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) 1999-2000 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 "ex.h"
  29 #include "constant.h"
  30 #include "pseries.h"
  31 #include "numeric.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "utils.h"
  36
  37 #ifndef NO_NAMESPACE_GINAC
  38 namespace GiNaC {
  39 #endif // ndef NO_NAMESPACE_GINAC
  40
  41 //////////
  42 // Logarithm of Gamma function
  43 //////////
  44
  45 static ex lgamma_evalf(const ex & x)
  46 {
  47         BEGIN_TYPECHECK
  48                 TYPECHECK(x,numeric)
  49         END_TYPECHECK(lgamma(x))
  50
  51         return lgamma(ex_to_numeric(x));
  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.exadd(_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; p<=m; ++p)
 107                 recur += log(arg+p);
 108         cout << recur << endl;
 109         return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
 110 }
 111
 112
 113 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 114                                                   evalf_func(lgamma_evalf).
 115                                                   derivative_func(lgamma_deriv).
 116                                                   series_func(lgamma_series));
 117
 118
 119 //////////
 120 // true Gamma function
 121 //////////
 122
 123 static ex tgamma_evalf(const ex & x)
 124 {
 125         BEGIN_TYPECHECK
 126                 TYPECHECK(x,numeric)
 127         END_TYPECHECK(tgamma(x))
 128
 129         return tgamma(ex_to_numeric(x));
 130 }
 131
 132
 133 /** Evaluation of tgamma(x), the true Gamma function.  Knows about integer
 134  *  arguments, half-integer arguments and that's it. Somebody ought to provide
 135  *  some good numerical evaluation some day...
 136  *
 137  *  @exception pole_error("tgamma_eval(): simple pole",0) */
 138 static ex tgamma_eval(const ex & x)
 139 {
 140         if (x.info(info_flags::numeric)) {
 141                 // trap integer arguments:
 142                 if (x.info(info_flags::integer)) {
 143                         // tgamma(n) -> (n-1)! for postitive n
 144                         if (x.info(info_flags::posint)) {
 145                                 return factorial(ex_to_numeric(x).sub(_num1()));
 146                         } else {
 147                                 throw (pole_error("tgamma_eval(): simple pole",1));
 148                         }
 149                 }
 150                 // trap half integer arguments:
 151                 if ((x*2).info(info_flags::integer)) {
 152                         // trap positive x==(n+1/2)
 153                         // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
 154                         if ((x*_ex2()).info(info_flags::posint)) {
 155                                 numeric n = ex_to_numeric(x).sub(_num1_2());
 156                                 numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
 157                                 coefficient = coefficient.div(pow(_num2(),n));
 158                                 return coefficient * pow(Pi,_ex1_2());
 159                         } else {
 160                                 // trap negative x==(-n+1/2)
 161                                 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
 162                                 numeric n = abs(ex_to_numeric(x).sub(_num1_2()));
 163                                 numeric coefficient = pow(_num_2(), n);
 164                                 coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
 165                                 return coefficient*power(Pi,_ex1_2());
 166                         }
 167                 }
 168                 //  tgamma_evalf should be called here once it becomes available
 169         }
 170
 171         return tgamma(x).hold();
 172 }
 173
 174
 175 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 176 {
 177         GINAC_ASSERT(deriv_param==0);
 178
 179         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 180         return psi(x)*tgamma(x);
 181 }
 182
 183
 184 static ex tgamma_series(const ex & arg,
 185                                                 const relational & rel,
 186                                                 int order,
 187                                                 unsigned options)
 188 {
 189         // method:
 190         // Taylor series where there is no pole falls back to psi function
 191         // evaluation.
 192         // On a pole at -m use the recurrence relation
 193         //   tgamma(x) == tgamma(x+1) / x
 194         // from which follows
 195         //   series(tgamma(x),x==-m,order) ==
 196         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1);
 197         const ex arg_pt = arg.subs(rel);
 198         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 199                 throw do_taylor();  // caught by function::series()
 200         // if we got here we have to care for a simple pole at -m:
 201         numeric m = -ex_to_numeric(arg_pt);
 202         ex ser_denom = _ex1();
 203         for (numeric p; p<=m; ++p)
 204                 ser_denom *= arg+p;
 205         return (tgamma(arg+m+_ex1())/ser_denom).series(rel, order+1, options);
 206 }
 207
 208
 209 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 210                                                   evalf_func(tgamma_evalf).
 211                                                   derivative_func(tgamma_deriv).
