/** @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 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 "ex.h"
+#include "constant.h"
+#include "series.h"
+#include "numeric.h"
+#include "power.h"
+#include "relational.h"
+#include "symbol.h"
+#include "utils.h"
+
+#ifndef NO_GINAC_NAMESPACE
+namespace GiNaC {
+#endif // ndef NO_GINAC_NAMESPACE
//////////
-// gamma function
+// Gamma-function
//////////
+static ex gamma_evalf(ex const & x)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(gamma(x))
+
+ return gamma(ex_to_numeric(x));
+}
+
/** 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...
*
- * @exception fail_numeric("complex_infinity") or something similar... */
-ex gamma_eval(ex const & x)
+ * @exception std::domain_error("gamma_eval(): simple pole") */
+static ex gamma_eval(ex const & x)
{
- if ( x.info(info_flags::numeric) ) {
-
+ if (x.info(info_flags::numeric)) {
// trap integer arguments:
- if ( x.info(info_flags::integer) ) {
+ 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()));
+ if (x.info(info_flags::posint)) {
+ return factorial(ex_to_numeric(x).sub(_num1()));
} else {
- return numZERO(); // Infinity. Throw? What?
+ throw (std::domain_error("gamma_eval(): simple pole"));
}
}
// trap half integer arguments:
- if ( (x*2).info(info_flags::integer) ) {
- // trap positive x=(n+1/2)
+ 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()));
+ if ((x*2).info(info_flags::posint)) {
+ numeric n = ex_to_numeric(x).sub(_num1_2());
+ numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
+ coefficient = coefficient.div(_num2().power(n));
+ return coefficient * pow(Pi,_num1_2());
} else {
- // trap negative x=(-n+1/2)
+ // 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 n = abs(ex_to_numeric(x).sub(_num1_2()));
numeric coefficient = numeric(-2).power(n);
- coefficient = coefficient.div(doublefactorial(n.mul(numTWO()).sub(numONE())));;
- return mul(coefficient,power(Pi,numHALF()));
+ coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
+ return coefficient*sqrt(Pi);
}
}
}
return gamma(x).hold();
}
+
+static ex gamma_diff(ex const & x, unsigned diff_param)
+{
+ GINAC_ASSERT(diff_param==0);
-ex gamma_evalf(ex const & x)
+ // d/dx log(gamma(x)) -> psi(x)
+ // d/dx gamma(x) -> psi(x)*gamma(x)
+ return psi(x)*gamma(x);
+}
+
+static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
+{
+ // method:
+ // Taylor series where there is no pole falls back to psi function evaluation.
+ // On a pole at -m use the recurrence relation
+ // gamma(x) == gamma(x+1) / x
+ // from which follows
+ // series(gamma(x),x,-m,order) ==
+ // series(gamma(x+m+1)/(x*(x+1)...*(x+m)),x,-m,order+1);
+ ex xpoint = x.subs(s==point);
+ if (!xpoint.info(info_flags::integer) || xpoint.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:
+ numeric m = -ex_to_numeric(xpoint);
+ ex ser_numer = gamma(x+m+_ex1());
+ ex ser_denom = _ex1();
+ for (numeric p; p<=m; ++p)
+ ser_denom *= x+p;
+ return (ser_numer/ser_denom).series(s, point, order+1);
+}
+
+REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+
+//////////
+// Beta-function
+//////////
+
+static ex beta_evalf(ex const & x, ex const & y)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
- END_TYPECHECK(gamma(x))
+ TYPECHECK(y,numeric)
+ END_TYPECHECK(beta(x,y))
- return gamma(ex_to_numeric(x));
+ return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y))
+ / gamma(ex_to_numeric(x+y));
}
-ex gamma_diff(ex const & x, unsigned diff_param)
+static ex beta_eval(ex const & x, ex const & y)
{
- ASSERT(diff_param==0);
+ if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
+ numeric nx(ex_to_numeric(x));
+ numeric ny(ex_to_numeric(y));
+ // treat all problematic x and y that may not be passed into gamma,
+ // because they would throw there although beta(x,y) is well-defined:
+ if (nx.is_real() && nx.is_integer() &&
+ ny.is_real() && ny.is_integer()) {
+ if (nx.is_negative()) {
+ if (nx<=-ny)
+ return _num_1().power(ny)*beta(1-x-y, y);
+ else
+ throw (std::domain_error("beta_eval(): simple pole"));
+ }
+ if (ny.is_negative()) {
+ if (ny<=-nx)
+ return _num_1().power(nx)*beta(1-y-x, x);
+ else
+ throw (std::domain_error("beta_eval(): simple pole"));
+ }
+ return gamma(x)*gamma(y)/gamma(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();
+ return gamma(x)*gamma(y)/gamma(x+y);
+ }
+ return beta(x,y).hold();
+}
- return power(x, -1); //!!
