/** @file inifcns_gamma.cpp
*
- * Implementation of Gamma function and some related stuff. */
+ * Implementation of Gamma-function, Beta-function, Polygamma-functions, and
+ * some related stuff. */
+
+/*
+ * GiNaC Copyright (C) 1999-2001 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ */
#include <vector>
#include <stdexcept>
-#include "ginac.h"
+#include "inifcns.h"
+#include "constant.h"
+#include "pseries.h"
+#include "numeric.h"
+#include "power.h"
+#include "relational.h"
+#include "symbol.h"
+#include "symmetry.h"
+#include "utils.h"
+
+namespace GiNaC {
//////////
-// gamma function
+// Logarithm of Gamma function
//////////
-/** Evaluation of gamma(x). Knows about integer arguments, half-integer
- * arguments and that's it. Somebody ought to provide some good numerical
- * evaluation some day...
+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 fail_numeric("complex_infinity") or something similar... */
-ex gamma_eval(ex const & x)
-{
- if ( x.info(info_flags::numeric) ) {
-
- // trap integer arguments:
- if ( x.info(info_flags::integer) ) {
- // gamma(n+1) -> n! for postitive n
- if ( x.info(info_flags::posint) ) {
- return factorial(ex_to_numeric(x).sub(numONE()));
- } else {
- return numZERO(); // Infinity. Throw? What?
- }
- }
- // trap half integer arguments:
- if ( (x*2).info(info_flags::integer) ) {
- // trap positive x=(n+1/2)
- // gamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
- if ( (x*2).info(info_flags::posint) ) {
- numeric n = ex_to_numeric(x).sub(numHALF());
- numeric coefficient = doublefactorial(n.mul(numTWO()).sub(numONE()));
- coefficient = coefficient.div(numTWO().power(n));
- return mul(coefficient,power(Pi,numHALF()));
- } else {
- // trap negative x=(-n+1/2)
- // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
- numeric n = abs(ex_to_numeric(x).sub(numHALF()));
- numeric coefficient = numeric(-2).power(n);
- coefficient = coefficient.div(doublefactorial(n.mul(numTWO()).sub(numONE())));;
- return mul(coefficient,power(Pi,numHALF()));
- }
- }
- }
- return gamma(x).hold();
-}
-
-ex gamma_evalf(ex const & x)
+ * @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);
+ 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*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));
+ } 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);
+ return (doublefactorial(n.mul(_num2).sub(_num1)).div(pow(_num2,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));
+ return (pow(_num_2, n).div(doublefactorial(n.mul(_num2).sub(_num1))))*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+1);
+ const ex arg_pt = arg.subs(rel);
+ 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+1, 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)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(gamma(x))
-
- return gamma(ex_to_numeric(x));
+ 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();
}
-ex gamma_diff(ex const & x, unsigned diff_param)
+
+static ex beta_eval(const ex & x, const ex & y)
{
- ASSERT(diff_param==0);
+ 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, 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, 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();
+}
- return power(x, -1); //!!
+
+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;
}
-ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
+
+static ex beta_series(const ex & arg1,
+ const ex & arg2,
+ const relational & rel,
+ int order,
+ unsigned options)
{
- //!! Only handle one special case for now...
- if (x.is_equal(s) && point.is_zero()) {
- ex e = 1 / s - EulerGamma + s * (power(Pi, 2) / 12 + power(EulerGamma, 2) / 2) + Order(power(s, 2));
- return e.series(s, point, order);
- } else
- throw(std::logic_error("don't know the series expansion of this particular gamma function"));
+ // 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);
+ const ex arg2_pt = arg2.subs(rel);
+ GINAC_ASSERT(is_exactly_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).series(rel, order, options);
+ else
+ arg1_ser = tgamma(arg1).series(rel,order);
+ // 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).series(rel, order, options);
+ else
+ arg2_ser = tgamma(arg2).series(rel,order);
+ // 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).series(rel, order, options);
+ else
+ arg1arg2_ser = tgamma(arg2+arg1).series(rel,order);
+ // compose the result (expanding all the terms):
+ return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
-REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+
+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); 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*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)*_num2; i>0; i-=_num2)
+ rat += _num2*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);
+ 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);
+ 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);
+}
+
+const unsigned function_index_psi1 =
+ function::register_new(function_options("psi").
+ 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))
+ // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
+ return pow(_num_1,nn+_num1)*factorial(nn)*zeta(ex(nn+_num1));
+ 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);
+ recur *= factorial(nn)*pow(_num_1, nn);
+ return recur+psi(n,_ex1);
+ } else {
+ // for non-positive integers there is a pole:
+ throw (pole_error("psi2_eval(): pole",1));
+ }
+ }
+ if ((_num2*nx).is_integer()) {
+ // half integer case
+ if (nx.is_equal(_num1_2))
+ // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
+ return pow(_num_1,nn+_num1)*factorial(nn)*(pow(_num2,nn+_num1) + _num_1)*zeta(ex(nn+_num1));
+ if (nx.is_positive()) {
+ const numeric m = nx - _num1_2;
+ // 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*m)*pow(_num2,nn+_num1)-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);
+ recur *= factorial(nn)*pow(_num_1, nn+_num_1);
+ 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);
+ 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);
+}
+
+const unsigned function_index_psi2 =
+ function::register_new(function_options("psi").
+ eval_func(psi2_eval).
+ evalf_func(psi2_evalf).
+ derivative_func(psi2_deriv).
+ series_func(psi2_series).
+ latex_name("\\psi").
+ overloaded(2));
+
+
+} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff