Macros
#define	SYMBOL_ABEL_POISSON(k, h) (pow(h,k))

#define	SYMBOL_SINGULARITY(k, h) ((2.0/(2k+1))pow(h,k))

#define	KT_ABEL_POISSON (0)
	Abel-Poisson kernel.

#define	KT_SINGULARITY (1)
	Singularity kernel.

#define	KT_LOC_SUPP (2)
	Locally supported kernel.

#define	KT_GAUSSIAN (3)
	Gaussian kernel.

Enumerations
enum	pvalue { NO = 0, YES = 1, BOTH = 2 }
	Enumeration type for yes/no/both-type parameters.

Functions
static int	scaled_modified_bessel_i_series (const R x, const R alpha, const int nb, const int ize, R *b)

static void	scaled_modified_bessel_i_normalize (const R x, const R alpha, const int nb, const int ize, R *b, const R sum_)

static int	smbi (const R x, const R alpha, const int nb, const int ize, R *b)
	Calculates the modified bessel function $I_{n+\alpha}(x)$ , possibly scaled by $\mathrm{e}^{-x}$ , for real non-negative with $0 \le \alpha < 1$ , and $n=0,1,\ldots,nb-1$ . More...

static double	innerProduct (const double phi1, const double theta1, const double phi2, const double theta2)
	Computes the $\mathbb{R}^3$ standard inner product between two vectors on the unit sphere $\mathbb{S}^2$ given in spherical coordinates. More...

static double	poissonKernel (const double x, const double h)
	Evaluates the Poisson kernel $Q_h: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ . More...

static double	singularityKernel (const double x, const double h)
	Evaluates the singularity kernel $S_h: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ . More...

static double	locallySupportedKernel (const double x, const double h, const double lambda)
	Evaluates the locally supported kernel $L_{h,\lambda}: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ . More...

static double	gaussianKernel (const double x, const double sigma)
	Evaluates the spherical Gaussian kernel $G_\sigma: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ . More...

int	main (int argc, char **argv)
	The main program. More...

Detailed Description

Function Documentation

static int smbi	(	const R	x,
		const R	alpha,
		const int	nb,
		const int	ize,
		R *	b
	)

static

Calculates the modified bessel function $I_{n+\alpha}(x)$ , possibly scaled by $\mathrm{e}^{-x}$ , for real non-negative $x,alpha$ with $0 \le \alpha < 1$ , and $n=0,1,\ldots,nb-1$ .

[in] x non-negative real number in $I_{n+\alpha}(x)$
[in] alpha non-negative real number with $0 \le \alpha < 1$ in $I_{n+\alpha}(x)$
[in] nb number of functions to be calculated
[in] ize switch between no scaling (ize = 1) and exponential scaling (ize = 2)
[out] b real output vector to contain ,
Returns
error indicator. Only if this value is identical to nb, then all values in b have been calculated to full accuracy. If not, errors are indicated using the following scheme:
- ncalc < 0: At least one of the arguments was out of range (e.g. nb <= 0, ize neither equals 1 nor 2, $|x| \ge exparg$ ). In this case, the output vector b is not calculated and ncalc is set to $\min(nb,0)-1$ .
- 0 < ncalc < nb: Not all requested functions could be calculated to full accuracy. This can occur when nb is much larger than |x|. in this case, the values $I_{n+\alpha}(x)$ are calculated to full accuracy for $n=0,1,\ldots,ncalc$ . The rest of the values up to $n=0,1,\ldots,nb-1$ is calculated to a lower accuracy.

This program is based on a program written by David J. Sookne [2] that computes values of the Bessel functions $J_{\nu}(x)$ or $I_{\nu}(x)$ for real argument $x$ and integer order $\nu$ . modifications include the restriction of the computation to the Bessel function $I_{\nu}(x)$ for non-negative real argument, the extension of the computation to arbitrary non-negative orders $\nu$ , and the elimination of most underflow.

References: [1] F. W. J. Olver and D. J. Sookne, A note on backward recurrence algorithms", Math. Comput. (26), 1972, pp 125 -- 132. [2] D. J. Sookne, "Bessel functions of real argument and int order", NBS Jour. of Res. B. (77B), 1973, pp. 125 – 132.

