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 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_TransformDirection Input: 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: $m$ – Integer Input: On entry: $m$ , the first dimension of the transform.

Constraint: $m \geq 1$ .
3: $n$ – Integer Input: On entry: $n$ , the second dimension of the transform.

Constraint: $n \geq 1$ .
4: $x [m \times n]$ – Complex Input/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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: 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 subsequent inverse transform 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.

c06puc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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

m n \times \log (m n)

, but also depends on the factorization of the individual dimensions

m

and

n

. 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.

c06pu: FL CL CPP AD PY MB

NAG CL Interfacec06puc (fft_​complex_​2d)

▸▿ 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

NAG CL Interface
c06puc (fft_complex_2d)