 212                                                   series_func(tgamma_series));
 213
 214
 215 //////////
 216 // beta-function
 217 //////////
 218
 219 static ex beta_evalf(const ex & x, const ex & y)
 220 {
 221         BEGIN_TYPECHECK
 222                 TYPECHECK(x,numeric)
 223                 TYPECHECK(y,numeric)
 224         END_TYPECHECK(beta(x,y))
 225
 226         return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y));
 227 }
 228
 229
 230 static ex beta_eval(const ex & x, const ex & y)
 231 {
 232         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 233                 // treat all problematic x and y that may not be passed into tgamma,
 234                 // because they would throw there although beta(x,y) is well-defined
 235                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 236                 numeric nx(ex_to_numeric(x));
 237                 numeric ny(ex_to_numeric(y));
 238                 if (nx.is_real() && nx.is_integer() &&
 239                         ny.is_real() && ny.is_integer()) {
 240                         if (nx.is_negative()) {
 241                                 if (nx<=-ny)
 242                                         return pow(_num_1(), ny)*beta(1-x-y, y);
 243                                 else
 244                                         throw (pole_error("beta_eval(): simple pole",1));
 245                         }
 246                         if (ny.is_negative()) {
 247                                 if (ny<=-nx)
 248                                         return pow(_num_1(), nx)*beta(1-y-x, x);
 249                                 else
 250                                         throw (pole_error("beta_eval(): simple pole",1));
 251                         }
 252                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 253                 }
 254                 // no problem in numerator, but denominator has pole:
 255                 if ((nx+ny).is_real() &&
 256                         (nx+ny).is_integer() &&
 257                         !(nx+ny).is_positive())
 258                          return _ex0();
 259                 // everything is ok:
 260                 return tgamma(x)*tgamma(y)/tgamma(x+y);
 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_ex_exactly_of_type(rel.lhs(),symbol));
 295         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 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
 325
 326 //////////
 327 // Psi-function (aka digamma-function)
 328 //////////
 329
 330 static ex psi1_evalf(const ex & x)
 331 {
 332         BEGIN_TYPECHECK
 333                 TYPECHECK(x,numeric)
 334         END_TYPECHECK(psi(x))
 335
 336         return psi(ex_to_numeric(x));
 337 }
 338
 339 /** Evaluation of digamma-function psi(x).
 340  *  Somebody ought to provide some good numerical evaluation some day... */
 341 static ex psi1_eval(const ex & x)
 342 {
 343         if (x.info(info_flags::numeric)) {
 344                 numeric nx = ex_to_numeric(x);
 345                 if (nx.is_integer()) {
 346                         // integer case
 347                         if (nx.is_positive()) {
 348                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 349                                 numeric rat(0);
 350                                 for (numeric i(nx+_num_1()); i.is_positive(); --i)
 351                                         rat += i.inverse();
 352                                 return rat-Euler;
 353                         } else {
 354                                 // for non-positive integers there is a pole:
 355                                 throw (pole_error("psi_eval(): simple pole",1));
 356                         }
 357                 }
 358                 if ((_num2()*nx).is_integer()) {
 359                         // half integer case
 360                         if (nx.is_positive()) {
 361                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 362                                 numeric rat(0);
 363                                 for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
 364                                                                           rat += _num2()*i.inverse();
 365                                                                           return rat-Euler-_ex2()*log(_ex2());
 366                         } else {
 367                                 // use the recurrence relation
 368                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 369                                 // to relate psi(-m-1/2) to psi(1/2):
 370                                 //   psi(-m-1/2) == psi(1/2) + r
 371                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 372                                 numeric recur(0);
 373                                 for (numeric p(nx); p<0; ++p)
 374                                         recur -= pow(p, _num_1());
 375                                 return recur+psi(_ex1_2());
 376                         }
 377                 }
 378                 //  psi1_evalf should be called here once it becomes available
 379         }
 380
 381         return psi(x).hold();
 382 }
 383
 384 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 385 {
 386         GINAC_ASSERT(deriv_param==0);
 387
 388         // d/dx psi(x) -> psi(1,x)
 389         return psi(_ex1(), x);
 390 }
 391
 392 static ex psi1_series(const ex & arg,
 393                                           const relational & rel,
 394                                           int order,
 395                                           unsigned options)
 396 {
 397         // method:
 398         // Taylor series where there is no pole falls back to polygamma function
 399         // evaluation.
 400         // On a pole at -m use the recurrence relation
 401         //   psi(x) == psi(x+1) - 1/z
 402         // from which follows
 403         //   series(psi(x),x==-m,order) ==
 404         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 405         const ex arg_pt = arg.subs(rel);
 406         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 407                 throw do_taylor();  // caught by function::series()
 408         // if we got here we have to care for a simple pole at -m:
 409         numeric m = -ex_to_numeric(arg_pt);
 410         ex recur;
 411         for (numeric p; p<=m; ++p)
 412                 recur += power(arg+p,_ex_1());
 413         return (psi(arg+m+_ex1())-recur).series(rel, order, options);
 414 }
 415
 416 const unsigned function_index_psi1 =
 417         function::register_new(function_options("psi").
