seminaive.c File Reference

Source code for seminaive discrete Legendre transform. More...

#include "s2kit/seminaive.h"
#include <math.h>
#include <stdio.h>
#include <string.h>
#include <fftw3.h>
#include "s2kit/cospml.h"

Go to the source code of this file.

Functions
void	InvDLTSemi (double *coeffs, const int bw, const int m, double *result, double *trans_cos_pml_table, double *sin_values, double *workspace, fftw_plan *plan)
	Computes the inverse Legendre transform using the transposed seminaive algorithm. More...
void	DLTSemi (double *data, const int bw, const int m, double *result, double *workspace, double *cos_pml_table, double *weights, fftw_plan *plan)
	Computes the Legendre transform of data using the seminaive algorithm. More...

Detailed Description

Source code for seminaive discrete Legendre transform.

Source code for computing the Legendre transform where projections are carried out in cosine space, i.e., the seminaive algorithm. For a description, see the related paper or Sean's thesis. // TODO link

Definition in file seminaive.c.

Function Documentation

DLTSemi()

void DLTSemi	(	double *	data,
		const int	bw,
		const int	m,
		double *	result,
		double *	workspace,
		double *	cos_pml_table,
		double *	weights,
		fftw_plan *	plan
	)

Computes the Legendre transform of data using the seminaive algorithm.

Note

See an example of use in test_DLT_semi.c

The forward transform looks like:

l = PCWf 

where f is the data vector, W is a quadrature matrix, C is a cosine transform matrix, P is a matrix full of coefficients of the cosine series representation of each Pml function P(m,m) P(m,m+1) ... P(m,bw-1), and l is the (associated) Legendre series representation of f.

Parameters

data	array of size 2*bw containing function to be transformed, assumes sampling at Chebyshev nodes
bw	bandwidth of the problem
m	order of the problem
result	array of length bw for returning computed Legendre coefficients, contains bw-m coeffs, with the <f,P(m,m)> coefficient located in result[0]
workspace	space for computations of size 4*bw
cos_pml_table	array containing the cosine series coefficients of the Pmls (or Gmls) for this problem
weights	array of size 4*bw which holds the weights for both even (starting at weights[0]) and odd (weights[2*bw]) transforms
plan	pointer to fftw_plan with input array being weighted_data and output being cos_data

Note

This function was written to be a part of the full spherical harmonic transform, so a lot of precomputation has been assumed.

cos_pml_table should be generated with GenerateCosPmlTable(). The offset for a particular Pml can be found with TableOffset(), the size of the table - with the TableSize(). Since the cosine series are always zero-striped, the zeroes have been removed.

weights should be generated by GenerateWeightsForDLT().

Warning

We use the guru interface to execute inverse DFT so we can apply the same plan to different arrays. Thus, the plan should be:
fftw_plan_r2r_1d(2*bw, weighted_data, cos_data, FFTW_REDFT10, FFTW_ESTIMATE)

Definition at line 153 of file seminaive.c.

InvDLTSemi()

void InvDLTSemi	(	double *	coeffs,
		const int	bw,
		const int	m,
		double *	result,
		double *	trans_cos_pml_table,
		double *	sin_values,
		double *	workspace,
		fftw_plan *	plan
	)

Computes the inverse Legendre transform using the transposed seminaive algorithm.

Note

See an example of use in test_DLT_semi.c

Because the Legendre transform is orthogonal, the inverse can be computed by transposing the matrix formulation of the problem.

(see the description of DLTSemi() for forward Legendre transform)

Then to do the inverse:

f = trans(C) trans(P) l 

so we need to transpose the matrix P from the forward transform and then do a cosine series evaluation. No quadrature matrix is necessary. If order m is odd, then there is also a sin factor that needs to be taken into account.

Parameters

coeffs	array of length bw-m containing associated Legendre series coefficients, assumed that first entry contains the P(m,m) coefficient
bw	problem bandwidth
m	order of the associated Legendre functions
result	array of 2*bw samples representing the evaluation of the Legendre series at 2*bw Chebyshev nodes
trans_cos_pml_table	array of the linearized form of trans(P) from the description of this function
sin_values	array is used when m is odd in generation of the values in trans(P)
workspace	space for computations of size 2*bw
plan	pointer to fftw_plan with input array being fcos and output being result

Note

This function was written to be a part of the full spherical harmonic transform, so a lot of precomputation has been assumed.

trans_cos_pml_table should be generated by TransposeCosPmlTable().

Warning

We use the guru interface to execute inverse DFT so we can apply the same plan to different arrays. Thus, the plan should be:
fftw_plan_r2r_1d(2*bw, fcos, result, FFTW_REDFT01, FFTW_ESTIMATE)

Definition at line 56 of file seminaive.c.