271 }
 272
 273
 274 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
 275 {
 276         GINAC_ASSERT(deriv_param<2);
 277         ex retval;
 278
 279         // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
 280         if (deriv_param==0)
 281                 retval = (psi(x)-psi(x+y))*beta(x,y);
 282         // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
 283         if (deriv_param==1)
 284                 retval = (psi(y)-psi(x+y))*beta(x,y);
 285         return retval;
 286 }
 287
 288
 289 static ex beta_series(const ex & arg1,
 290                       const ex & arg2,
 291                       const relational & rel,
 292                       int order,
 293                       unsigned options)
 294 {
 295         // method:
 296         // Taylor series where there is no pole of one of the tgamma functions
 297         // falls back to beta function evaluation.  Otherwise, fall back to
 298         // tgamma series directly.
 299         const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern);
 300         const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
 301         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 302         const symbol &s = ex_to<symbol>(rel.lhs());
 303         ex arg1_ser, arg2_ser, arg1arg2_ser;
 304         if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
 305             (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
 306                 throw do_taylor();  // caught by function::series()
 307         // trap the case where arg1 is on a pole:
 308         if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
 309                 arg1_ser = tgamma(arg1+s);
 310         else
 311                 arg1_ser = tgamma(arg1);
 312         // trap the case where arg2 is on a pole:
 313         if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
 314                 arg2_ser = tgamma(arg2+s);
 315         else
 316                 arg2_ser = tgamma(arg2);
 317         // trap the case where arg1+arg2 is on a pole:
 318         if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
 319                 arg1arg2_ser = tgamma(arg2+arg1+s);
 320         else
 321                 arg1arg2_ser = tgamma(arg2+arg1);
 322         // compose the result (expanding all the terms):
 323         return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 324 }
 325
 326
 327 REGISTER_FUNCTION(beta, eval_func(beta_eval).
 328                         evalf_func(beta_evalf).
 329                         derivative_func(beta_deriv).
 330                         series_func(beta_series).
 331                         latex_name("\\mathrm{B}").
 332                                                 set_symmetry(sy_symm(0, 1)));
 333
 334
 335 //////////
 336 // Psi-function (aka digamma-function)
 337 //////////
 338
 339 static ex psi1_evalf(const ex & x)
 340 {
 341         if (is_exactly_a<numeric>(x)) {
 342                 try {
 343                         return psi(ex_to<numeric>(x));
 344                 } catch (const dunno &e) { }
 345         }
 346
 347         return psi(x).hold();
 348 }
 349
 350 /** Evaluation of digamma-function psi(x).
 351  *  Somebody ought to provide some good numerical evaluation some day... */
 352 static ex psi1_eval(const ex & x)
 353 {
 354         if (x.info(info_flags::numeric)) {
 355                 const numeric &nx = ex_to<numeric>(x);
 356                 if (nx.is_integer()) {
 357                         // integer case
 358                         if (nx.is_positive()) {
 359                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 360                                 numeric rat = 0;
 361                                 for (numeric i(nx+(*_num_1_p)); i>0; --i)
 362                                         rat += i.inverse();
 363                                 return rat-Euler;
 364                         } else {
 365                                 // for non-positive integers there is a pole:
 366                                 throw (pole_error("psi_eval(): simple pole",1));
 367                         }
 368                 }
 369                 if (((*_num2_p)*nx).is_integer()) {
 370                         // half integer case
 371                         if (nx.is_positive()) {
 372                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 373                                 numeric rat = 0;
 374                                 for (numeric i = (nx+(*_num_1_p))*(*_num2_p); i>0; i-=(*_num2_p))
 375                                         rat += (*_num2_p)*i.inverse();
 376                                 return rat-Euler-_ex2*log(_ex2);
 377                         } else {
 378                                 // use the recurrence relation
 379                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 380                                 // to relate psi(-m-1/2) to psi(1/2):
 381                                 //   psi(-m-1/2) == psi(1/2) + r
 382                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 383                                 numeric recur = 0;
 384                                 for (numeric p = nx; p<0; ++p)
 385                                         recur -= pow(p, *_num_1_p);
 386                                 return recur+psi(_ex1_2);
 387                         }
 388                 }
 389                 //  psi1_evalf should be called here once it becomes available
 390         }
 391
 392         return psi(x).hold();
 393 }
 394
 395 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 396 {
 397         GINAC_ASSERT(deriv_param==0);
 398
 399         // d/dx psi(x) -> psi(1,x)
 400         return psi(_ex1, x);
 401 }
 402
 403 static ex psi1_series(const ex & arg,
 404                       const relational & rel,
 405                       int order,
 406                       unsigned options)
 407 {
 408         // method:
 409         // Taylor series where there is no pole falls back to polygamma function
 410         // evaluation.
 411         // On a pole at -m use the recurrence relation
 412         //   psi(x) == psi(x+1) - 1/z
 413         // from which follows
 414         //   series(psi(x),x==-m,order) ==
 415         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 416         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 417         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 418                 throw do_taylor();  // caught by function::series()
 419         // if we got here we have to care for a simple pole at -m:
 420         const numeric m = -ex_to<numeric>(arg_pt);
 421         ex recur;
 422         for (numeric p; p<=m; ++p)
 423                 recur += power(arg+p,_ex_1);
 424         return (psi(arg+m+_ex1)-recur).series(rel, order, options);
 425 }
 426
 427 unsigned psi1_SERIAL::serial =
 428         function::register_new(function_options("psi", 1).
