crandall.c File Reference

Calculates the summand function G and related functions in Crandall's formula. More...

#include "crandall.h"
#include "gamma.h"
#include "stdbool.h"
#include "tools.h"
#include <complex.h>
#include <float.h>
#include <math.h>

Include dependency graph for crandall.c:

Macros
#define	EPS   ldexp(1, -30)
	epsilon for the cutoff around nu = dimension.
#define	EPS_ZERO_PIY   (M_PI * 1e-64)
	epsilon for the cutoff around arguments of the crandall function around zero.

Functions
double	assignzArgBound (double nu)
	Calculates bounds on when to use asymptotic expansion of the upper incomplete gamma function, depending on the value of nu.
double complex	crandall_g (unsigned int dim, double nu, const double *z, double prefactor, double zArgBound)
	Calculates the upper Crandall function.
double complex	crandall_gReg_nuequalsdimplus2k (double s, double arg, double k, double lambda)
	Calculates the regularization of the zero summand in the second sum in Crandall's formula in the special case of nu = dim + 2k for some natural number k.
double complex	crandall_gReg (unsigned int dim, double s, const double *z, double prefactor)
	Calculates the regularization of the zero summand in the second sum in Crandall's formula.
double complex	crandall_g_lower (unsigned int dim, double nu, const double *z, double prefactor)
	Calculates the lower Crandall function.
double	crandall_gReg_harmonic (int k, int n, unsigned int dim, double s, const double *z, double prefactor)
	Calculates the regularization of the zero summand in the second sum in Crandall's formula for the harmonic method.
double	polynomial_y_der (unsigned int k, unsigned int dim, const double *z, const unsigned int *alpha, unsigned int alphaAbs, unsigned int n)
	Calculates the derivatives of Y_k(z) / n! = (pi * z**2)**k / n! where n <= k.

Detailed Description

Calculates the summand function G and related functions in Crandall's formula.

Function Documentation

assignzArgBound()

double assignzArgBound

(

double

nu

)

Calculates bounds on when to use asymptotic expansion of the upper incomplete gamma function, depending on the value of nu.

calculates bounds on when to use asymptotic expansion of the upper incomplete gamma function, depending on the value of nu.

Parameters

[in]

nu

exponent of the regularized Epstein zeta function.

Returns

minimum value of z, when to use the fast asymptotic expansion in the calculation of the incomplete upper gamma function upperGamma(nu, z).

crandall_g()

double complex crandall_g	(	unsigned int	dim,
		double	nu,
		const double *	z,
		double	prefactor,
		double	zArgBound )

Calculates the upper Crandall function.

Parameters

[in]	dim	dimension of the input vectors.
[in]	nu	exponent of the regularized Epstein zeta function.
[in]	z	input vector of the function.
[in]	prefactor	prefactor of the vector, e. g. lambda or 1/lambda in Crandall's formula
[in]	zArgBound	minimum value of pi * z**2, when to use the fast asymptotic expansion in the calculation of the Crandall function.

Returns

upperGamma(nu / 2,pi prefactor * z**2) / (pi * prefactor z**2)^(nu / 2) if |z| > 0 and - 2 / nu otherwise.

crandall_g_lower()

double complex crandall_g_lower	(	unsigned int	dim,
		double	nu,
		const double *	z,
		double	prefactor )

Calculates the lower Crandall function.

Parameters

[in]	dim	dimension of the input vectors.
[in]	nu	exponent of the regularized Epstein zeta function.
[in]	z	input vector of the function.
[in]	prefactor	prefactor of the vector, e. g. lambda or 1/lambda in Crandall's formula
[in]	zArgBound	minimum value of pi * z**2, when to use the fast asymptotic expansion in the calculation of the Crandall function.

Returns

lowerGamma(nu / 2,pi prefactor * z**2) / (pi * prefactor z**2)^(nu / 2) if |z| > 0 and 2 / nu otherwise.

crandall_gReg()

double complex crandall_gReg	(	unsigned int	dim,
		double	s,
		const double *	z,
		double	prefactor )

Calculates the regularization of the zero summand in the second sum in Crandall's formula.

Parameters

[in]	dim	dimension of the input vectors.
[in]	s	dimension minus exponent of the regularized Epstein zeta function, that is d - nu.
[in]	z	input vector of the function.
[in]	prefactor	prefactor of the vector, e. g. lambda.

Returns

- gamma(s/2) * gammaStar(s/2, pi * prefactor * z**2), where gammaStar is the twice regularized lower incomplete gamma function if s is not equal to - 2k and (pi * prefactor * y ** 2) ** (- s / 2) (gamma(s / 2, pi * prefactor * z ** 2) + ((-1)^k / k! ) * (log(pi * y ** 2) - log(prefactor ** 2))) if s is equal to - 2k for non negative natural number k.

crandall_gReg_harmonic()

double crandall_gReg_harmonic	(	int	k,
		int	n,
		unsigned int	dim,
		double	s,
		const double *	z,
		double	prefactor )

Calculates the regularization of the zero summand in the second sum in Crandall's formula for the harmonic method.

Parameters

[in]	k	specifies degree |alpha| - 2k.
[in]	n	|alpha|.
[in]	s	dimension minus exponent of the regularized Epstein zeta function, that is d - nu.
[in]	dim	dimension of the input vectors.
[in]	z	input vector of the function.
[in]	prefactor	prefactor of the vector, e. g. lambda.

Returns

- gamma(s/2) * gammaStar(s/2, pi * prefactor * z**2), where gammaStar is the twice regularized lower incomplete gamma function if s is and the special definition if s is equal to - 2k for non negative natural number k <= l.

crandall_gReg_nuequalsdimplus2k()

double complex crandall_gReg_nuequalsdimplus2k	(	double	s,
		double	arg,
		double	k,
		double	lambda )

Calculates the regularization of the zero summand in the second sum in Crandall's formula in the special case of nu = dim + 2k for some natural number k.

Parameters

[in]	s	dimension minus exponent of the regularized Epstein zeta function.
[in]	arg	input of the function.
[in]	k	k = - s / 2 = (nu - d) / 2 as an integer.
[in]	lambda	scaling parameter of crandalls formula.

Returns

arg ** (- s / 2) * (gamma(s / 2, arg) + ((-1)^k / k! ) * (log(arg) - log(lambda ** 2)).

polynomial_y_der()

double polynomial_y_der	(	unsigned int	k,
		unsigned int	dim,
		const double *	z,
		const unsigned int *	alpha,
		unsigned int	alphaAbs,
		unsigned int	n )

Calculates the derivatives of Y_k(z) / n! = (pi * z**2)**k / n! where n <= k.

Parameters

[in]	k	integer power.
[in]	dim	dimension of z.
[in]	y	vector of the polynomial. @parma[in] alpha: multi-index for the derivative.
[in]	n	factorial divisor smaller than k.

Returns

partial derivative of Y_k(z) / n!.