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         BEGIN_TYPECHECK
  46                 TYPECHECK(x,numeric)
  47         END_TYPECHECK(lgamma(x))
  48
  49         return lgamma(ex_to<numeric>(x));
  50 }
  51
  52
  53 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
  54  *  Knows about integer arguments and that's it.  Somebody ought to provide
  55  *  some good numerical evaluation some day...
  56  *
  57  *  @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
  58 static ex lgamma_eval(const ex & x)
  59 {
  60         if (x.info(info_flags::numeric)) {
  61                 // trap integer arguments:
  62                 if (x.info(info_flags::integer)) {
  63                         // lgamma(n) -> log((n-1)!) for postitive n
  64                         if (x.info(info_flags::posint))
  65                                 return log(factorial(x + _ex_1()));
  66                         else
  67                                 throw (pole_error("lgamma_eval(): logarithmic pole",0));
  68                 }
  69                 //  lgamma_evalf should be called here once it becomes available
  70         }
  71
  72         return lgamma(x).hold();
  73 }
  74
  75
  76 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
  77 {
  78         GINAC_ASSERT(deriv_param==0);
  79
  80         // d/dx  lgamma(x) -> psi(x)
  81         return psi(x);
  82 }
  83
  84
  85 static ex lgamma_series(const ex & arg,
  86                         const relational & rel,
  87                         int order,
  88                         unsigned options)
  89 {
  90         // method:
  91         // Taylor series where there is no pole falls back to psi function
  92         // evaluation.
  93         // On a pole at -m we could use the recurrence relation
  94         //   lgamma(x) == lgamma(x+1)-log(x)
  95         // from which follows
  96         //   series(lgamma(x),x==-m,order) ==
  97         //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
  98         const ex arg_pt = arg.subs(rel);
  99         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 100                 throw do_taylor();  // caught by function::series()
 101         // if we got here we have to care for a simple pole of tgamma(-m):
 102         numeric m = -ex_to<numeric>(arg_pt);
 103         ex recur;
 104         for (numeric p; p<=m; ++p)
 105                 recur += log(arg+p);
 106         return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
 107 }
 108
 109
 110 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 111                           evalf_func(lgamma_evalf).
 112                           derivative_func(lgamma_deriv).
 113                           series_func(lgamma_series).
 114                           latex_name("\\log \\Gamma"));
 115
 116
 117 //////////
 118 // true Gamma function
 119 //////////
 120
 121 static ex tgamma_evalf(const ex & x)
 122 {
 123         BEGIN_TYPECHECK
 124                 TYPECHECK(x,numeric)
 125         END_TYPECHECK(tgamma(x))
 126
 127         return tgamma(ex_to<numeric>(x));
 128 }
 129
 130
 131 /** Evaluation of tgamma(x), the true Gamma function.  Knows about integer
 132  *  arguments, half-integer arguments and that's it. Somebody ought to provide
 133  *  some good numerical evaluation some day...
 134  *
 135  *  @exception pole_error("tgamma_eval(): simple pole",0) */
 136 static ex tgamma_eval(const ex & x)
 137 {
 138         if (x.info(info_flags::numeric)) {
 139                 // trap integer arguments:
 140                 if (x.info(info_flags::integer)) {
 141                         // tgamma(n) -> (n-1)! for postitive n
 142                         if (x.info(info_flags::posint)) {
 143                                 return factorial(ex_to<numeric>(x).sub(_num1()));
 144                         } else {
 145                                 throw (pole_error("tgamma_eval(): simple pole",1));
 146                         }
 147                 }
 148                 // trap half integer arguments:
 149                 if ((x*2).info(info_flags::integer)) {
 150                         // trap positive x==(n+1/2)
 151                         // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
 152                         if ((x*_ex2()).info(info_flags::posint)) {
 153                                 numeric n = ex_to<numeric>(x).sub(_num1_2());
 154                                 numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
 155                                 coefficient = coefficient.div(pow(_num2(),n));
 156                                 return coefficient * pow(Pi,_ex1_2());
 157                         } else {
 158                                 // trap negative x==(-n+1/2)
 159                                 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
 160                                 numeric n = abs(ex_to<numeric>(x).sub(_num1_2()));
 161                                 numeric coefficient = pow(_num_2(), n);
 162                                 coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
 163                                 return coefficient*power(Pi,_ex1_2());
 164                         }
 165                 }
 166                 //  tgamma_evalf should be called here once it becomes available
 167         }
 168
 169         return tgamma(x).hold();
 170 }
 171
 172
 173 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 174 {
 175         GINAC_ASSERT(deriv_param==0);
 176
 177         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 178         return psi(x)*tgamma(x);
 179 }
 180
 181
 182 static ex tgamma_series(const ex & arg,
 183                         const relational & rel,
 184                         int order,
 185                         unsigned options)
 186 {
 187         // method:
 188         // Taylor series where there is no pole falls back to psi function
 189         // evaluation.
