* some related stuff. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2014 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
*
* 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
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <vector>
-#include <stdexcept>
-
#include "inifcns.h"
-#include "ex.h"
#include "constant.h"
#include "pseries.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
+#include "operators.h"
#include "symbol.h"
+#include "symmetry.h"
#include "utils.h"
-#ifndef NO_NAMESPACE_GINAC
+#include <stdexcept>
+#include <vector>
+
namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
//////////
// Logarithm of Gamma function
static ex lgamma_evalf(const ex & x)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(lgamma(x))
-
- return lgamma(ex_to_numeric(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...
+ * Handles integer arguments as a special case.
*
- * @exception std::domain_error("lgamma_eval(): simple pole") */
+ * @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.exadd(_ex_1())));
- } else {
- throw (std::domain_error("lgamma_eval(): logarithmic pole"));
- }
- }
- // lgamma_evalf should be called here once it becomes available
- }
-
- return lgamma(x).hold();
+ 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));
+ }
+ if (!ex_to<numeric>(x).is_rational())
+ return lgamma(ex_to<numeric>(x));
+ }
+
+ 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);
+ 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, subs_options::no_pattern);
+ 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);
+}
+
+
+static ex lgamma_conjugate(const ex & x)
+{
+ // conjugate(lgamma(x))==lgamma(conjugate(x)) unless on the branch cut
+ // which runs along the negative real axis.
+ if (x.info(info_flags::positive)) {
+ return lgamma(x);
+ }
+ if (is_exactly_a<numeric>(x) &&
+ !x.imag_part().is_zero()) {
+ return lgamma(x.conjugate());
+ }
+ return conjugate_function(lgamma(x)).hold();
}
-// need to implement lgamma_series.
REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
evalf_func(lgamma_evalf).
- derivative_func(lgamma_deriv));
+ derivative_func(lgamma_deriv).
+ series_func(lgamma_series).
+ conjugate_func(lgamma_conjugate).
+ latex_name("\\log \\Gamma"));
//////////
static ex tgamma_evalf(const ex & x)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(tgamma(x))
-
- return tgamma(ex_to_numeric(x));
+ if (is_exactly_a<numeric>(x)) {
+ try {
+ return tgamma(ex_to<numeric>(x));
+ } catch (const dunno &e) { }
+ }
+
+ return tgamma(x).hold();
}
* arguments, half-integer arguments and that's it. Somebody ought to provide
* some good numerical evaluation some day...
*
- * @exception std::domain_error("tgamma_eval(): simple pole") */
+ * @exception pole_error("tgamma_eval(): simple pole",0) */
static ex tgamma_eval(const ex & x)
{
- if (x.info(info_flags::numeric)) {
- // trap integer arguments:
- if (x.info(info_flags::integer)) {
- // tgamma(n) -> (n-1)! for postitive n
- if (x.info(info_flags::posint)) {
- return factorial(ex_to_numeric(x).sub(_num1()));
- } else {
- throw (std::domain_error("tgamma_eval(): simple pole"));
- }
- }
- // trap half integer arguments:
- if ((x*2).info(info_flags::integer)) {
- // trap positive x==(n+1/2)
- // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
- if ((x*_ex2()).info(info_flags::posint)) {
- numeric n = ex_to_numeric(x).sub(_num1_2());
- numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1()));
- coefficient = coefficient.div(pow(_num2(),n));
- return coefficient * pow(Pi,_ex1_2());
- } else {
- // trap negative x==(-n+1/2)
- // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
- numeric n = abs(ex_to_numeric(x).sub(_num1_2()));
- numeric coefficient = pow(_num_2(), n);
- coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));;
- return coefficient*power(Pi,_ex1_2());
- }
- }
- // tgamma_evalf should be called here once it becomes available
- }
-
- return tgamma(x).hold();
+ if (x.info(info_flags::numeric)) {
+ // trap integer arguments:
+ const numeric two_x = (*_num2_p)*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_p));
+ } 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_p);
+ return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,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_p));
+ return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi);
+ }
+ }
+ if (!ex_to<numeric>(x).is_rational())
+ return tgamma(ex_to<numeric>(x));
+ }
+
+ 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);
+ 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);
+ const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
+ 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, options);
}
-static ex tgamma_series(const ex & x, const relational & r, int order)
+static ex tgamma_conjugate(const ex & x)
{
- // 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 x_pt = x.subs(r);
- 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(x_pt);
- ex ser_denom = _ex1();
- for (numeric p; p<=m; ++p)
- ser_denom *= x+p;
- return (tgamma(x+m+_ex1())/ser_denom).series(r, order+1);
+ // conjugate(tgamma(x))==tgamma(conjugate(x))
+ return tgamma(x.conjugate());
}
REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
evalf_func(tgamma_evalf).
derivative_func(tgamma_deriv).
- series_func(tgamma_series));
+ series_func(tgamma_series).
+ conjugate_func(tgamma_conjugate).
