* some related stuff. */
/*
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2000 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
#include "inifcns.h"
#include "ex.h"
#include "constant.h"
-#include "series.h"
+#include "pseries.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
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...
return coefficient*power(Pi,_ex1_2());
}
}
+ // gamma_evalf should be called here once it becomes available
}
+
return gamma(x).hold();
}
+
static ex gamma_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
return psi(x)*gamma(x);
}
-static ex gamma_series(const ex & x, const symbol & s, const ex & point, int order)
+
+static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to psi function
// 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))
+ // series(gamma(x+m+1)/(x*(x+1)*...*(x+m)),x,-m,order+1);
+ const ex x_pt = x.subs(s==pt);
+ if (!x_pt.info(info_flags::integer) || x_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:
- numeric m = -ex_to_numeric(xpoint);
- ex ser_numer = gamma(x+m+_ex1());
+ numeric m = -ex_to_numeric(x_pt);
ex ser_denom = _ex1();
for (numeric p; p<=m; ++p)
ser_denom *= x+p;
- return (ser_numer/ser_denom).series(s, point, order+1);
+ return (gamma(x+m+_ex1())/ser_denom).series(s, pt, order+1);
}
+
REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+
//////////
// Beta-function
//////////
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));
+ return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y))/gamma(ex_to_numeric(x+y));
}
+
static ex beta_eval(const ex & x, const ex & 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
// using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
+ numeric nx(ex_to_numeric(x));
+ numeric ny(ex_to_numeric(y));
if (nx.is_real() && nx.is_integer() &&
ny.is_real() && ny.is_integer()) {
if (nx.is_negative()) {
return beta(x,y).hold();
}
+
static ex beta_diff(const ex & x, const ex & y, unsigned diff_param)
{
GINAC_ASSERT(diff_param<2);
return retval;
}
-static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & point, int order)
+
+static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order)
{
// method:
- // Taylor series where there is no pole falls back to beta 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);
- ex ypoint = y.subs(s==point);
- if ((!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive)) &&
- (!ypoint.info(info_flags::integer) || ypoint.info(info_flags::positive)))
+ // Taylor series where there is no pole of one of the gamma functions
+ // falls back to beta function evaluation. Otherwise, fall back to
+ // gamma series directly.
+ // FIXME: this could need some testing, maybe it's wrong in some cases?
+ const ex x_pt = x.subs(s==pt);
+ const ex y_pt = y.subs(s==pt);
+ ex x_ser, y_ser, xy_ser;
+ if ((!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) &&
+ (!y_pt.info(info_flags::integer) || y_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:
- throw (std::domain_error("beta_series(): please code me"));
- /*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);*/
+ // trap the case where x is on a pole directly:
+ if (x.info(info_flags::integer) && !x.info(info_flags::positive))
+ x_ser = gamma(x+s).series(s,pt,order);
+ else
+ x_ser = gamma(x).series(s,pt,order);
+ // trap the case where y is on a pole directly:
+ if (y.info(info_flags::integer) && !y.info(info_flags::positive))
+ y_ser = gamma(y+s).series(s,pt,order);
+ else
+ y_ser = gamma(y).series(s,pt,order);
+ // trap the case where y is on a pole directly:
+ if ((x+y).info(info_flags::integer) && !(x+y).info(info_flags::positive))
+ xy_ser = gamma(y+x+s).series(s,pt,order);
+ else
+ xy_ser = gamma(y+x).series(s,pt,order);
+ // compose the result:
+ return (x_ser*y_ser/xy_ser).series(s,pt,order);
}
+
REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series);
+
//////////
// Psi-function (aka digamma-function)
//////////
static ex psi1_eval(const ex & 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 nx = ex_to_numeric(x);
+ if (nx.is_integer()) {
+ // integer case
+ if (nx.is_positive()) {
+ // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma
numeric rat(0);
- for (numeric i(ex_to_numeric(x)-_num1()); i.is_positive(); --i)
+ for (numeric i(nx+_num_1()); i.is_positive(); --i)
rat += i.inverse();
return rat-EulerGamma;
+ } else {
+ // for non-positive integers there is a pole:
+ throw (std::domain_error("psi_eval(): simple pole"));
}
- // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2)
- if ((_ex2()*x).info(info_flags::integer)) {
+ }
+ if ((_num2()*nx).is_integer()) {
+ // half integer case
+ if (nx.is_positive()) {
+ // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2)
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
+ for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
+ rat += _num2()*i.inverse();
+ return rat-EulerGamma-_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();
}
return psi(_ex1(), x);
}
-static ex psi1_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// 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))
+ const ex x_pt = x.subs(s==pt);
+ if (!x_pt.info(info_flags::integer) || x_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:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex recur;
for (numeric p; p<=m; ++p)
recur += power(x+p,_ex_1());
- return (psi(x+m+_ex1())-recur).series(s, point, order);
+ return (psi(x+m+_ex1())-recur).series(s, pt, order);
}
const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);
numeric nn = ex_to_numeric(n);
numeric nx = ex_to_numeric(x);
if (nx.is_integer()) {
+ // integer case
if (nx.is_equal(_num1()))
- return pow(_num_1(), nn+_num1())*factorial(nn)*zeta(ex(nn+_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;
+ 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 (std::domain_error("psi2_eval(): pole"));
}
- // for non-positive integers there is a pole:
- throw (std::domain_error("psi2_eval(): pole"));
}
+ 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()) {
+ 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();
}
return psi(n+_ex1(), x);
}
-static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & point, int order)
+static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// 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))
+ const ex x_pt = x.subs(s==pt);
+ if (!x_pt.info(info_flags::integer) || x_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:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
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);
+ return (psi(n, x+m+_ex1())-recur).series(s, pt, order);
}
const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);
git://www.ginac.de / ginac.git / blobdiff