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fastsumS2_matlab
[Fast summation of radial functions on the sphere]


Defines

#define SYMBOL_ABEL_POISSON(k, h)   (pow(h,k))
 Macro for the Fourier-Legendre coefficients of the Abel-Poisson kernel.
#define SYMBOL_SINGULARITY(k, h)   ((2.0/(2*k+1))*pow(h,k))
 Macro for the Fourier-Legendre coefficients of the singularity kernel.
#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

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.
double poissonKernel (const double x, const double h)
 Evaluates the Poisson kernel $Q_h: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .
double singularityKernel (const double x, const double h)
 Evaluates the singularity kernel $S_h: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .
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 $x \in [-1,1]$ .
double gaussianKernel (const double x, const double sigma)
 Evaluates the spherical Gaussian kernel $G_\sigma: [-1,1] \rightarrow \mathbb{R}$ at a node $x \in [-1,1]$ .
int main (int argc, char **argv)
 The main program.

Function Documentation

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.

  • 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 76 of file fastsumS2.c.

Referenced by main().

double poissonKernel (  const double  x,
const double  h
) 
 

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

  • x The node $x \in [-1,1]$
  • h The parameter $h \in (0,1)$
Returns:
The value of the Poisson kernel $Q_h(x)$ at the node $x$
Author:
Jens Keiner

Definition at line 93 of file fastsumS2.c.

References PI.

Referenced by main().

double singularityKernel (  const double  x,
const double  h
) 
 

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

  • x The node $x \in [-1,1]$
  • h The parameter $h \in (0,1)$
Returns:
The value of the Poisson kernel $S_h(x)$ at the node $x$
Author:
Jens Keiner

Definition at line 109 of file fastsumS2.c.

References PI.

Referenced by main().

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 $x \in [-1,1]$ .

  • 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 $x$
Author:
Jens Keiner

Definition at line 127 of file fastsumS2.c.

Referenced by main().

double gaussianKernel (  const double  x,
const double  sigma
) 
 

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

  • 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 $x$
Author:
Jens Keiner

Definition at line 145 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 160 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(), ndsft_adjoint(), ndsft_trafo(), nfft_error_l_infty_1_complex(), NFFT_MAX, nfft_second(), nfsft_adjoint(), NFSFT_F_HAT_SIZE, nfsft_finalize(), nfsft_forget(), NFSFT_INDEX, nfsft_init_guru(), NFSFT_NO_FAST_ALGORITHM, nfsft_precompute(), nfsft_precompute_x(), nfsft_trafo(), NFSFT_USE_DPT, NFSFT_USE_NDFT, PI, poissonKernel(), PRE_PHI_HUT, PRE_PSI, singularityKernel(), SYMBOL_ABEL_POISSON, SYMBOL_SINGULARITY, and nfsft_plan::x.

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