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) 1999-2001 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 "symbol.h"
  34 #include "symmetry.h"
  35 #include "utils.h"
  36
  37 namespace GiNaC {
  38
  39 //////////
  40 // Logarithm of Gamma function
  41 //////////
  42
  43 static ex lgamma_evalf(const ex & x)
  44 {
  45         if (is_exactly_a<numeric>(x)) {
  46                 try {
  47                         return lgamma(ex_to<numeric>(x));
  48                 } catch (const dunno &e) { }
  49         }
  50
  51         return lgamma(x).hold();
  52 }
  53
  54
  55 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
  56  *  Knows about integer arguments and that's it.  Somebody ought to provide
  57  *  some good numerical evaluation some day...
  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                 //  lgamma_evalf should be called here once it becomes available
  72         }
  73
  74         return lgamma(x).hold();
  75 }
  76
  77
  78 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
  79 {
  80         GINAC_ASSERT(deriv_param==0);
  81
  82         // d/dx  lgamma(x) -> psi(x)
  83         return psi(x);
  84 }
  85
  86
  87 static ex lgamma_series(const ex & arg,
  88                         const relational & rel,
  89                         int order,
  90                         unsigned options)
  91 {
  92         // method:
  93         // Taylor series where there is no pole falls back to psi function
  94         // evaluation.
  95         // On a pole at -m we could use the recurrence relation
  96         //   lgamma(x) == lgamma(x+1)-log(x)
  97         // from which follows
  98         //   series(lgamma(x),x==-m,order) ==
  99         //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
 100         const ex arg_pt = arg.subs(rel);
 101         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 102                 throw do_taylor();  // caught by function::series()
 103         // if we got here we have to care for a simple pole of tgamma(-m):
 104         numeric m = -ex_to<numeric>(arg_pt);
 105         ex recur;
 106         for (numeric p = 0; p<=m; ++p)
 107                 recur += log(arg+p);
 108         return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
 109 }
 110
 111
 112 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 113                           evalf_func(lgamma_evalf).
 114                           derivative_func(lgamma_deriv).
 115                           series_func(lgamma_series).
 116                           latex_name("\\log \\Gamma"));
 117
 118
 119 //////////
 120 // true Gamma function
 121 //////////
 122
 123 static ex tgamma_evalf(const ex & x)
 124 {
 125         if (is_exactly_a<numeric>(x)) {
 126                 try {
 127                         return tgamma(ex_to<numeric>(x));
 128                 } catch (const dunno &e) { }
 129         }
 130
 131         return tgamma(x).hold();
 132 }
 133
 134
 135 /** Evaluation of tgamma(x), the true Gamma function.  Knows about integer
 136  *  arguments, half-integer arguments and that's it. Somebody ought to provide
 137  *  some good numerical evaluation some day...
 138  *
 139  *  @exception pole_error("tgamma_eval(): simple pole",0) */
 140 static ex tgamma_eval(const ex & x)
 141 {
 142         if (x.info(info_flags::numeric)) {
 143                 // trap integer arguments:
 144                 const numeric two_x = _num2()*ex_to<numeric>(x);
 145                 if (two_x.is_even()) {
 146                         // tgamma(n) -> (n-1)! for postitive n
 147                         if (two_x.is_positive()) {
 148                                 return factorial(ex_to<numeric>(x).sub(_num1()));
 149                         } else {
 150                                 throw (pole_error("tgamma_eval(): simple pole",1));
 151                         }
 152                 }
 153                 // trap half integer arguments:
 154                 if (two_x.is_integer()) {
 155                         // trap positive x==(n+1/2)
 156                         // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
 157                         if (two_x.is_positive()) {
 158                                 const numeric n = ex_to<numeric>(x).sub(_num1_2());
 159                                 return (doublefactorial(n.mul(_num2()).sub(_num1())).div(pow(_num2(),n))) * pow(Pi,_ex1_2());
 160                         } else {
 161                                 // trap negative x==(-n+1/2)
 162                                 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
 163                                 const numeric n = abs(ex_to<numeric>(x).sub(_num1_2()));
 164                                 return (pow(_num_2(), n).div(doublefactorial(n.mul(_num2()).sub(_num1()))))*power(Pi,_ex1_2());
 165                         }
 166                 }
 167                 //  tgamma_evalf should be called here once it becomes available
 168         }
 169
 170         return tgamma(x).hold();
 171 }
 172
 173
 174 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 175 {
 176         GINAC_ASSERT(deriv_param==0);
 177
 178         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 179         return psi(x)*tgamma(x);
 180 }
 181
 182
 183 static ex tgamma_series(const ex & arg,
 184                         const relational & rel,
 185                         int order,
 186                         unsigned options)
 187 {
 188         // method:
 189         // Taylor series where there is no pole falls back to psi function
 190         // evaluation.
