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_1d (c06pcc) with

direct = Nag_ForwardTransform

followed by a call with

direct = Nag_BackwardTransform

will restore the original data.

nag_sum_fft_complex_1d (c06pcc) 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). If

n

is a large prime number or if

n

contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see Bluestein (1968)), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.

4

References

Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219

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: $x [n]$ – ComplexInput/Output: On entry: $x [j]$ must contain $z_{j}$ , for $j = 0, 1, \dots, n - 1$ .

On exit: the components of the discrete Fourier transform. ${\hat{z}}_{k}$ is contained in $x [k]$ , for $0 \leq k \leq n - 1$ .
3: $n$ – IntegerInput: On entry: $n$ , the number of data values.

Constraint: $n \geq 1$ .
4: $fail$ – 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.

$〈value〉$ is an invalid value of direct.
NE_INT: 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.

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_1d (c06pcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sum_fft_complex_1d (c06pcc) 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

n

. nag_sum_fft_complex_1d (c06pcc) is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

. This function internally allocates a workspace of

2 n + 15

Complex values. When the Bluestein's FFT algorithm is in use, an additional Complex workspace of size approximately

8 n

is allocated.

10

Example

This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by nag_sum_fft_complex_1d (c06pcc) with

direct = Nag_ForwardTransform

). It then performs an inverse transform using nag_sum_fft_complex_1d (c06pcc) with

direct = Nag_BackwardTransform

, and prints the sequence so obtained alongside the original data values.

NAG Library Function Document

nag_sum_fft_complex_1d (c06pcc)

▸▿ 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

2

Specification

3

Description