+static ex beta_diff(ex const & x, ex const & y, unsigned diff_param)
+{
+ GINAC_ASSERT(diff_param<2);
+ ex retval;
+
+ // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
+ if (diff_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 (diff_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)
+REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, NULL);
+
+//////////
+// Psi-function (aka digamma-function)
+//////////
+
+static ex psi1_evalf(ex const & x)
{
- //!! 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"));
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(psi(x))
+
+ return psi(ex_to_numeric(x));
}
-REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+/** Evaluation of digamma-function psi(x).
+ * Somebody ought to provide some good numerical evaluation some day... */
+static ex psi1_eval(ex const & x)
+{
+ if (x.info(info_flags::numeric)) {
+ if (x.info(info_flags::integer) && !x.info(info_flags::positive))
+ throw (std::domain_error("psi_eval(): simple pole"));
+ if (x.info(info_flags::positive)) {
+ // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma
+ if (x.info(info_flags::integer)) {
+ numeric rat(0);
+ for (numeric i(ex_to_numeric(x)-_num1()); i.is_positive(); --i)
+ rat += i.inverse();
+ return rat-EulerGamma;
+ }
+ // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2)
+ if ((_ex2()*x).info(info_flags::integer)) {
+ numeric rat(0);
+ for (numeric i((ex_to_numeric(x)-_num1())*_num2()); i.is_positive(); i-=_num2())
+ rat += _num2()*i.inverse();
+ return rat-EulerGamma-_ex2()*log(_ex2());
+ }
+ if (x.compare(_ex1())==1) {
+ // should call numeric, since >1
+ }
+ }
+ }
+ return psi(x).hold();
+}
+
+static ex psi1_diff(ex const & x, unsigned diff_param)
+{
+ GINAC_ASSERT(diff_param==0);
+
+ // d/dx psi(x) -> psi(1,x)
+ return psi(_ex1(), x);
+}
+
+static ex psi1_series(ex const & x, symbol const & s, ex const & point, int order)
+{
+ // 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);
+ ex xpoint = x.subs(s==point);
+ if (!xpoint.info(info_flags::integer) || xpoint.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:
+ numeric m = -ex_to_numeric(xpoint);
+ ex recur;
+ for (numeric p; p<=m; ++p)
+ recur += power(x+p,_ex_1());
+ return (psi(x+m+_ex1())-recur).series(s, point, order);
+}
+
+const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);
+
+//////////
+// Psi-functions (aka polygamma-functions) psi(0,x)==psi(x)
+//////////
+
+static ex psi2_evalf(ex const & n, ex const & x)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(n,numeric)
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(psi(n,x))
+
+ return psi(ex_to_numeric(n), ex_to_numeric(x));
+}
+
+/** Evaluation of polygamma-function psi(n,x).
+ * Somebody ought to provide some good numerical evaluation some day... */
+static ex psi2_eval(ex const & n, ex const & x)
+{
+ // psi(0,x) -> psi(x)
+ if (n.is_zero())
+ return psi(x);
+ // psi(-1,x) -> log(gamma(x))
+ if (n.is_equal(_ex_1()))
+ return log(gamma(x));
+ if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
+ x.info(info_flags::numeric)) {
+ numeric nn = ex_to_numeric(n);
+ numeric nx = ex_to_numeric(x);
+ if (x.is_equal(_ex1()))
+ return _num_1().power(nn+_num1())*factorial(nn)*zeta(ex(nn+_num1()));
+ }
+ return psi(n, x).hold();
+}
+
+static ex psi2_diff(ex const & n, ex const & x, unsigned diff_param)
+{
+ GINAC_ASSERT(diff_param<2);
+
+ if (diff_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+1, x);
+}
+
+static ex psi2_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order)
+{
+ // 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! / z^(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);
+ ex xpoint = x.subs(s==point);
+ if (!xpoint.info(info_flags::integer) || xpoint.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:
+ numeric m = -ex_to_numeric(xpoint);
+ ex recur;
+ for (numeric p; p<=m; ++p)
+ recur += power(x+p,-n+_ex_1());
+ recur *= factorial(n)*power(_ex_1(),n);
+ return (psi(n, x+m+_ex1())-recur).series(s, point, order);
+}
+
+const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);
+
+#ifndef NO_GINAC_NAMESPACE
+} // namespace GiNaC
+#endif // ndef NO_GINAC_NAMESPACE
git://www.ginac.de / ginac.git / blobdiff