+ latex_name("\\Gamma"));
//////////
static ex beta_evalf(const ex & x, const ex & y)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- TYPECHECK(y,numeric)
- END_TYPECHECK(beta(x,y))
-
- return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y));
+ if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
+ try {
+ return exp(lgamma(ex_to<numeric>(x))+lgamma(ex_to<numeric>(y))-lgamma(ex_to<numeric>(x+y)));
+ } catch (const dunno &e) { }
+ }
+
+ return beta(x,y).hold();
}
static ex beta_eval(const ex & x, const ex & y)
{
- 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)
- 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()) {
- if (nx<=-ny)
- return pow(_num_1(), ny)*beta(1-x-y, y);
- else
- throw (std::domain_error("beta_eval(): simple pole"));
- }
- if (ny.is_negative()) {
- if (ny<=-nx)
- return pow(_num_1(), nx)*beta(1-y-x, x);
- else
- throw (std::domain_error("beta_eval(): simple pole"));
- }
- 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();
- // everything is ok:
- return tgamma(x)*tgamma(y)/tgamma(x+y);
- }
-
- return beta(x,y).hold();
+ if (x.is_equal(_ex1))
+ return 1/y;
+ if (y.is_equal(_ex1))
+ return 1/x;
+ 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_p, 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_p, 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;
+ if (!ex_to<numeric>(x).is_rational() || !ex_to<numeric>(x).is_rational())
+ return evalf(beta(x, y).hold());
+ }
+
+ return beta(x,y).hold();
}
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;
+ 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;
}
-static ex beta_series(const ex & x, const ex & y, const relational & r, int order)
+static ex beta_series(const ex & arg1,
+ const ex & arg2,
+ const relational & rel,
+ int order,
+ unsigned options)
{
- // 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.
- // FIXME: this could need some testing, maybe it's wrong in some cases?
- const ex x_pt = x.subs(r);
- const ex y_pt = y.subs(r);
- GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
- const symbol *s = static_cast<symbol *>(r.lhs().bp);
- 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()
- // trap the case where x is on a pole directly:
- if (x.info(info_flags::integer) && !x.info(info_flags::positive))
- x_ser = tgamma(x+*s).series(r,order);
- else
- x_ser = tgamma(x).series(r,order);
- // trap the case where y is on a pole directly:
- if (y.info(info_flags::integer) && !y.info(info_flags::positive))
- y_ser = tgamma(y+*s).series(r,order);
- else
- y_ser = tgamma(y).series(r,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 = tgamma(y+x+*s).series(r,order);
- else
- xy_ser = tgamma(y+x).series(r,order);
- // compose the result:
- return (x_ser*y_ser/xy_ser).series(r,order);
+ // 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, subs_options::no_pattern);
+ const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
+ GINAC_ASSERT(is_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);
+ else
+ arg1_ser = tgamma(arg1);
+ // 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);
+ else
+ arg2_ser = tgamma(arg2);
+ // 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);
+ else
+ arg1arg2_ser = tgamma(arg2+arg1);
+ // compose the result (expanding all the terms):
+ return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
}
REGISTER_FUNCTION(beta, eval_func(beta_eval).
evalf_func(beta_evalf).
derivative_func(beta_deriv).
- series_func(beta_series));
+ series_func(beta_series).
+ latex_name("\\mathrm{B}").
+ set_symmetry(sy_symm(0, 1)));
//////////
static ex psi1_evalf(const ex & x)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(psi(x))
-
- return psi(ex_to_numeric(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)) {
- numeric nx = ex_to_numeric(x);
- if (nx.is_integer()) {
- // integer case
- if (nx.is_positive()) {
- // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - gamma
- numeric rat(0);
- for (numeric i(nx+_num_1()); i.is_positive(); --i)
- rat += i.inverse();
- return rat-gamma;
- } else {
- // for non-positive integers there is a pole:
- throw (std::domain_error("psi_eval(): simple pole"));
- }
- }
- if ((_num2()*nx).is_integer()) {
- // half integer case
- if (nx.is_positive()) {
- // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - gamma - 2log(2)
- numeric rat(0);
- for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2())
- rat += _num2()*i.inverse();
- return rat-gamma-_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();
+ 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_p)); 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_p)*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_p))*(*_num2_p); i>0; i-=(*_num2_p))
+ rat += (*_num2_p)*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_p);
+ 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);
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx psi(x) -> psi(1,x)
+ return psi(_ex1, x);
}
-static ex psi1_series(const ex & x, const relational & r, int order)
+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 x_pt = x.subs(r);
- 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(x_pt);
- ex recur;
- for (numeric p; p<=m; ++p)
- recur += power(x+p,_ex_1());
- return (psi(x+m+_ex1())-recur).series(r, 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);
+ const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
+ 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).
- overloaded(2));
+unsigned psi1_SERIAL::serial =
+ function::register_new(function_options("psi", 1).
+ 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)
{
- BEGIN_TYPECHECK
- TYPECHECK(n,numeric)
- TYPECHECK(x,numeric)
- END_TYPECHECK(psi(n,x))
-
- return psi(ex_to_numeric(n), ex_to_numeric(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)) {
- numeric nn = ex_to_numeric(n);
- 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 (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();
+ // 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_p))
+ // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
+ return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
+ 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_p));
+ recur *= factorial(nn)*pow((*_num_1_p), nn);
+ return recur+psi(n,_ex1);
+ } else {
+ // for non-positive integers there is a pole:
+ throw (pole_error("psi2_eval(): pole",1));
+ }
+ }
+ if (((*_num2_p)*nx).is_integer()) {
+ // half integer case
+ if (nx.is_equal(*_num1_2_p))
+ // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
+ return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
+ if (nx.is_positive()) {
+ const numeric m = nx - (*_num1_2_p);
+ // 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_p)*m)*pow((*_num2_p),nn+(*_num1_p))-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_p));
+ recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
+ 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);
+ 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 & x, const relational & r, int order)
+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 x_pt = x.subs(r);
- 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(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(r, 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! / 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, subs_options::no_pattern);
+ 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).
- overloaded(2));
+unsigned psi2_SERIAL::serial =
+ function::register_new(function_options("psi", 2).
+ eval_func(psi2_eval).
+ evalf_func(psi2_evalf).
+ derivative_func(psi2_deriv).
+ series_func(psi2_series).
+ latex_name("\\psi").
+ overloaded(2));
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff