NAG CL Interface
c06pvc (fft_​real_​2d)

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1 Purpose

c06pvc computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values.

2 Specification

#include <nag.h>
void  c06pvc (Integer m, Integer n, const double x[], Complex y[], NagError *fail)
The function may be called by the names: c06pvc or nag_sum_fft_real_2d.

3 Description

c06pvc computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values xj1j2, for j1=0,1,,m-1 and j2=0,1,,n-1.
The discrete Fourier transform is here defined by
z^ k1 k2 = 1mn j1=0 m-1 j2=0 n-1 x j1 j2 × exp(-2πi( j1 k1 m + j2 k2 n )) ,  
where k1=0,1,,m-1 and k2=0,1,,n-1. (Note the scale factor of 1mn in this definition.)
The transformed values z^ k1 k2 are complex. Because of conjugate symmetry (i.e., z^ k1 k2 is the complex conjugate of z^ (m-k1) (n-k2) ), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of c06pvc followed by a call of c06pwc will restore the original data.
This function performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).

4 References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5 Arguments

1: m Integer Input
On entry: m, the first dimension of the transform.
Constraint: m1.
2: n Integer Input
On entry: n, the second dimension of the transform.
Constraint: n1.
3: x[ m×n ] const double Input
On entry: the real input dataset x, where x j1 j2 is stored in x[ j2 × m+ j1], for j1=0,1,,m-1 and j2=0,1,,n-1.
4: y[ (m/2+1)×n ] Complex Output
On exit: the complex output dataset z^, where z^ k1 k2 is stored in y[ k2 × (m/2+1) + k1], for k1=0,1,,m/2 and k2=0,1,,n-1. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Some indication of accuracy can be obtained by performing a forward transform using c06pvc and a backward transform using c06pwc, and comparing the results with the original sequence (in exact arithmetic they would be identical).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
c06pvc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pvc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by c06pvc is approximately proportional to mn log(mn) , but also depends on the factors of m and n. c06pvc is fastest if the only prime factors of m and n are 2, 3 and 5, and is particularly slow if m or n is a large prime, or has large prime factors.
Workspace is internally allocated by c06pvc. The total size of these arrays is approximately proportional to mn.

10 Example

This example reads in a bivariate sequence of real data values and prints their discrete Fourier transforms as computed by c06pvc. Inverse transforms are then calculated by calling c06pwc showing that the original sequences are restored.

10.1 Program Text

Program Text (c06pvce.c)

10.2 Program Data

Program Data (c06pvce.d)

10.3 Program Results

Program Results (c06pvce.r)