 429                                eval_func(psi1_eval).
 430                                evalf_func(psi1_evalf).
 431                                derivative_func(psi1_deriv).
 432                                series_func(psi1_series).
 433                                latex_name("\\psi").
 434                                overloaded(2));
 435
 436 //////////
 437 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 438 //////////
 439
 440 static ex psi2_evalf(const ex & n, const ex & x)
 441 {
 442         if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
 443                 try {
 444                         return psi(ex_to<numeric>(n),ex_to<numeric>(x));
 445                 } catch (const dunno &e) { }
 446         }
 447
 448         return psi(n,x).hold();
 449 }
 450
 451 /** Evaluation of polygamma-function psi(n,x).
 452  *  Somebody ought to provide some good numerical evaluation some day... */
 453 static ex psi2_eval(const ex & n, const ex & x)
 454 {
 455         // psi(0,x) -> psi(x)
 456         if (n.is_zero())
 457                 return psi(x);
 458         // psi(-1,x) -> log(tgamma(x))
 459         if (n.is_equal(_ex_1))
 460                 return log(tgamma(x));
 461         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 462                 x.info(info_flags::numeric)) {
 463                 const numeric &nn = ex_to<numeric>(n);
 464                 const numeric &nx = ex_to<numeric>(x);
 465                 if (nx.is_integer()) {
 466                         // integer case
 467                         if (nx.is_equal(*_num1_p))
 468                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 469                                 return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
 470                         if (nx.is_positive()) {
 471                                 // use the recurrence relation
 472                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 473                                 // to relate psi(n,m) to psi(n,1):
 474                                 //   psi(n,m) == psi(n,1) + r
 475                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 476                                 numeric recur = 0;
 477                                 for (numeric p = 1; p<nx; ++p)
 478                                         recur += pow(p, -nn+(*_num_1_p));
 479                                 recur *= factorial(nn)*pow((*_num_1_p), nn);
 480                                 return recur+psi(n,_ex1);
 481                         } else {
 482                                 // for non-positive integers there is a pole:
 483                                 throw (pole_error("psi2_eval(): pole",1));
 484                         }
 485                 }
 486                 if (((*_num2_p)*nx).is_integer()) {
 487                         // half integer case
 488                         if (nx.is_equal(*_num1_2_p))
 489                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 490                                 return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
 491                         if (nx.is_positive()) {
 492                                 const numeric m = nx - (*_num1_2_p);
 493                                 // use the multiplication formula
 494                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 495                                 // to revert to positive integer case
 496                                 return psi(n,(*_num2_p)*m)*pow((*_num2_p),nn+(*_num1_p))-psi(n,m);
 497                         } else {
 498                                 // use the recurrence relation
 499                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 500                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 501                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 502                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 503                                 numeric recur = 0;
 504                                 for (numeric p = nx; p<0; ++p)
 505                                         recur += pow(p, -nn+(*_num_1_p));
 506                                 recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
 507                                 return recur+psi(n,_ex1_2);
 508                         }
 509                 }
 510                 //  psi2_evalf should be called here once it becomes available
 511         }
 512
 513         return psi(n, x).hold();
 514 }
 515
 516 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 517 {
 518         GINAC_ASSERT(deriv_param<2);
 519
 520         if (deriv_param==0) {
 521                 // d/dn psi(n,x)
 522                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 523         }
 524         // d/dx psi(n,x) -> psi(n+1,x)
 525         return psi(n+_ex1, x);
 526 }
 527
 528 static ex psi2_series(const ex & n,
 529                       const ex & arg,
 530                       const relational & rel,
 531                       int order,
 532                       unsigned options)
 533 {
 534         // method:
 535         // Taylor series where there is no pole falls back to polygamma function
 536         // evaluation.
 537         // On a pole at -m use the recurrence relation
 538         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 539         // from which follows
 540         //   series(psi(x),x==-m,order) ==
 541         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 542         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 543         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 544         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 545                 throw do_taylor();  // caught by function::series()
 546         // if we got here we have to care for a pole of order n+1 at -m:
 547         const numeric m = -ex_to<numeric>(arg_pt);
 548         ex recur;
 549         for (numeric p; p<=m; ++p)
 550                 recur += power(arg+p,-n+_ex_1);
 551         recur *= factorial(n)*power(_ex_1,n);
 552         return (psi(n, arg+m+_ex1)-recur).series(rel, order, options);
 553 }
 554
 555 unsigned psi2_SERIAL::serial =
 556         function::register_new(function_options("psi", 2).
 557                                eval_func(psi2_eval).
 558                                evalf_func(psi2_evalf).
 559                                derivative_func(psi2_deriv).
 560                                series_func(psi2_series).
 561                                latex_name("\\psi").
 562                                overloaded(2));
 563
 564
 565 } // namespace GiNaC