/** @file inifcns_gamma.cpp
*
- * Implementation of Gamma-function, Polygamma-functions, 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
#include "power.h"
#include "symbol.h"
+#ifndef NO_GINAC_NAMESPACE
namespace GiNaC {
+#endif // ndef NO_GINAC_NAMESPACE
//////////
// Gamma-function
{
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) ) {
+ if (x.info(info_flags::posint)) {
return factorial(ex_to_numeric(x).sub(numONE()));
} 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) ) {
+ 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) ) {
+ 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 gamma(x).hold();
}
-
+
static ex gamma_evalf(ex const & x)
{
BEGIN_TYPECHECK
{
GINAC_ASSERT(diff_param==0);
- return psi(exZERO(),x)*gamma(x); // diff(log(gamma(x)),x)==psi(0,x)
+ return psi(x)*gamma(x); // diff(log(gamma(x)),x)==psi(x)
}
static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+//////////
+// Beta-function
+//////////
+
+static ex beta_eval(ex const & x, ex const & y)
+{
+ 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 numMINUSONE().power(ny)*beta(1-x-y, y);
+ else
+ throw (std::domain_error("beta_eval(): simple pole"));
+ }
+ if (ny.is_negative()) {
+ if (ny<=-nx)
+ return numMINUSONE().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 exZERO();
+ return gamma(x)*gamma(y)/gamma(x+y);
+ }
+ return beta(x,y).hold();
+}
+
+static ex beta_evalf(ex const & x, ex const & y)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ TYPECHECK(y,numeric)
+ END_TYPECHECK(beta(x,y))
+
+ return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y))
+ / gamma(ex_to_numeric(x+y));
+}
+
+static ex beta_diff(ex const & x, ex const & y, unsigned diff_param)
+{
+ GINAC_ASSERT(diff_param<2);
+ ex retval;
+
+ if (diff_param==0) // d/dx beta(x,y)
+ retval = (psi(x)-psi(x+y))*beta(x,y);
+ if (diff_param==1) // d/dy beta(x,y)
+ retval = (psi(y)-psi(x+y))*beta(x,y);
+ return retval;
+}
+
+static ex beta_series(ex const & x, ex const & y, symbol const & s, ex const & point, int order)
+{
+ if (x.is_equal(s) && point.is_zero()) {
+ ex e = 1 / s - EulerGamma + s * (pow(Pi, 2) / 12 + pow(EulerGamma, 2) / 2) + Order(pow(s, 2));
+ return e.series(s, point, order);
+ } else
+ throw(std::logic_error("don't know the series expansion of this particular beta function"));
+}
+
+REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series);
+
//////////
// Psi-function (aka polygamma-function)
//////////
+/** Evaluation of polygamma-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)-numONE()); 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 ((exTWO()*x).info(info_flags::integer)) {
+ numeric rat(0);
+ for (numeric i((ex_to_numeric(x)-numONE())*numTWO()); i.is_positive(); i-=numTWO())
+ rat += numTWO()*i.inverse();
+ return rat-EulerGamma-exTWO()*log(exTWO());
+ }
+ if (x.compare(exONE())==1) {
+ // should call numeric, since >1
+ }
+ }
+ }
+ return psi(x).hold();
+}
+
+static ex psi1_evalf(ex const & x)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(psi(x))
+
+ return psi(ex_to_numeric(x));
+}
+
+static ex psi1_diff(ex const & x, unsigned diff_param)
+{
+ GINAC_ASSERT(diff_param==0);
+
+ return psi(exONE(), x);
+}
+
+const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, NULL);
+
+//////////
+// Psi-functions (aka polygamma-functions) psi(0,x)==psi(x)
+//////////
+
/** Evaluation of polygamma-function psi(n,x).
* Somebody ought to provide some good numerical evaluation some day... */
-static ex psi_eval(ex const & n, ex const & x)
+static ex psi2_eval(ex const & n, ex const & x)
{
+ // psi(0,x) -> psi(x)
+ if (n.is_zero())
+ return psi(x);
if (n.info(info_flags::numeric) && x.info(info_flags::numeric)) {
// do some stuff...
}
return psi(n, x).hold();
}
-
-static ex psi_evalf(ex const & n, ex const & x)
+
+static ex psi2_evalf(ex const & n, ex const & x)
{
BEGIN_TYPECHECK
TYPECHECK(n,numeric)
return psi(ex_to_numeric(n), ex_to_numeric(x));
}
-static ex psi_diff(ex const & n, ex const & x, unsigned diff_param)
+static ex psi2_diff(ex const & n, ex const & x, unsigned diff_param)
{
- GINAC_ASSERT(diff_param==0);
+ 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)
return psi(n+1, x);
}
-static ex psi_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order)
-{
- throw(std::logic_error("Nobody told me how to series expand the psi function. :-("));
-}
-
-REGISTER_FUNCTION(psi, psi_eval, psi_evalf, psi_diff, psi_series);
+const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, NULL);
+#ifndef NO_GINAC_NAMESPACE
} // namespace GiNaC
+#endif // ndef NO_GINAC_NAMESPACE
git://www.ginac.de / ginac.git / blobdiff