/** @file inifcns_gamma.cpp
*
* Implementation of Gamma-function, Beta-function, Polygamma-functions, and
* some related stuff. */
/*
* GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <vector>
#include <stdexcept>
#include "inifcns.h"
#include "constant.h"
#include "pseries.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
#include "operators.h"
#include "symbol.h"
#include "symmetry.h"
#include "utils.h"
namespace GiNaC {
//////////
// Logarithm of Gamma function
//////////
static ex lgamma_evalf(const ex & x)
{
if (is_exactly_a<numeric>(x)) {
try {
return lgamma(ex_to<numeric>(x));
} catch (const dunno &e) { }
}
return lgamma(x).hold();
}
/** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
* Knows about integer arguments and that's it. Somebody ought to provide
* some good numerical evaluation some day...
*
* @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
static ex lgamma_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// trap integer arguments:
if (x.info(info_flags::integer)) {
// lgamma(n) -> log((n-1)!) for postitive n
if (x.info(info_flags::posint))
return log(factorial(x + _ex_1));
else
throw (pole_error("lgamma_eval(): logarithmic pole",0));
}
// lgamma_evalf should be called here once it becomes available
}
return lgamma(x).hold();
}
static ex lgamma_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
// d/dx lgamma(x) -> psi(x)
return psi(x);
}
static ex lgamma_series(const ex & arg,
const relational & rel,
int order,
unsigned options)
{
// method:
// Taylor series where there is no pole falls back to psi function
// evaluation.
// On a pole at -m we could use the recurrence relation
// lgamma(x) == lgamma(x+1)-log(x)
// from which follows
// series(lgamma(x),x==-m,order) ==
// series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole of tgamma(-m):
numeric m = -ex_to<numeric>(arg_pt);
ex recur;
for (numeric p = 0; p<=m; ++p)
recur += log(arg+p);
return (lgamma(arg+m+_ex1)-recur).series(rel, order, options);
}
REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
evalf_func(lgamma_evalf).
derivative_func(lgamma_deriv).
series_func(lgamma_series).
latex_name("\\log \\Gamma"));
//////////
// true Gamma function
//////////
static ex tgamma_evalf(const ex & x)
{
if (is_exactly_a<numeric>(x)) {
try {
return tgamma(ex_to<numeric>(x));
} catch (const dunno &e) { }
}
return tgamma(x).hold();
}
/** Evaluation of tgamma(x), the true Gamma function. Knows about integer
* arguments, half-integer arguments and that's it. Somebody ought to provide
* some good numerical evaluation some day...
*
* @exception pole_error("tgamma_eval(): simple pole",0) */
static ex tgamma_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// trap integer arguments:
const numeric two_x = (*_num2_p)*ex_to<numeric>(x);
if (two_x.is_even()) {
// tgamma(n) -> (n-1)! for postitive n
if (two_x.is_positive()) {
return factorial(ex_to<numeric>(x).sub(*_num1_p));
} else {
throw (pole_error("tgamma_eval(): simple pole",1));
}
}
// trap half integer arguments:
if (two_x.is_integer()) {
// trap positive x==(n+1/2)
// tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
if (two_x.is_positive()) {
const numeric n = ex_to<numeric>(x).sub(*_num1_2_p);
return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,n))) * sqrt(Pi);
} else {
// trap negative x==(-n+1/2)
// tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
const numeric n = abs(ex_to<numeric>(x).sub(*_num1_2_p));
return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi);
}
}
// tgamma_evalf should be called here once it becomes available
}
return tgamma(x).hold();
}
static ex tgamma_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
// d/dx tgamma(x) -> psi(x)*tgamma(x)
return psi(x)*tgamma(x);
}
static ex tgamma_series(const ex & arg,
const relational & rel,
int order,
unsigned options)
{
// method:
// Taylor series where there is no pole falls back to psi function
// evaluation.