 191         // On a pole at -m use the recurrence relation
 192         //   tgamma(x) == tgamma(x+1) / x
 193         // from which follows
 194         //   series(tgamma(x),x==-m,order) ==
 195         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
 196         const ex arg_pt = arg.subs(rel);
 197         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 198                 throw do_taylor();  // caught by function::series()
 199         // if we got here we have to care for a simple pole at -m:
 200         const numeric m = -ex_to<numeric>(arg_pt);
 201         ex ser_denom = _ex1();
 202         for (numeric p; p<=m; ++p)
 203                 ser_denom *= arg+p;
 204         return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
 205 }
 206
 207
 208 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 209                           evalf_func(tgamma_evalf).
 210                           derivative_func(tgamma_deriv).
 211                           series_func(tgamma_series).
 212                           latex_name("\\Gamma"));
 213
 214
 215 //////////
 216 // beta-function
 217 //////////
 218
 219 static ex beta_evalf(const ex & x, const ex & y)
 220 {
 221         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
 222                 try {
 223                         return tgamma(ex_to<numeric>(x))*tgamma(ex_to<numeric>(y))/tgamma(ex_to<numeric>(x+y));
 224                 } catch (const dunno &e) { }
 225         }
 226
 227         return beta(x,y).hold();
 228 }
 229
 230
 231 static ex beta_eval(const ex & x, const ex & y)
 232 {
 233         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 234                 // treat all problematic x and y that may not be passed into tgamma,
 235                 // because they would throw there although beta(x,y) is well-defined
 236                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 237                 const numeric nx = ex_to<numeric>(x);
 238                 const numeric ny = ex_to<numeric>(y);
 239                 if (nx.is_real() && nx.is_integer() &&
 240                         ny.is_real() && ny.is_integer()) {
 241                         if (nx.is_negative()) {
 242                                 if (nx<=-ny)
 243                                         return pow(_num_1(), ny)*beta(1-x-y, y);
 244                                 else
 245                                         throw (pole_error("beta_eval(): simple pole",1));
 246                         }
 247                         if (ny.is_negative()) {
 248                                 if (ny<=-nx)
 249                                         return pow(_num_1(), nx)*beta(1-y-x, x);
 250                                 else
 251                                         throw (pole_error("beta_eval(): simple pole",1));
 252                         }
 253                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 254                 }
 255                 // no problem in numerator, but denominator has pole:
 256                 if ((nx+ny).is_real() &&
 257                     (nx+ny).is_integer() &&
 258                    !(nx+ny).is_positive())
 259                          return _ex0();
 260                 // beta_evalf should be called here once it becomes available
 261         }
 262
 263         return beta(x,y).hold();
 264 }
 265
 266
 267 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
 268 {
 269         GINAC_ASSERT(deriv_param<2);
 270         ex retval;
 271
 272         // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
 273         if (deriv_param==0)
 274                 retval = (psi(x)-psi(x+y))*beta(x,y);
 275         // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
 276         if (deriv_param==1)
 277                 retval = (psi(y)-psi(x+y))*beta(x,y);
 278         return retval;
 279 }
 280
 281
 282 static ex beta_series(const ex & arg1,
 283                       const ex & arg2,
 284                       const relational & rel,
 285                       int order,
 286                       unsigned options)
 287 {
 288         // method:
 289         // Taylor series where there is no pole of one of the tgamma functions
 290         // falls back to beta function evaluation.  Otherwise, fall back to
 291         // tgamma series directly.
