NAG Library Function Document

nag_sum_fft_complex_2d (c06puc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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_{j_{1} j_{2}}

, for

j_{1} = 0, 1, \dots, m - 1

and

j_{2} = 0, 1, \dots, n - 1

The discrete Fourier transform is here defined by

{\hat{z}}_{k_{1} k_{2}} = \frac{1}{\sqrt{m n}} \sum_{j_{1} = 0}^{m - 1} \sum_{j_{2} = 0}^{n - 1} z_{j_{1} j_{2}} \times \exp (\pm 2 π i (\frac{j_{1} k_{1}}{m} + \frac{j_{2} k_{2}}{n})),

where

k_{1} = 0, 1, \dots, m - 1

and

k_{2} = 0, 1, \dots, n - 1

(Note the scale factor of

\frac{1}{\sqrt{m n}}

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, then direct must be set equal to $Nag_ForwardTransform$ .
If the backward transform is to be computed then 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: $m \geq 1$ .
3: n – IntegerInput: On entry: $n$ , the second dimension of the transform.
Constraint: $n \geq 1$ .
4: x[ $m \times n$ ] – ComplexInput/Output: On entry: the complex data values. $x [m \times j_{2} + j_{1}]$ must contain $z_{j_{1} j_{2}}$ , for $j_{1} = 0, 1, \dots, m - 1$ and $j_{2} = 0, 1, \dots, n - 1$ .

On exit: the corresponding elements of the computed transform.
5: fail – NagError *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.
$"⟨value⟩"$ is an invalid value of direct.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
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 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken is approximately proportional to

m n \times \log (m n)

, 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

5

; and fastest of all if they are powers of

2

. This function internally allocates a workspace of

m n + 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.

NAG Library Function Documentnag_sum_fft_complex_2d (c06puc)