 190         // On a pole at -m use the recurrence relation
 191         //   tgamma(x) == tgamma(x+1) / x
 192         // from which follows
 193         //   series(tgamma(x),x==-m,order) ==
 194         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
 195         const ex arg_pt = arg.subs(rel);
 196         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 197                 throw do_taylor();  // caught by function::series()
 198         // if we got here we have to care for a simple pole at -m:
 199         numeric m = -ex_to<numeric>(arg_pt);
 200         ex ser_denom = _ex1();
 201         for (numeric p; p<=m; ++p)
 202                 ser_denom *= arg+p;
 203         return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
 204 }
 205
 206
 207 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 208                           evalf_func(tgamma_evalf).
 209                           derivative_func(tgamma_deriv).
 210                           series_func(tgamma_series).
 211                           latex_name("\\Gamma"));
 212
 213
 214 //////////
 215 // beta-function
 216 //////////
 217
 218 static ex beta_evalf(const ex & x, const ex & y)
 219 {
 220         BEGIN_TYPECHECK
 221                 TYPECHECK(x,numeric)
 222                 TYPECHECK(y,numeric)
 223         END_TYPECHECK(beta(x,y))
 224
 225         return tgamma(ex_to<numeric>(x))*tgamma(ex_to<numeric>(y))/tgamma(ex_to<numeric>(x+y));
 226 }
 227
 228
 229 static ex beta_eval(const ex & x, const ex & y)
 230 {
 231         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 232                 // treat all problematic x and y that may not be passed into tgamma,
 233                 // because they would throw there although beta(x,y) is well-defined
 234                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 235                 numeric nx(ex_to<numeric>(x));
 236                 numeric ny(ex_to<numeric>(y));
 237                 if (nx.is_real() && nx.is_integer() &&
 238                         ny.is_real() && ny.is_integer()) {
 239                         if (nx.is_negative()) {
 240                                 if (nx<=-ny)
 241                                         return pow(_num_1(), ny)*beta(1-x-y, y);
 242                                 else
 243                                         throw (pole_error("beta_eval(): simple pole",1));
 244                         }
 245                         if (ny.is_negative()) {
 246                                 if (ny<=-nx)
 247                                         return pow(_num_1(), nx)*beta(1-y-x, x);
 248                                 else
 249                                         throw (pole_error("beta_eval(): simple pole",1));
 250                         }
 251                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 252                 }
 253                 // no problem in numerator, but denominator has pole:
 254                 if ((nx+ny).is_real() &&
 255                         (nx+ny).is_integer() &&
 256                         !(nx+ny).is_positive())
 257                          return _ex0();
 258                 // everything is ok:
 259                 return tgamma(x)*tgamma(y)/tgamma(x+y);
 260         }
 261
 262         return beta(x,y).hold();
 263 }
 264
 265
 266 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
 267 {
 268         GINAC_ASSERT(deriv_param<2);
 269         ex retval;
 270
 271         // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
 272         if (deriv_param==0)
 273                 retval = (psi(x)-psi(x+y))*beta(x,y);
 274         // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
 275         if (deriv_param==1)
 276                 retval = (psi(y)-psi(x+y))*beta(x,y);
 277         return retval;
 278 }
 279
 280
 281 static ex beta_series(const ex & arg1,
 282                       const ex & arg2,
 283                       const relational & rel,
 284                       int order,
 285                       unsigned options)
 286 {
 287         // method:
 288         // Taylor series where there is no pole of one of the tgamma functions
 289         // falls back to beta function evaluation.  Otherwise, fall back to
 290         // tgamma series directly.
