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 "utils.h"
  35
  36 namespace GiNaC {
  37
  38 //////////
  39 // Logarithm of Gamma function
  40 //////////
  41
  42 static ex lgamma_evalf(const ex & x)
  43 {
  44         BEGIN_TYPECHECK
  45                 TYPECHECK(x,numeric)
  46         END_TYPECHECK(lgamma(x))
  47
  48         return lgamma(ex_to_numeric(x));
  49 }
  50
  51
  52 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
  53  *  Knows about integer arguments and that's it.  Somebody ought to provide
  54  *  some good numerical evaluation some day...
  55  *
  56  *  @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
  57 static ex lgamma_eval(const ex & x)
  58 {
  59         if (x.info(info_flags::numeric)) {
  60                 // trap integer arguments:
  61                 if (x.info(info_flags::integer)) {
  62                         // lgamma(n) -> log((n-1)!) for postitive n
  63                         if (x.info(info_flags::posint))
  64                                 return log(factorial(x.exadd(_ex_1())));
  65                         else
  66                                 throw (pole_error("lgamma_eval(): logarithmic pole",0));
  67                 }
  68                 //  lgamma_evalf should be called here once it becomes available
  69         }
  70
  71         return lgamma(x).hold();
  72 }
  73
  74
  75 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
  76 {
  77         GINAC_ASSERT(deriv_param==0);
  78
  79         // d/dx  lgamma(x) -> psi(x)
  80         return psi(x);
  81 }
  82
  83
  84 static ex lgamma_series(const ex & arg,
  85                         const relational & rel,
  86                         int order,
  87                         unsigned options)
  88 {
  89         // method:
  90         // Taylor series where there is no pole falls back to psi function
  91         // evaluation.
  92         // On a pole at -m we could use the recurrence relation
  93         //   lgamma(x) == lgamma(x+1)-log(x)
  94         // from which follows
  95         //   series(lgamma(x),x==-m,order) ==
  96         //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
  97         const ex arg_pt = arg.subs(rel);
  98         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
  99                 throw do_taylor();  // caught by function::series()
 100         // if we got here we have to care for a simple pole of tgamma(-m):
 101         numeric m = -ex_to_numeric(arg_pt);
 102         ex recur;
 103         for (numeric p; p<=m; ++p)
 104                 recur += log(arg+p);
 105         return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
 106 }
 107
 108
 109 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 110                           evalf_func(lgamma_evalf).
 111                           derivative_func(lgamma_deriv).
 112                           series_func(lgamma_series));
 113
 114
 115 //////////
 116 // true Gamma function
 117 //////////
 118
 119 static ex tgamma_evalf(const ex & x)
 120 {
 121         BEGIN_TYPECHECK
 122                 TYPECHECK(x,numeric)
 123         END_TYPECHECK(tgamma(x))
 124
 125         return tgamma(ex_to_numeric(x));
 126 }
 127
 128
 129 /** Evaluation of tgamma(x), the true Gamma function.  Knows about integer
 130  *  arguments, half-integer arguments and that's it. Somebody ought to provide
 131  *  some good numerical evaluation some day...
 132  *
 133  *  @exception pole_error("tgamma_eval(): simple pole",0) */
 134 static ex tgamma_eval(const ex & x)
 135 {
 136         if (x.info(info_flags::numeric)) {
 137                 // trap integer arguments:
 138                 if (x.info(info_flags::integer)) {
 139                         // tgamma(n) -> (n-1)! for postitive n
 140                         if (x.info(info_flags::posint)) {
 141                                 return factorial(ex_to_numeric(x).sub(_num1()));
 142                         } else {
 143                                 throw (pole_error("tgamma_eval(): simple pole",1));
 144                         }
 145                 }
 146                 // trap half integer arguments:
 147                 if ((x*2).info(info_flags::integer)) {
 148                         // trap positive x==(n+1/2)
 149                         // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
 150                         if ((x*_ex2()).info(info_flags::posint)) {
 151                                 numeric n = ex_to_numeric(x).sub(_num1_2());
 152                                 numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
 153                                 coefficient = coefficient.div(pow(_num2(),n));
 154                                 return coefficient * pow(Pi,_ex1_2());
 155                         } else {
 156                                 // trap negative x==(-n+1/2)
 157                                 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
 158                                 numeric n = abs(ex_to_numeric(x).sub(_num1_2()));
 159                                 numeric coefficient = pow(_num_2(), n);
 160                                 coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
 161                                 return coefficient*power(Pi,_ex1_2());
 162                         }
 163                 }
 164                 //  tgamma_evalf should be called here once it becomes available
 165         }
 166
 167         return tgamma(x).hold();
 168 }
 169
 170
 171 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 172 {
 173         GINAC_ASSERT(deriv_param==0);
 174
 175         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 176         return psi(x)*tgamma(x);
 177 }
 178
 179
 180 static ex tgamma_series(const ex & arg,
 181                         const relational & rel,
 182                         int order,
 183                         unsigned options)
 184 {
 185         // method:
 186         // Taylor series where there is no pole falls back to psi function
 187         // evaluation.
