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 #ifndef NO_NAMESPACE_GINAC
  37 namespace GiNaC {
  38 #endif // ndef NO_NAMESPACE_GINAC
  39
  40 //////////
  41 // Logarithm of Gamma function
  42 //////////
  43
  44 static ex lgamma_evalf(const ex & x)
  45 {
  46         BEGIN_TYPECHECK
  47                 TYPECHECK(x,numeric)
  48         END_TYPECHECK(lgamma(x))
  49
  50         return lgamma(ex_to_numeric(x));
  51 }
  52
  53
  54 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
  55  *  Knows about integer arguments and that's it.  Somebody ought to provide
  56  *  some good numerical evaluation some day...
  57  *
  58  *  @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
  59 static ex lgamma_eval(const ex & x)
  60 {
  61         if (x.info(info_flags::numeric)) {
  62                 // trap integer arguments:
  63                 if (x.info(info_flags::integer)) {
  64                         // lgamma(n) -> log((n-1)!) for postitive n
  65                         if (x.info(info_flags::posint))
  66                                 return log(factorial(x.exadd(_ex_1())));
  67                         else
  68                                 throw (pole_error("lgamma_eval(): logarithmic pole",0));
  69                 }
  70                 //  lgamma_evalf should be called here once it becomes available
  71         }
  72
  73         return lgamma(x).hold();
  74 }
  75
  76
  77 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
  78 {
  79         GINAC_ASSERT(deriv_param==0);
  80
  81         // d/dx  lgamma(x) -> psi(x)
  82         return psi(x);
  83 }
  84
  85
  86 static ex lgamma_series(const ex & arg,
  87                         const relational & rel,
  88                         int order,
  89                         unsigned options)
  90 {
  91         // method:
  92         // Taylor series where there is no pole falls back to psi function
  93         // evaluation.
  94         // On a pole at -m we could use the recurrence relation
  95         //   lgamma(x) == lgamma(x+1)-log(x)
  96         // from which follows
  97         //   series(lgamma(x),x==-m,order) ==
  98         //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
  99         const ex arg_pt = arg.subs(rel);
 100         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 101                 throw do_taylor();  // caught by function::series()
 102         // if we got here we have to care for a simple pole of tgamma(-m):
 103         numeric m = -ex_to_numeric(arg_pt);
 104         ex recur;
 105         for (numeric p; p<=m; ++p)
 106                 recur += log(arg+p);
 107         return (lgamma(arg+m+_ex1())-recur).series(rel, order, options);
 108 }
 109
 110
 111 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 112                           evalf_func(lgamma_evalf).
 113                           derivative_func(lgamma_deriv).
 114                           series_func(lgamma_series));
 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
 212
 213 //////////
 214 // beta-function
 215 //////////
 216
 217 static ex beta_evalf(const ex & x, const ex & y)
 218 {
 219         BEGIN_TYPECHECK
 220                 TYPECHECK(x,numeric)
 221                 TYPECHECK(y,numeric)
 222         END_TYPECHECK(beta(x,y))
 223
 224         return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y));
 225 }
 226
 227
 228 static ex beta_eval(const ex & x, const ex & y)
 229 {
 230         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 231                 // treat all problematic x and y that may not be passed into tgamma,
 232                 // because they would throw there although beta(x,y) is well-defined
 233                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 234                 numeric nx(ex_to_numeric(x));
 235                 numeric ny(ex_to_numeric(y));
 236                 if (nx.is_real() && nx.is_integer() &&
 237                         ny.is_real() && ny.is_integer()) {
 238                         if (nx.is_negative()) {
 239                                 if (nx<=-ny)
 240                                         return pow(_num_1(), ny)*beta(1-x-y, y);
 241                                 else
 242                                         throw (pole_error("beta_eval(): simple pole",1));
 243                         }
 244                         if (ny.is_negative()) {
 245                                 if (ny<=-nx)
 246                                         return pow(_num_1(), nx)*beta(1-y-x, x);
 247                                 else
 248                                         throw (pole_error("beta_eval(): simple pole",1));
 249                         }
 250                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 251                 }
 252                 // no problem in numerator, but denominator has pole:
 253                 if ((nx+ny).is_real() &&
 254                         (nx+ny).is_integer() &&
 255                         !(nx+ny).is_positive())
 256                          return _ex0();
 257                 // everything is ok:
 258                 return tgamma(x)*tgamma(y)/tgamma(x+y);
 259         }
 260
 261         return beta(x,y).hold();
 262 }
 263
 264
 265 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
 266 {
 267         GINAC_ASSERT(deriv_param<2);
 268         ex retval;
 269
 270         // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
 271         if (deriv_param==0)
 272                 retval = (psi(x)-psi(x+y))*beta(x,y);
 273         // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
 274         if (deriv_param==1)
 275                 retval = (psi(y)-psi(x+y))*beta(x,y);
 276         return retval;
 277 }
 278
 279
 280 static ex beta_series(const ex & arg1,
 281                       const ex & arg2,
 282                       const relational & rel,
 283                       int order,
 284                       unsigned options)
 285 {
 286         // method:
 287         // Taylor series where there is no pole of one of the tgamma functions
 288         // falls back to beta function evaluation.  Otherwise, fall back to
 289         // tgamma series directly.
