nag_sum_fft_hermitian_2d (c06pwc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_sum_fft_hermitian_2d (c06pwc)

+ Contents

    1  Purpose
    7  Accuracy
    10  Example

1  Purpose

nag_sum_fft_hermitian_2d (c06pwc) computes the two-dimensional inverse discrete Fourier transform of a bivariate Hermitian sequence of complex data values.

2  Specification

#include <nag.h>
#include <nagc06.h>
void  nag_sum_fft_hermitian_2d (Integer m, Integer n, const Complex y[], double x[], NagError *fail)

3  Description

nag_sum_fft_hermitian_2d (c06pwc) computes the two-dimensional inverse discrete Fourier transform of a bivariate Hermitian sequence of complex data values zj1j2, for j1=0,1,,m-1 and j2=0,1,,n-1.
The discrete Fourier transform is here defined by
x^ 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.)
Because the input data satisfies conjugate symmetry (i.e., z k1 k2  is the complex conjugate of z m-k1 k2 , the transformed values x^ k1 k2  are real.
A call of nag_sum_fft_real_2d (c06pvc) followed by a call of nag_sum_fft_hermitian_2d (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:     mIntegerInput
On entry: m, the first dimension of the transform.
Constraint: m1.
2:     nIntegerInput
On entry: n, the second dimension of the transform.
Constraint: n1.
3:     y[ m/2+1×n ]const ComplexInput
On entry: the Hermitian sequence of complex input dataset z, where z j1 j2  is stored in y[ j2 × m/2+1 + j1] , for j1=0,1,,m/2 and j2=0,1,,n-1.
4:     x[ m×n ]doubleOutput
On exit: the real output dataset x^, where x^ k1 k2  is stored in x[ k2 × m+ k1], for k1=0,1,,m-1 and k2=0,1,,n-1.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.

7  Accuracy

Some indication of accuracy can be obtained by performing a forward transform using nag_sum_fft_real_2d (c06pvc) and a backward transform using nag_sum_fft_hermitian_2d (c06pwc), and comparing the results with the original sequence (in exact arithmetic they would be identical).

8  Parallelism and Performance

nag_sum_fft_hermitian_2d (c06pwc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_hermitian_2d (c06pwc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The time taken by nag_sum_fft_hermitian_2d (c06pwc) is approximately proportional to mn logmn , but also depends on the factors of m and n. nag_sum_fft_hermitian_2d (c06pwc) 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 nag_sum_fft_hermitian_2d (c06pwc). The total size of these arrays is approximately proportional to mn.

10  Example

See Section 10 in nag_sum_fft_real_2d (c06pvc).

nag_sum_fft_hermitian_2d (c06pwc) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014