 292         const ex arg1_pt = arg1.subs(rel);
 293         const ex arg2_pt = arg2.subs(rel);
 294         GINAC_ASSERT(is_exactly_a<symbol>(rel.lhs()));
 295         const symbol &s = ex_to<symbol>(rel.lhs());
 296         ex arg1_ser, arg2_ser, arg1arg2_ser;
 297         if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
 298             (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
 299                 throw do_taylor();  // caught by function::series()
 300         // trap the case where arg1 is on a pole:
 301         if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
 302                 arg1_ser = tgamma(arg1+s).series(rel, order, options);
 303         else
 304                 arg1_ser = tgamma(arg1).series(rel,order);
 305         // trap the case where arg2 is on a pole:
 306         if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
 307                 arg2_ser = tgamma(arg2+s).series(rel, order, options);
 308         else
 309                 arg2_ser = tgamma(arg2).series(rel,order);
 310         // trap the case where arg1+arg2 is on a pole:
 311         if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
 312                 arg1arg2_ser = tgamma(arg2+arg1+s).series(rel, order, options);
 313         else
 314                 arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
 315         // compose the result (expanding all the terms):
 316         return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 317 }
 318
 319
 320 REGISTER_FUNCTION(beta, eval_func(beta_eval).
 321                         evalf_func(beta_evalf).
 322                         derivative_func(beta_deriv).
 323                         series_func(beta_series).
 324                         latex_name("\\mbox{B}").
 325                                                 set_symmetry(sy_symm(0, 1)));
 326
 327
 328 //////////
 329 // Psi-function (aka digamma-function)
 330 //////////
 331
 332 static ex psi1_evalf(const ex & x)
 333 {
 334         if (is_exactly_a<numeric>(x)) {
 335                 try {
 336                         return psi(ex_to<numeric>(x));
 337                 } catch (const dunno &e) { }
 338         }
 339
 340         return psi(x).hold();
 341 }
 342
 343 /** Evaluation of digamma-function psi(x).
 344  *  Somebody ought to provide some good numerical evaluation some day... */
 345 static ex psi1_eval(const ex & x)
 346 {
 347         if (x.info(info_flags::numeric)) {
 348                 const numeric nx = ex_to<numeric>(x);
 349                 if (nx.is_integer()) {
 350                         // integer case
 351                         if (nx.is_positive()) {
 352                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 353                                 numeric rat = 0;
 354                                 for (numeric i(nx+_num_1()); i>0; --i)
 355                                         rat += i.inverse();
 356                                 return rat-Euler;
 357                         } else {
 358                                 // for non-positive integers there is a pole:
 359                                 throw (pole_error("psi_eval(): simple pole",1));
 360                         }
 361                 }
 362                 if ((_num2()*nx).is_integer()) {
 363                         // half integer case
 364                         if (nx.is_positive()) {
 365                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 366                                 numeric rat = 0;
 367                                 for (numeric i = (nx+_num_1())*_num2(); i>0; i-=_num2())
 368                                         rat += _num2()*i.inverse();
 369                                 return rat-Euler-_ex2()*log(_ex2());
 370                         } else {
 371                                 // use the recurrence relation
 372                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 373                                 // to relate psi(-m-1/2) to psi(1/2):
 374                                 //   psi(-m-1/2) == psi(1/2) + r
 375                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 376                                 numeric recur = 0;
 377                                 for (numeric p = nx; p<0; ++p)
 378                                         recur -= pow(p, _num_1());
 379                                 return recur+psi(_ex1_2());
 380                         }
 381                 }
 382                 //  psi1_evalf should be called here once it becomes available
 383         }
 384
 385         return psi(x).hold();
 386 }
 387
 388 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 389 {
 390         GINAC_ASSERT(deriv_param==0);
 391
 392         // d/dx psi(x) -> psi(1,x)
 393         return psi(_ex1(), x);
 394 }
 395
 396 static ex psi1_series(const ex & arg,
 397                       const relational & rel,
 398                       int order,
 399                       unsigned options)
 400 {
 401         // method:
 402         // Taylor series where there is no pole falls back to polygamma function
 403         // evaluation.
 404         // On a pole at -m use the recurrence relation
 405         //   psi(x) == psi(x+1) - 1/z
 406         // from which follows
 407         //   series(psi(x),x==-m,order) ==
 408         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 409         const ex arg_pt = arg.subs(rel);
 410         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 411                 throw do_taylor();  // caught by function::series()
 412         // if we got here we have to care for a simple pole at -m:
 413         const numeric m = -ex_to<numeric>(arg_pt);
 414         ex recur;
 415         for (numeric p; p<=m; ++p)
 416                 recur += power(arg+p,_ex_1());
 417         return (psi(arg+m+_ex1())-recur).series(rel, order, options);
 418 }
 419
 420 const unsigned function_index_psi1 =
 421         function::register_new(function_options("psi").
 422                                eval_func(psi1_eval).
 423                                evalf_func(psi1_evalf).
 424                                derivative_func(psi1_deriv).