Modified by W. J. Cody, Applied Mathematics Division, Argonne National Laboratory, Argonne, IL, 60439, USA

Modified by Jens Keiner, Institute of Mathematics, University of Lübeck, 23560 Lübeck, Germany

Definition at line 192 of file fastsumS2.c.

Referenced by main().

static double innerProduct	(	const double	phi1,
		const double	theta1,
		const double	phi2,
		const double	theta2
	)

inlinestatic

Computes the $\mathbb{R}^3$ standard inner product between two vectors on the unit sphere $\mathbb{S}^2$ given in spherical coordinates.

phi1 The angle $\varphi_1 \in [-\pi,\pi)$ of the first vector
theta1 The angle $\vartheta_1 \in [0,\pi]$ of the first vector
phi2 The angle $\varphi_2 \in [-\pi,\pi)$ of the second vector
theta2 The angle $\vartheta_2 \in [0,\pi]$ of the second vector

Returns: The inner product $\cos \vartheta_1 \cos \vartheta_2 + \sin \vartheta_1 \sin(\vartheta_2 \cos(\varphi_1 - \varphi_2)$

Author: Jens Keiner

Definition at line 449 of file fastsumS2.c.

Referenced by main().

static double poissonKernel	(	const double	x,
		const double	h
	)

inlinestatic

Evaluates the Poisson kernel $Q_h: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .

x The node $x \in [-1,1]$
h The parameter $h \in (0,1)$

Returns: The value of the Poisson kernel at the node

Author: Jens Keiner

Definition at line 468 of file fastsumS2.c.

Referenced by main().

static double singularityKernel	(	const double	x,
		const double	h
	)

inlinestatic

Evaluates the singularity kernel $S_h: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .

x The node $x \in [-1,1]$
h The parameter $h \in (0,1)$

Returns: The value of the Poisson kernel at the node

Author: Jens Keiner

Definition at line 484 of file fastsumS2.c.

Referenced by main().

static double locallySupportedKernel	(	const double	x,
		const double	h,
		const double	lambda
	)

inlinestatic

Evaluates the locally supported kernel $L_{h,\lambda}: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .

x The node $x \in [-1,1]$
h The parameter $h \in (0,1)$
lambda The parameter $\lambda \in \mathbb{N}_0$

Returns: The value of the locally supported kernel $L_{h,\lambda}(x)$ at the node

Author: Jens Keiner

Definition at line 502 of file fastsumS2.c.

Referenced by main().

static double gaussianKernel	(	const double	x,
		const double	sigma
	)

inlinestatic

Evaluates the spherical Gaussian kernel $G_\sigma: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .

x The node $x \in [-1,1]$
sigma The parameter $\sigma \in \mathbb{R}_+$

Returns: The value of the pherical Gaussian kernel $G_\sigma(x)$ at the node

Author: Jens Keiner

Definition at line 520 of file fastsumS2.c.

Referenced by main().

int main	(	int	argc,
		char **	argv
	)

The main program.

Parameters

argc	The number of arguments
argv	An array containing the arguments as C-strings

Returns: Exit code

Author: Jens Keiner

Definition at line 535 of file fastsumS2.c.

References nfsft_plan::f, nfsft_plan::f_hat, FFT_OUT_OF_PLACE, FFTW_INIT, gaussianKernel(), innerProduct(), KT_ABEL_POISSON, KT_GAUSSIAN, KT_LOC_SUPP, KT_SINGULARITY, locallySupportedKernel(), nfft_elapsed_seconds(), nfft_free(), nfft_malloc(), nfsft_adjoint(), NFSFT_F_HAT_SIZE, nfsft_finalize(), nfsft_forget(), NFSFT_INDEX, NFSFT_NO_FAST_ALGORITHM, nfsft_precompute(), nfsft_trafo(), NFSFT_USE_DPT, NFSFT_USE_NDFT, poissonKernel(), PRE_PHI_HUT, PRE_PSI, singularityKernel(), smbi(), X, and nfsft_plan::x.

Macros

Enumerations

Functions

Detailed Description

Function Documentation