 291         const ex arg1_pt = arg1.subs(rel);
 292         const ex arg2_pt = arg2.subs(rel);
 293         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 294         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 295         ex arg1_ser, arg2_ser, arg1arg2_ser;
 296         if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
 297             (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
 298                 throw do_taylor();  // caught by function::series()
 299         // trap the case where arg1 is on a pole:
 300         if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
 301                 arg1_ser = tgamma(arg1+*s).series(rel, order, options);
 302         else
 303                 arg1_ser = tgamma(arg1).series(rel,order);
 304         // trap the case where arg2 is on a pole:
 305         if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
 306                 arg2_ser = tgamma(arg2+*s).series(rel, order, options);
 307         else
 308                 arg2_ser = tgamma(arg2).series(rel,order);
 309         // trap the case where arg1+arg2 is on a pole:
 310         if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
 311                 arg1arg2_ser = tgamma(arg2+arg1+*s).series(rel, order, options);
 312         else
 313                 arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
 314         // compose the result (expanding all the terms):
 315         return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 316 }
 317
 318
 319 REGISTER_FUNCTION(beta, eval_func(beta_eval).
 320                         evalf_func(beta_evalf).
 321                         derivative_func(beta_deriv).
 322                         series_func(beta_series).
 323                         latex_name("\\mbox{B}").
 324                                                 set_symmetry(sy_symm(0, 1)));
 325
 326
 327 //////////
 328 // Psi-function (aka digamma-function)
 329 //////////
 330
 331 static ex psi1_evalf(const ex & x)
 332 {
 333         BEGIN_TYPECHECK
 334                 TYPECHECK(x,numeric)
 335         END_TYPECHECK(psi(x))
 336
 337         return psi(ex_to<numeric>(x));
 338 }
 339
 340 /** Evaluation of digamma-function psi(x).
 341  *  Somebody ought to provide some good numerical evaluation some day... */
 342 static ex psi1_eval(const ex & x)
 343 {
 344         if (x.info(info_flags::numeric)) {
 345                 numeric nx = ex_to<numeric>(x);
 346                 if (nx.is_integer()) {
 347                         // integer case
 348                         if (nx.is_positive()) {
 349                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 350                                 numeric rat(0);
 351                                 for (numeric i(nx+_num_1()); i.is_positive(); --i)
 352                                         rat += i.inverse();
 353                                 return rat-Euler;
 354                         } else {
 355                                 // for non-positive integers there is a pole:
 356                                 throw (pole_error("psi_eval(): simple pole",1));
 357                         }
 358                 }
 359                 if ((_num2()*nx).is_integer()) {
 360                         // half integer case
 361                         if (nx.is_positive()) {
 362                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 363                                 numeric rat(0);
 364                                 for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
 365                                         rat += _num2()*i.inverse();
 366                                 return rat-Euler-_ex2()*log(_ex2());
 367                         } else {
 368                                 // use the recurrence relation
 369                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 370                                 // to relate psi(-m-1/2) to psi(1/2):
 371                                 //   psi(-m-1/2) == psi(1/2) + r
 372                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 373                                 numeric recur(0);
 374                                 for (numeric p(nx); p<0; ++p)
 375                                         recur -= pow(p, _num_1());
 376                                 return recur+psi(_ex1_2());
 377                         }
 378                 }
 379                 //  psi1_evalf should be called here once it becomes available
 380         }
 381
 382         return psi(x).hold();
 383 }
 384
 385 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 386 {
 387         GINAC_ASSERT(deriv_param==0);
 388
 389         // d/dx psi(x) -> psi(1,x)
 390         return psi(_ex1(), x);
 391 }
 392
 393 static ex psi1_series(const ex & arg,
 394                       const relational & rel,
 395                       int order,
 396                       unsigned options)
 397 {
 398         // method:
 399         // Taylor series where there is no pole falls back to polygamma function
 400         // evaluation.
 401         // On a pole at -m use the recurrence relation
 402         //   psi(x) == psi(x+1) - 1/z
 403         // from which follows
 404         //   series(psi(x),x==-m,order) ==
 405         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 406         const ex arg_pt = arg.subs(rel);
 407         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 408                 throw do_taylor();  // caught by function::series()
 409         // if we got here we have to care for a simple pole at -m:
 410         numeric m = -ex_to<numeric>(arg_pt);
 411         ex recur;
 412         for (numeric p; p<=m; ++p)
 413                 recur += power(arg+p,_ex_1());
 414         return (psi(arg+m+_ex1())-recur).series(rel, order, options);
 415 }
 416
 417 const unsigned function_index_psi1 =
 418         function::register_new(function_options("psi").
 419                                eval_func(psi1_eval).
 420                                evalf_func(psi1_evalf).
 421                                derivative_func(psi1_deriv).
 422                                series_func(psi1_series).