 425                                series_func(psi1_series).
 426                                latex_name("\\psi").
 427                                overloaded(2));
 428
 429 //////////
 430 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 431 //////////
 432
 433 static ex psi2_evalf(const ex & n, const ex & x)
 434 {
 435         if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
 436                 try {
 437                         return psi(ex_to<numeric>(n),ex_to<numeric>(x));
 438                 } catch (const dunno &e) { }
 439         }
 440
 441         return psi(n,x).hold();
 442 }
 443
 444 /** Evaluation of polygamma-function psi(n,x).
 445  *  Somebody ought to provide some good numerical evaluation some day... */
 446 static ex psi2_eval(const ex & n, const ex & x)
 447 {
 448         // psi(0,x) -> psi(x)
 449         if (n.is_zero())
 450                 return psi(x);
 451         // psi(-1,x) -> log(tgamma(x))
 452         if (n.is_equal(_ex_1()))
 453                 return log(tgamma(x));
 454         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 455                 x.info(info_flags::numeric)) {
 456                 const numeric nn = ex_to<numeric>(n);
 457                 const numeric nx = ex_to<numeric>(x);
 458                 if (nx.is_integer()) {
 459                         // integer case
 460                         if (nx.is_equal(_num1()))
 461                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 462                                 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
 463                         if (nx.is_positive()) {
 464                                 // use the recurrence relation
 465                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 466                                 // to relate psi(n,m) to psi(n,1):
 467                                 //   psi(n,m) == psi(n,1) + r
 468                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 469                                 numeric recur = 0;
 470                                 for (numeric p = 1; p<nx; ++p)
 471                                         recur += pow(p, -nn+_num_1());
 472                                 recur *= factorial(nn)*pow(_num_1(), nn);
 473                                 return recur+psi(n,_ex1());
 474                         } else {
 475                                 // for non-positive integers there is a pole:
 476                                 throw (pole_error("psi2_eval(): pole",1));
 477                         }
 478                 }
 479                 if ((_num2()*nx).is_integer()) {
 480                         // half integer case
 481                         if (nx.is_equal(_num1_2()))
 482                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 483                                 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
 484                         if (nx.is_positive()) {
 485                                 const numeric m = nx - _num1_2();
 486                                 // use the multiplication formula
 487                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 488                                 // to revert to positive integer case
 489                                 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
 490                         } else {
 491                                 // use the recurrence relation
 492                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 493                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 494                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 495                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 496                                 numeric recur = 0;
 497                                 for (numeric p = nx; p<0; ++p)
 498                                         recur += pow(p, -nn+_num_1());
 499                                 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
 500                                 return recur+psi(n,_ex1_2());
 501                         }
 502                 }
 503                 //  psi2_evalf should be called here once it becomes available
 504         }
 505
 506         return psi(n, x).hold();
 507 }
 508
 509 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 510 {
 511         GINAC_ASSERT(deriv_param<2);
 512
 513         if (deriv_param==0) {
 514                 // d/dn psi(n,x)
 515                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 516         }
 517         // d/dx psi(n,x) -> psi(n+1,x)
 518         return psi(n+_ex1(), x);
 519 }
 520
 521 static ex psi2_series(const ex & n,
 522                       const ex & arg,
 523                       const relational & rel,
 524                       int order,
 525                       unsigned options)
 526 {
 527         // method:
 528         // Taylor series where there is no pole falls back to polygamma function
 529         // evaluation.
 530         // On a pole at -m use the recurrence relation
 531         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 532         // from which follows
 533         //   series(psi(x),x==-m,order) ==
 534         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 535         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 536         const ex arg_pt = arg.subs(rel);
 537         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 538                 throw do_taylor();  // caught by function::series()
 539         // if we got here we have to care for a pole of order n+1 at -m:
 540         const numeric m = -ex_to<numeric>(arg_pt);
 541         ex recur;
 542         for (numeric p; p<=m; ++p)
 543                 recur += power(arg+p,-n+_ex_1());
 544         recur *= factorial(n)*power(_ex_1(),n);
 545         return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
 546 }
 547
 548 const unsigned function_index_psi2 =
 549         function::register_new(function_options("psi").
 550                                eval_func(psi2_eval).
 551                                evalf_func(psi2_evalf).
 552                                derivative_func(psi2_deriv).
 553                                series_func(psi2_series).
 554                                latex_name("\\psi").
 555                                overloaded(2));
 556
 557
 558 } // namespace GiNaC