 290         const ex arg1_pt = arg1.subs(rel);
 291         const ex arg2_pt = arg2.subs(rel);
 292         GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol));
 293         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
 294         ex arg1_ser, arg2_ser, arg1arg2_ser;
 295         if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
 296             (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
 297                 throw do_taylor();  // caught by function::series()
 298         // trap the case where arg1 is on a pole:
 299         if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
 300                 arg1_ser = tgamma(arg1+*s).series(rel, order, options);
 301         else
 302                 arg1_ser = tgamma(arg1).series(rel,order);
 303         // trap the case where arg2 is on a pole:
 304         if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
 305                 arg2_ser = tgamma(arg2+*s).series(rel, order, options);
 306         else
 307                 arg2_ser = tgamma(arg2).series(rel,order);
 308         // trap the case where arg1+arg2 is on a pole:
 309         if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
 310                 arg1arg2_ser = tgamma(arg2+arg1+*s).series(rel, order, options);
 311         else
 312                 arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
 313         // compose the result (expanding all the terms):
 314         return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 315 }
 316
 317
 318 REGISTER_FUNCTION(beta, eval_func(beta_eval).
 319                         evalf_func(beta_evalf).
 320                         derivative_func(beta_deriv).
 321                         series_func(beta_series));
 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                 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.is_positive(); --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.is_positive(); 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         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                                overloaded(2));
 421
 422 //////////
 423 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 424 //////////
 425
 426 static ex psi2_evalf(const ex & n, const ex & x)
 427 {
 428         BEGIN_TYPECHECK
 429                 TYPECHECK(n,numeric)
 430                 TYPECHECK(x,numeric)
 431         END_TYPECHECK(psi(n,x))
 432
 433         return psi(ex_to_numeric(n), ex_to_numeric(x));
 434 }
 435
 436 /** Evaluation of polygamma-function psi(n,x).
 437  *  Somebody ought to provide some good numerical evaluation some day... */
 438 static ex psi2_eval(const ex & n, const ex & x)
 439 {
 440         // psi(0,x) -> psi(x)
 441         if (n.is_zero())
 442                 return psi(x);
 443         // psi(-1,x) -> log(tgamma(x))
 444         if (n.is_equal(_ex_1()))
 445                 return log(tgamma(x));
 446         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 447                 x.info(info_flags::numeric)) {
 448                 numeric nn = ex_to_numeric(n);
 449                 numeric nx = ex_to_numeric(x);
 450                 if (nx.is_integer()) {
 451                         // integer case
 452                         if (nx.is_equal(_num1()))
 453                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 454                                 return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
 455                         if (nx.is_positive()) {
 456                                 // use the recurrence relation
 457                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 458                                 // to relate psi(n,m) to psi(n,1):
 459                                 //   psi(n,m) == psi(n,1) + r
 460                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 461                                 numeric recur(0);
 462                                 for (numeric p(1); p<nx; ++p)
 463                                         recur += pow(p, -nn+_num_1());
 464                                 recur *= factorial(nn)*pow(_num_1(), nn);
 465                                 return recur+psi(n,_ex1());
 466                         } else {
 467                                 // for non-positive integers there is a pole:
 468                                 throw (pole_error("psi2_eval(): pole",1));
 469                         }
 470                 }
 471                 if ((_num2()*nx).is_integer()) {
 472                         // half integer case
 473                         if (nx.is_equal(_num1_2()))
 474                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 475                                 return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
 476                         if (nx.is_positive()) {
 477                                 numeric m = nx - _num1_2();
 478                                 // use the multiplication formula
 479                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 480                                 // to revert to positive integer case
 481                                 return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
 482                         } else {
 483                                 // use the recurrence relation
 484                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 485                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 486                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 487                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 488                                 numeric recur(0);
 489                                 for (numeric p(nx); p<0; ++p)
 490                                         recur += pow(p, -nn+_num_1());
 491                                 recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
 492                                 return recur+psi(n,_ex1_2());
 493                         }
 494                 }
 495                 //  psi2_evalf should be called here once it becomes available
 496         }
 497
 498         return psi(n, x).hold();
 499 }
 500
 501 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 502 {
 503         GINAC_ASSERT(deriv_param<2);
 504
 505         if (deriv_param==0) {
 506                 // d/dn psi(n,x)
 507                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 508         }
 509         // d/dx psi(n,x) -> psi(n+1,x)
 510         return psi(n+_ex1(), x);
 511 }
 512
 513 static ex psi2_series(const ex & n,
 514                       const ex & arg,
 515                       const relational & rel,
 516                       int order,
 517                       unsigned options)
 518 {
 519         // method:
 520         // Taylor series where there is no pole falls back to polygamma function
 521         // evaluation.
 522         // On a pole at -m use the recurrence relation
 523         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 524         // from which follows
 525         //   series(psi(x),x==-m,order) ==
 526         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 527         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 528         const ex arg_pt = arg.subs(rel);
 529         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 530                 throw do_taylor();  // caught by function::series()
 531         // if we got here we have to care for a pole of order n+1 at -m:
 532         numeric m = -ex_to_numeric(arg_pt);
 533         ex recur;
 534         for (numeric p; p<=m; ++p)
 535                 recur += power(arg+p,-n+_ex_1());
 536         recur *= factorial(n)*power(_ex_1(),n);
 537         return (psi(n, arg+m+_ex1())-recur).series(rel, order, options);
 538 }
 539
 540 const unsigned function_index_psi2 =
 541         function::register_new(function_options("psi").
 542                                eval_func(psi2_eval).
 543                                evalf_func(psi2_evalf).
 544                                derivative_func(psi2_deriv).
 545                                series_func(psi2_series).
 546                                overloaded(2));
 547
 548
 549 #ifndef NO_NAMESPACE_GINAC
 550 } // namespace GiNaC
 551 #endif // ndef NO_NAMESPACE_GINAC