 418                                                    eval_func(psi1_eval).
 419                                                    evalf_func(psi1_evalf).
 420                            derivative_func(psi1_deriv).
 421                            series_func(psi1_series).
 422                            overloaded(2));
 423
 424 //////////
 425 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 426 //////////
 427
 428 static ex psi2_evalf(const ex & n, const ex & x)
 429 {
 430         BEGIN_TYPECHECK
 431                 TYPECHECK(n,numeric)
 432                 TYPECHECK(x,numeric)
 433         END_TYPECHECK(psi(n,x))
 434
 435         return psi(ex_to_numeric(n), ex_to_numeric(x));
 436 }
 437
 438 /** Evaluation of polygamma-function psi(n,x).
 439  *  Somebody ought to provide some good numerical evaluation some day... */
 440 static ex psi2_eval(const ex & n, const ex & x)
 441 {
 442         // psi(0,x) -> psi(x)
 443         if (n.is_zero())
 444                 return psi(x);
 445         // psi(-1,x) -> log(tgamma(x))
 446         if (n.is_equal(_ex_1()))
 447                 return log(tgamma(x));
 448         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 449                 x.info(info_flags::numeric)) {
 450                 numeric nn = ex_to_numeric(n);
 451                 numeric nx = ex_to_numeric(x);
 452                 if (nx.is_integer()) {
 453                         // integer case
 454                         if (nx.is_equal(_num1()))
 455                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 456                                 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
 457                         if (nx.is_positive()) {
 458                                 // use the recurrence relation
 459                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 460                                 // to relate psi(n,m) to psi(n,1):
 461                                 //   psi(n,m) == psi(n,1) + r
 462                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 463                                 numeric recur(0);
 464                                 for (numeric p(1); p<nx; ++p)
 465                                         recur += pow(p, -nn+_num_1());
 466                                 recur *= factorial(nn)*pow(_num_1(), nn);
 467                                 return recur+psi(n,_ex1());
 468                         } else {
 469                                 // for non-positive integers there is a pole:
 470                                 throw (pole_error("psi2_eval(): pole",1));
 471                         }
 472                 }
 473                 if ((_num2()*nx).is_integer()) {
 474                         // half integer case
 475                         if (nx.is_equal(_num1_2()))
 476                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 477                                 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
 478                         if (nx.is_positive()) {
 479                                 numeric m = nx - _num1_2();
 480                                 // use the multiplication formula
 481                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 482                                 // to revert to positive integer case
 483                                 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
 484                         } else {
 485                                 // use the recurrence relation
 486                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 487                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 488                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 489                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 490                                 numeric recur(0);
 491                                 for (numeric p(nx); p<0; ++p)
 492                                         recur += pow(p, -nn+_num_1());
 493                                 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
 494                                 return recur+psi(n,_ex1_2());
 495                         }
 496                 }
 497                 //  psi2_evalf should be called here once it becomes available
 498         }
 499
 500         return psi(n, x).hold();
 501 }
 502
 503 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 504 {
 505         GINAC_ASSERT(deriv_param<2);
 506
 507         if (deriv_param==0) {
 508                 // d/dn psi(n,x)
 509                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 510         }
 511         // d/dx psi(n,x) -> psi(n+1,x)
 512         return psi(n+_ex1(), x);
 513 }
 514
 515 static ex psi2_series(const ex & n,
 516                                           const ex & arg,
 517                                           const relational & rel,
 518                                           int order,
 519                                           unsigned options)
 520 {
 521         // method:
 522         // Taylor series where there is no pole falls back to polygamma function
 523         // evaluation.
 524         // On a pole at -m use the recurrence relation
 525         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 526         // from which follows
 527         //   series(psi(x),x==-m,order) ==
 528         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 529         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 530         const ex arg_pt = arg.subs(rel);
 531         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 532                 throw do_taylor();  // caught by function::series()
 533         // if we got here we have to care for a pole of order n+1 at -m:
 534         numeric m = -ex_to_numeric(arg_pt);
 535         ex recur;
 536         for (numeric p; p<=m; ++p)
 537                 recur += power(arg+p,-n+_ex_1());
 538         recur *= factorial(n)*power(_ex_1(),n);
 539         return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
 540 }
 541
 542 const unsigned function_index_psi2 =
 543         function::register_new(function_options("psi").
 544                                                    eval_func(psi2_eval).
 545                                                    evalf_func(psi2_evalf).
 546                            derivative_func(psi2_deriv).
 547                            series_func(psi2_series).
 548                            overloaded(2));
 549
 550
 551 #ifndef NO_NAMESPACE_GINAC
 552 } // namespace GiNaC
 553 #endif // ndef NO_NAMESPACE_GINAC