zeta.c File Reference

Calculates the (regularized) Epstein zeta function. More...

#include <complex.h>
#include <math.h>
#include <stdbool.h>
#include <stdlib.h>
#include "crandall.h"
#include "tools.h"
#include "zeta.h"

Include dependency graph for zeta.c:

Macros
#define	G_BOUND   3.2
	Smallest value z such that G(nu, z) is negligible for nu < 10.
#define	EPS   ldexp(1, -30)
	epsilon for the cutoff around nu = dimension.
#define	EPS_ZERO_Y   1e-64
	epsilon for the cutoff around x = 0 and y = 0

Functions
double complex	sum_real (double nu, unsigned int dim, double lambda, const double *m, const double *x, const double *y, const int cutoffs[], double zArgBound)
	calculates the first sum in Crandall's formula.
double complex	sum_fourier (double nu, unsigned int dim, double lambda, const double *m_invt, const double *x, const double *y, const int cutoffs[], double zArgBound)
	calculates the second sum in Crandall's formula.
double *	vectorProj (unsigned int dim, const double *m, const double *m_invt, const double *v)
	calculate projection of vector to elementary lattice cell.
double complex	epsteinZetaInternal (double nu, unsigned int dim, const double *m, const double *x, const double *y, double lambda, int reg)
	calculates the (regularized) Epstein Zeta function.

Detailed Description

Calculates the (regularized) Epstein zeta function.

Function Documentation

epsteinZetaInternal()

double complex epsteinZetaInternal	(	double	nu,
		unsigned int	dim,
		const double *	m,
		const double *	x,
		const double *	y,
		double	lambda,
		int	reg )

calculates the (regularized) Epstein Zeta function.

Parameters

[in]	nu	exponent for the Epstein zeta function.
[in]	dim	dimension of the input vectors.
[in]	m	matrix that transforms the lattice in the Epstein Zeta function.
[in]	x	x vector of the Epstein Zeta function.
[in]	y	y vector of the Epstein Zeta function.
[in]	lambda	relative weight of the sums in Crandall's formula.
[in]	reg	0 for no regularization, > 0 for the regularization.

Returns

function value of the regularized Epstein zeta.

sum_fourier()

double complex sum_fourier	(	double	nu,
		unsigned int	dim,
		double	lambda,
		const double *	m_invt,
		const double *	x,
		const double *	y,
		const int	cutoffs[],
		double	zArgBound )

calculates the second sum in Crandall's formula.

Parameters

[in]	nu	exponent for the Epstein zeta function.
[in]	dim	dimension of the input vectors.
[in]	lambda	parameters that decides the weight of each sum.
[in]	m	matrix that transforms the lattice in the Epstein Zeta function.
[in]	x	projection of x vector to elementary lattice cell.
[in]	y	projection of y vector to elementary lattice cell.
[in]	cutoffs	how many summands in each direction are considered.
[in]	zArgBound	global bound on when to use the asymptotic expansion in the incomplete gamma evaluation.

Returns

helper function for the second sum in crandalls formula. Calculates sum_{k in m_invt whole_numbers ** dim without zero} G_{dim - nu}(lambda * (k + y)) X exp(-2 * PI * I * x * (k + y))

sum_real()

double complex sum_real	(	double	nu,
		unsigned int	dim,
		double	lambda,
		const double *	m,
		const double *	x,
		const double *	y,
		const int	cutoffs[],
		double	zArgBound )

calculates the first sum in Crandall's formula.

Parameters

[in]	nu	exponent for the Epstein zeta function.
[in]	dim	dimension of the input vectors.
[in]	lambda	parameters that decides the weight of each sum.
[in]	m	matrix that transforms the lattice in the Epstein Zeta function.
[in]	x	projection of x vector to elementary lattice cell.
[in]	y	projection of y vector to elementary lattice cell.
[in]	cutoffs	how many summands in each direction are considered.
[in]	zArgBound	global bound on when to use the asymptotic expansion in the incomplete gamma evaluation.

Returns

helper function for the first sum in crandalls formula. Calculates sum_{z in m whole_numbers ** dim} G_{nu}((z - x) / lambda)) X exp(-2 * PI * I * z * y)

vectorProj()

double * vectorProj	(	unsigned int	dim,
		const double *	m,
		const double *	m_invt,
		const double *	v )

calculate projection of vector to elementary lattice cell.

Parameters

[in]	dim	dimension of the input vectors
[in]	m	matrix that transforms the lattice in the function.
[in]	m_invt	inverse of m.
[in]	v	vector for which the projection to the elementary lattice cell is needet.

Returns

projection of v to the elementary lattice cell.