 188         // On a pole at -m use the recurrence relation
 189         //   tgamma(x) == tgamma(x+1) / x
 190         // from which follows
 191         //   series(tgamma(x),x==-m,order) ==
 192         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
 193         const ex arg_pt = arg.subs(rel);
 194         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 195                 throw do_taylor();  // caught by function::series()
 196         // if we got here we have to care for a simple pole at -m:
 197         numeric m = -ex_to_numeric(arg_pt);
 198         ex ser_denom = _ex1();
 199         for (numeric p; p<=m; ++p)
 200                 ser_denom *= arg+p;
 201         return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
 202 }
 203
 204
 205 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 206                           evalf_func(tgamma_evalf).
 207                           derivative_func(tgamma_deriv).
 208                           series_func(tgamma_series));
 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                 numeric nx(ex_to_numeric(x));
 233                 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
 321
 322 //////////
 323 // Psi-function (aka digamma-function)
 324 //////////
 325
 326 static ex psi1_evalf(const ex & x)
 327 {
 328         BEGIN_TYPECHECK
 329                 TYPECHECK(x,numeric)
 330         END_TYPECHECK(psi(x))
 331
 332         return psi(ex_to_numeric(x));
 333 }
 334
 335 /** Evaluation of digamma-function psi(x).
 336  *  Somebody ought to provide some good numerical evaluation some day... */
 337 static ex psi1_eval(const ex & x)
 338 {
 339         if (x.info(info_flags::numeric)) {
 340                 numeric nx = ex_to_numeric(x);
 341                 if (nx.is_integer()) {
 342                         // integer case
 343                         if (nx.is_positive()) {
 344                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 345                                 numeric rat(0);
 346                                 for (numeric i(nx+_num_1()); i.is_positive(); --i)
 347                                         rat += i.inverse();
 348                                 return rat-Euler;
 349                         } else {
 350                                 // for non-positive integers there is a pole:
 351                                 throw (pole_error("psi_eval(): simple pole",1));
 352                         }
 353                 }
 354                 if ((_num2()*nx).is_integer()) {
 355                         // half integer case
 356                         if (nx.is_positive()) {
 357                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 358                                 numeric rat(0);
 359                                 for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
 360                                                                           rat += _num2()*i.inverse();
 361                                                                           return rat-Euler-_ex2()*log(_ex2());
 362                         } else {
 363                                 // use the recurrence relation
 364                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 365                                 // to relate psi(-m-1/2) to psi(1/2):
 366                                 //   psi(-m-1/2) == psi(1/2) + r
 367                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 368                                 numeric recur(0);
 369                                 for (numeric p(nx); p<0; ++p)
 370                                         recur -= pow(p, _num_1());
 371                                 return recur+psi(_ex1_2());
 372                         }
 373                 }
 374                 //  psi1_evalf should be called here once it becomes available
 375         }
 376
 377         return psi(x).hold();
 378 }
 379
 380 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 381 {
 382         GINAC_ASSERT(deriv_param==0);
 383
 384         // d/dx psi(x) -> psi(1,x)
 385         return psi(_ex1(), x);
 386 }
 387
 388 static ex psi1_series(const ex & arg,
 389                       const relational & rel,
 390                       int order,
 391                       unsigned options)
 392 {
 393         // method:
 394         // Taylor series where there is no pole falls back to polygamma function
 395         // evaluation.
 396         // On a pole at -m use the recurrence relation
 397         //   psi(x) == psi(x+1) - 1/z
 398         // from which follows
 399         //   series(psi(x),x==-m,order) ==
 400         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 401         const ex arg_pt = arg.subs(rel);
 402         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 403                 throw do_taylor();  // caught by function::series()
 404         // if we got here we have to care for a simple pole at -m:
 405         numeric m = -ex_to_numeric(arg_pt);
 406         ex recur;
 407         for (numeric p; p<=m; ++p)
 408                 recur += power(arg+p,_ex_1());
 409         return (psi(arg+m+_ex1())-recur).series(rel, order, options);
 410 }
 411
 412 const unsigned function_index_psi1 =
 413         function::register_new(function_options("psi").