 423                                latex_name("\\psi").
 424                                overloaded(2));
 425
 426 //////////
 427 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 428 //////////
 429
 430 static ex psi2_evalf(const ex & n, const ex & x)
 431 {
 432         BEGIN_TYPECHECK
 433                 TYPECHECK(n,numeric)
 434                 TYPECHECK(x,numeric)
 435         END_TYPECHECK(psi(n,x))
 436
 437         return psi(ex_to<numeric>(n), ex_to<numeric>(x));
 438 }
 439
 440 /** Evaluation of polygamma-function psi(n,x).
 441  *  Somebody ought to provide some good numerical evaluation some day... */
 442 static ex psi2_eval(const ex & n, const ex & x)
 443 {
 444         // psi(0,x) -> psi(x)
 445         if (n.is_zero())
 446                 return psi(x);
 447         // psi(-1,x) -> log(tgamma(x))
 448         if (n.is_equal(_ex_1()))
 449                 return log(tgamma(x));
 450         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 451                 x.info(info_flags::numeric)) {
 452                 numeric nn = ex_to<numeric>(n);
 453                 numeric nx = ex_to<numeric>(x);
 454                 if (nx.is_integer()) {
 455                         // integer case
 456                         if (nx.is_equal(_num1()))
 457                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 458                                 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
 459                         if (nx.is_positive()) {
 460                                 // use the recurrence relation
 461                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 462                                 // to relate psi(n,m) to psi(n,1):
 463                                 //   psi(n,m) == psi(n,1) + r
 464                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 465                                 numeric recur(0);
 466                                 for (numeric p(1); p<nx; ++p)
 467                                         recur += pow(p, -nn+_num_1());
 468                                 recur *= factorial(nn)*pow(_num_1(), nn);
 469                                 return recur+psi(n,_ex1());
 470                         } else {
 471                                 // for non-positive integers there is a pole:
 472                                 throw (pole_error("psi2_eval(): pole",1));
 473                         }
 474                 }
 475                 if ((_num2()*nx).is_integer()) {
 476                         // half integer case
 477                         if (nx.is_equal(_num1_2()))
 478                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 479                                 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
 480                         if (nx.is_positive()) {
 481                                 numeric m = nx - _num1_2();
 482                                 // use the multiplication formula
 483                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 484                                 // to revert to positive integer case
 485                                 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
 486                         } else {
 487                                 // use the recurrence relation
 488                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 489                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 490                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 491                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 492                                 numeric recur(0);
 493                                 for (numeric p(nx); p<0; ++p)
 494                                         recur += pow(p, -nn+_num_1());
 495                                 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
 496                                 return recur+psi(n,_ex1_2());
 497                         }
 498                 }
 499                 //  psi2_evalf should be called here once it becomes available
 500         }
 501
 502         return psi(n, x).hold();
 503 }
 504
 505 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 506 {
 507         GINAC_ASSERT(deriv_param<2);
 508
 509         if (deriv_param==0) {
 510                 // d/dn psi(n,x)
 511                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 512         }
 513         // d/dx psi(n,x) -> psi(n+1,x)
 514         return psi(n+_ex1(), x);
 515 }
 516
 517 static ex psi2_series(const ex & n,
 518                       const ex & arg,
 519                       const relational & rel,
 520                       int order,
 521                       unsigned options)
 522 {
 523         // method:
 524         // Taylor series where there is no pole falls back to polygamma function
 525         // evaluation.
 526         // On a pole at -m use the recurrence relation
 527         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 528         // from which follows
 529         //   series(psi(x),x==-m,order) ==
 530         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 531         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 532         const ex arg_pt = arg.subs(rel);
 533         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 534                 throw do_taylor();  // caught by function::series()
 535         // if we got here we have to care for a pole of order n+1 at -m:
 536         numeric m = -ex_to<numeric>(arg_pt);
 537         ex recur;
 538         for (numeric p; p<=m; ++p)
 539                 recur += power(arg+p,-n+_ex_1());
 540         recur *= factorial(n)*power(_ex_1(),n);
 541         return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
 542 }
 543
 544 const unsigned function_index_psi2 =
 545         function::register_new(function_options("psi").
 546                                eval_func(psi2_eval).
 547                                evalf_func(psi2_evalf).
 548                                derivative_func(psi2_deriv).
 549                                series_func(psi2_series).
 550                                latex_name("\\psi").
 551                                overloaded(2));
 552
 553
 554 } // namespace GiNaC