NAG C Library Function Document

nag_sum_fft_complex_2d (c06puc)

1
Purpose

nag_sum_fft_complex_2d (c06puc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values (using complex data type).

2
Specification

#include <nag.h>
#include <nagc06.h>
void  nag_sum_fft_complex_2d (Nag_TransformDirection direct, Integer m, Integer n, Complex x[], NagError *fail)

3
Description

nag_sum_fft_complex_2d (c06puc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values z j1 j2 , 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 z 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 minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of nag_sum_fft_complex_2d (c06puc) with direct=Nag_ForwardTransform followed by a call with direct=Nag_BackwardTransform 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) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

5
Arguments

1:     direct Nag_TransformDirectionInput
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to Nag_ForwardTransform.
If the backward transform is to be computed, direct must be set equal to Nag_BackwardTransform.
Constraint: direct=Nag_ForwardTransform or Nag_BackwardTransform.
2:     m IntegerInput
On entry: m, the first dimension of the transform.
Constraint: m1.
3:     n IntegerInput
On entry: n, the second dimension of the transform.
Constraint: n1.
4:     x[ m×n ] ComplexInput/Output
On entry: the complex data values. x[m×j2+j1] must contain z j1 j2 , for j1=0,1,,m-1 and j2=0,1,,n-1.
On exit: the corresponding elements of the computed transform.
5:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
NE_INTERNAL_ERROR
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.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

8
Parallelism and Performance

nag_sum_fft_complex_2d (c06puc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_2d (c06puc) 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 is approximately proportional to mn × logmn , but also depends on the factorization of the individual dimensions m and n. nag_sum_fft_complex_2d (c06puc) is faster if the only prime factors are 2, 3 or 5; and fastest of all if they are powers of 2. This function internally allocates a workspace of mn+n+m+30 Complex values.

10
Example

This example reads in a bivariate sequence of complex data values and prints the two-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.

10.1
Program Text

Program Text (c06puce.c)

10.2
Program Data

Program Data (c06puce.d)

10.3
Program Results

Program Results (c06puce.r)