// On a pole at -m use the recurrence relation
// tgamma(x) == tgamma(x+1) / x
// from which follows
// series(tgamma(x),x==-m,order) ==
// series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order);
const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole at -m:
const numeric m = -ex_to<numeric>(arg_pt);
ex ser_denom = _ex1;
for (numeric p; p<=m; ++p)
ser_denom *= arg+p;
return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order, options);
}
REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
evalf_func(tgamma_evalf).
derivative_func(tgamma_deriv).
series_func(tgamma_series).
latex_name("\\Gamma"));
//////////
// beta-function
//////////
static ex beta_evalf(const ex & x, const ex & y)
{
if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
try {
return tgamma(ex_to<numeric>(x))*tgamma(ex_to<numeric>(y))/tgamma(ex_to<numeric>(x+y));
} catch (const dunno &e) { }
}
return beta(x,y).hold();
}
static ex beta_eval(const ex & x, const ex & y)
{
if (x.is_equal(_ex1))
return 1/y;
if (y.is_equal(_ex1))
return 1/x;
if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
// treat all problematic x and y that may not be passed into tgamma,
// because they would throw there although beta(x,y) is well-defined
// using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
const numeric &nx = ex_to<numeric>(x);
const numeric &ny = ex_to<numeric>(y);
if (nx.is_real() && nx.is_integer() &&
ny.is_real() && ny.is_integer()) {
if (nx.is_negative()) {
if (nx<=-ny)
return pow(*_num_1_p, ny)*beta(1-x-y, y);
else
throw (pole_error("beta_eval(): simple pole",1));
}
if (ny.is_negative()) {
if (ny<=-nx)
return pow(*_num_1_p, nx)*beta(1-y-x, x);
else
throw (pole_error("beta_eval(): simple pole",1));
}
return tgamma(x)*tgamma(y)/tgamma(x+y);
}
// no problem in numerator, but denominator has pole:
if ((nx+ny).is_real() &&
(nx+ny).is_integer() &&
!(nx+ny).is_positive())
return _ex0;
// beta_evalf should be called here once it becomes available
}
return beta(x,y).hold();
}
static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param<2);
ex retval;
// d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
if (deriv_param==0)
retval = (psi(x)-psi(x+y))*beta(x,y);
// d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
if (deriv_param==1)
retval = (psi(y)-psi(x+y))*beta(x,y);
return retval;
}
static ex beta_series(const ex & arg1,
const ex & arg2,
const relational & rel,
int order,
unsigned options)
{
// method:
// Taylor series where there is no pole of one of the tgamma functions
// falls back to beta function evaluation. Otherwise, fall back to
// tgamma series directly.
const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern);
const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
GINAC_ASSERT(is_a<symbol>(rel.lhs()));
const symbol &s = ex_to<symbol>(rel.lhs());
ex arg1_ser, arg2_ser, arg1arg2_ser;
if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
(!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
throw do_taylor(); // caught by function::series()
// trap the case where arg1 is on a pole:
if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
arg1_ser = tgamma(arg1+s);
else
arg1_ser = tgamma(arg1);
// trap the case where arg2 is on a pole:
if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
arg2_ser = tgamma(arg2+s);
else
arg2_ser = tgamma(arg2);
// trap the case where arg1+arg2 is on a pole:
if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
arg1arg2_ser = tgamma(arg2+arg1+s);
else
arg1arg2_ser = tgamma(arg2+arg1);
// compose the result (expanding all the terms):
return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
REGISTER_FUNCTION(beta, eval_func(beta_eval).
evalf_func(beta_evalf).
derivative_func(beta_deriv).
series_func(beta_series).
latex_name("\\mbox{B}").
set_symmetry(sy_symm(0, 1)));
//////////
// Psi-function (aka digamma-function)
//////////
static ex psi1_evalf(const ex & x)
{
if (is_exactly_a<numeric>(x)) {
try {
return psi(ex_to<numeric>(x));
} catch (const dunno &e) { }
}
return psi(x).hold();
}
/** Evaluation of digamma-function psi(x).
* Somebody ought to provide some good numerical evaluation some day... */
static ex psi1_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
const numeric &nx = ex_to<numeric>(x);
if (nx.is_integer()) {
// integer case
if (nx.is_positive()) {
// psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
numeric rat = 0;
for (numeric i(nx+(*_num_1_p)); i>0; --i)
rat += i.inverse();
return rat-Euler;
} else {
// for non-positive integers there is a pole:
throw (pole_error("psi_eval(): simple pole",1));
}
}
if (((*_num2_p)*nx).is_integer()) {
// half integer case
if (nx.is_positive()) {
// psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
numeric rat = 0;
for (numeric i = (nx+(*_num_1_p))*(*_num2_p); i>0; i-=(*_num2_p))
rat += (*_num2_p)*i.inverse();
return rat-Euler-_ex2*log(_ex2);
} else {
// use the recurrence relation
// psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
// to relate psi(-m-1/2) to psi(1/2):
// psi(-m-1/2) == psi(1/2) + r
// where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
numeric recur = 0;
for (numeric p = nx; p<0; ++p)
recur -= pow(p, *_num_1_p);
return recur+psi(_ex1_2);
}
}
// psi1_evalf should be called here once it becomes available
}
return psi(x).hold();
}
static ex psi1_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
// d/dx psi(x) -> psi(1,x)
return psi(_ex1, x);
}
static ex psi1_series(const ex & arg,
const relational & rel,
int order,
unsigned options)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// evaluation.
