. The plus or minus sign in the argument of the exponential terms in the above definition determine the direction of the transform: a minus sign defines the forward direction and a plus sign defines the backward direction.

The extension to higher dimensions is obvious. (Note the scale factor of

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

in this definition.)

A call of nag_fft_multid_full (c06pjc) with

direct = Nag_ForwardTransform

followed by a call with

direct = Nag_BackwardTransform

will restore the original data.

The data values must be supplied in a one-dimensional array using column-major storage ordering of multidimensional data (i.e., with the first subscript

j_{1}

varying most rapidly).

This function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in 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: $ndim$ – IntegerInput: On entry: $m$ , the number of dimensions (or variables) in the multivariate data.

Constraint: $ndim \geq 1$ .
3: $nd [ndim]$ – const IntegerInput: On entry: the elements of nd must contain the dimensions of the ndim variables; that is, $nd [i - 1]$ must contain the dimension of the $i$ th variable.

Constraint: $nd [i - 1] \geq 1$ , for $i = 1, 2, \dots, ndim$ .
4: $n$ – IntegerInput: On entry: $n$ , the total number of data values.

Constraint: n must equal the product of the first ndim elements of the array nd.
5: $x [n]$ – ComplexInput/Output: On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, $z_{j_{1} j_{2} \dots j_{m}}$ is stored in $x [j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3} + \dots]$ .

On exit: the corresponding elements of the computed transform.
6: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 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, $ndim = 〈value〉$ .
Constraint: $ndim \geq 1$ .
NE_INT_2: n must equal the product of the dimensions held in array nd: $n = 〈value〉$ , product of nd elements is $〈value〉$ .

On entry $nd [I - 1] = 〈value〉$ and $I = 〈value〉$ .
Constraint: $nd [I - 1] \geq 1$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
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_fft_multid_full (c06pjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_fft_multid_full (c06pjc) 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

n \times \log (n)

, but also depends on the factorization of the individual dimensions

nd [i - 1]

. nag_fft_multid_full (c06pjc) is faster if the only prime factors are

2

3

5

; and fastest of all if they are powers of

2

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_fft_multid_full (c06pjc)

▸▿ 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 Library Function Document

nag_fft_multid_full (c06pjc)