 414                                eval_func(psi1_eval).
 415                                evalf_func(psi1_evalf).
 416                                derivative_func(psi1_deriv).
 417                                series_func(psi1_series).
 418                                overloaded(2));
 419
 420 //////////
 421 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 422 //////////
 423
 424 static ex psi2_evalf(const ex & n, const ex & x)
 425 {
 426         BEGIN_TYPECHECK
 427                 TYPECHECK(n,numeric)
 428                 TYPECHECK(x,numeric)
 429         END_TYPECHECK(psi(n,x))
 430
 431         return psi(ex_to_numeric(n), ex_to_numeric(x));
 432 }
 433
 434 /** Evaluation of polygamma-function psi(n,x).
 435  *  Somebody ought to provide some good numerical evaluation some day... */
 436 static ex psi2_eval(const ex & n, const ex & x)
 437 {
 438         // psi(0,x) -> psi(x)
 439         if (n.is_zero())
 440                 return psi(x);
 441         // psi(-1,x) -> log(tgamma(x))
 442         if (n.is_equal(_ex_1()))
 443                 return log(tgamma(x));
 444         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 445                 x.info(info_flags::numeric)) {
 446                 numeric nn = ex_to_numeric(n);
 447                 numeric nx = ex_to_numeric(x);
 448                 if (nx.is_integer()) {
 449                         // integer case
 450                         if (nx.is_equal(_num1()))
 451                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 452                                 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
 453                         if (nx.is_positive()) {
 454                                 // use the recurrence relation
 455                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 456                                 // to relate psi(n,m) to psi(n,1):
 457                                 //   psi(n,m) == psi(n,1) + r
 458                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 459                                 numeric recur(0);
 460                                 for (numeric p(1); p<nx; ++p)
 461                                         recur += pow(p, -nn+_num_1());
 462                                 recur *= factorial(nn)*pow(_num_1(), nn);
 463                                 return recur+psi(n,_ex1());
 464                         } else {
 465                                 // for non-positive integers there is a pole:
 466                                 throw (pole_error("psi2_eval(): pole",1));
 467                         }
 468                 }
 469                 if ((_num2()*nx).is_integer()) {
 470                         // half integer case
 471                         if (nx.is_equal(_num1_2()))
 472                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 473                                 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
 474                         if (nx.is_positive()) {
 475                                 numeric m = nx - _num1_2();
 476                                 // use the multiplication formula
 477                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 478                                 // to revert to positive integer case
 479                                 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
 480                         } else {
 481                                 // use the recurrence relation
 482                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 483                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 484                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 485                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 486                                 numeric recur(0);
 487                                 for (numeric p(nx); p<0; ++p)
 488                                         recur += pow(p, -nn+_num_1());
 489                                 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
 490                                 return recur+psi(n,_ex1_2());
 491                         }
 492                 }
 493                 //  psi2_evalf should be called here once it becomes available
 494         }
 495
 496         return psi(n, x).hold();
 497 }
 498
 499 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 500 {
 501         GINAC_ASSERT(deriv_param<2);
 502
 503         if (deriv_param==0) {
 504                 // d/dn psi(n,x)
 505                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 506         }
 507         // d/dx psi(n,x) -> psi(n+1,x)
 508         return psi(n+_ex1(), x);
 509 }
 510
 511 static ex psi2_series(const ex & n,
 512                       const ex & arg,
 513                       const relational & rel,
 514                       int order,
 515                       unsigned options)
 516 {
 517         // method:
 518         // Taylor series where there is no pole falls back to polygamma function
 519         // evaluation.
 520         // On a pole at -m use the recurrence relation
 521         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 522         // from which follows
 523         //   series(psi(x),x==-m,order) ==
 524         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 525         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 526         const ex arg_pt = arg.subs(rel);
 527         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 528                 throw do_taylor();  // caught by function::series()
 529         // if we got here we have to care for a pole of order n+1 at -m:
 530         numeric m = -ex_to_numeric(arg_pt);
 531         ex recur;
 532         for (numeric p; p<=m; ++p)
 533                 recur += power(arg+p,-n+_ex_1());
 534         recur *= factorial(n)*power(_ex_1(),n);
 535         return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
 536 }
 537
 538 const unsigned function_index_psi2 =
 539         function::register_new(function_options("psi").
 540                                eval_func(psi2_eval).
 541                                evalf_func(psi2_evalf).
 542                                derivative_func(psi2_deriv).
 543                                series_func(psi2_series).
 544                                overloaded(2));
 545
 546
 547 } // namespace GiNaC