// On a pole at -m use the recurrence relation
// psi(x) == psi(x+1) - 1/z
// from which follows
// series(psi(x),x==-m,order) ==
// series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole at -m:
const numeric m = -ex_to<numeric>(arg_pt);
ex recur;
for (numeric p; p<=m; ++p)
recur += power(arg+p,_ex_1);
return (psi(arg+m+_ex1)-recur).series(rel, order, options);
}
unsigned psi1_SERIAL::serial =
function::register_new(function_options("psi", 1).
eval_func(psi1_eval).
evalf_func(psi1_evalf).
derivative_func(psi1_deriv).
series_func(psi1_series).
latex_name("\\psi").
overloaded(2));
//////////
// Psi-functions (aka polygamma-functions) psi(0,x)==psi(x)
//////////
static ex psi2_evalf(const ex & n, const ex & x)
{
if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
try {
return psi(ex_to<numeric>(n),ex_to<numeric>(x));
} catch (const dunno &e) { }
}
return psi(n,x).hold();
}
/** Evaluation of polygamma-function psi(n,x).
* Somebody ought to provide some good numerical evaluation some day... */
static ex psi2_eval(const ex & n, const ex & x)
{
// psi(0,x) -> psi(x)
if (n.is_zero())
return psi(x);
// psi(-1,x) -> log(tgamma(x))
if (n.is_equal(_ex_1))
return log(tgamma(x));
if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
x.info(info_flags::numeric)) {
const numeric &nn = ex_to<numeric>(n);
const numeric &nx = ex_to<numeric>(x);
if (nx.is_integer()) {
// integer case
if (nx.is_equal(*_num1_p))
// use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
if (nx.is_positive()) {
// use the recurrence relation
// psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
// to relate psi(n,m) to psi(n,1):
// psi(n,m) == psi(n,1) + r
// where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
numeric recur = 0;
for (numeric p = 1; p<nx; ++p)
recur += pow(p, -nn+(*_num_1_p));
recur *= factorial(nn)*pow((*_num_1_p), nn);
return recur+psi(n,_ex1);
} else {
// for non-positive integers there is a pole:
throw (pole_error("psi2_eval(): pole",1));
}
}
if (((*_num2_p)*nx).is_integer()) {
// half integer case
if (nx.is_equal(*_num1_2_p))
// use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
if (nx.is_positive()) {
const numeric m = nx - (*_num1_2_p);
// use the multiplication formula
// psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
// to revert to positive integer case
return psi(n,(*_num2_p)*m)*pow((*_num2_p),nn+(*_num1_p))-psi(n,m);
} else {
// use the recurrence relation
// psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
// to relate psi(n,-m-1/2) to psi(n,1/2):
// psi(n,-m-1/2) == psi(n,1/2) + r
// where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
numeric recur = 0;
for (numeric p = nx; p<0; ++p)
recur += pow(p, -nn+(*_num_1_p));
recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
return recur+psi(n,_ex1_2);
}
}
// psi2_evalf should be called here once it becomes available
}
return psi(n, x).hold();
}
static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param<2);
if (deriv_param==0) {
// d/dn psi(n,x)
throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
}
// d/dx psi(n,x) -> psi(n+1,x)
return psi(n+_ex1, x);
}
static ex psi2_series(const ex & n,
const ex & arg,
const relational & rel,
int order,
unsigned options)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// evaluation.
// On a pole at -m use the recurrence relation
// psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
// from which follows
// series(psi(x),x==-m,order) ==
// series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
// ... + (x+m)^(-n-1))),x==-m,order);
const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a pole of order n+1 at -m:
const numeric m = -ex_to<numeric>(arg_pt);
ex recur;
for (numeric p; p<=m; ++p)
recur += power(arg+p,-n+_ex_1);
recur *= factorial(n)*power(_ex_1,n);
return (psi(n, arg+m+_ex1)-recur).series(rel, order, options);
}
unsigned psi2_SERIAL::serial =
function::register_new(function_options("psi", 2).
eval_func(psi2_eval).
evalf_func(psi2_evalf).
derivative_func(psi2_deriv).
series_func(psi2_series).
latex_name("\\psi").
overloaded(2));
} // namespace GiNaC
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