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