/** @file inifcns_gamma.cpp
*
- * Implementation of Gamma function and some related stuff.
- *
+ * Implementation of Gamma-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
#include "power.h"
#include "symbol.h"
+#ifndef NO_GINAC_NAMESPACE
+namespace GiNaC {
+#endif // ndef NO_GINAC_NAMESPACE
+
//////////
-// gamma function
+// Gamma-function
//////////
/** Evaluation of gamma(x). Knows about integer arguments, half-integer
* evaluation some day...
*
* @exception fail_numeric("complex_infinity") or something similar... */
-ex gamma_eval(ex const & x)
+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) ) {
+ 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) ) {
+ 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 coefficient * power(Pi,numHALF());
+ return coefficient * pow(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 coefficient * power(Pi,numHALF());
+ return coefficient*sqrt(Pi);
}
}
}
return gamma(x).hold();
}
-
-ex gamma_evalf(ex const & x)
+
+static ex gamma_evalf(ex const & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return gamma(ex_to_numeric(x));
}
-ex gamma_diff(ex const & x, unsigned diff_param)
+static ex gamma_diff(ex const & x, unsigned diff_param)
{
- ASSERT(diff_param==0);
-
- return power(x, -1); //!!
+ GINAC_ASSERT(diff_param==0);
+
+ return psi(x)*gamma(x); // diff(log(gamma(x)),x)==psi(x)
}
-ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
{
- //!! Only handle one special case for now...
+ // FIXME: 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));
+ 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 gamma function"));
}
REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_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 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 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));
+}
+
+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)
+ return psi(n+1, x);
+}
+
+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