NAG Library Function Document

nag_fft_3d (c06pxc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_fft_3d (c06pxc) computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).

2 Specification

#include <nag.h>

#include <nagc06.h>

void	nag_fft_3d (Nag_TransformDirection direct, Integer n1, Integer n2, Integer n3, Complex x[], NagError *fail)

3 Description

nag_fft_3d (c06pxc) computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values

z_{j_{1} j_{2} j_{3}}

, for

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

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

and

j_{3} = 0, 1, \dots, n_{3} - 1

The discrete Fourier transform is here defined by

{\hat{z}}_{k_{1} k_{2} k_{3}} = \frac{1}{\sqrt{n_{1} n_{2} n_{3}}} \sum_{j_{1} = 0}^{n_{1} - 1} \sum_{j_{2} = 0}^{n_{2} - 1} \sum_{j_{3} = 0}^{n_{3} - 1} z_{j_{1} j_{2} j_{3}} \times \exp (\pm 2 π i (\frac{j_{1} k_{1}}{n_{1}} + \frac{j_{2} k_{2}}{n_{2}} + \frac{j_{3} k_{3}}{n_{3}})),

where

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

k_{2} 0, 1, \dots, n_{2} - 1

and

k_{3} 0, 1, \dots, n_{3} - 1

(Note the scale factor of

\frac{1}{\sqrt{n_{1} n_{2} n_{3}}}

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_fft_3d (c06pxc) 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 (see Brigham (1974)).

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: n1 – IntegerInput: On entry: $n_{1}$ , the first dimension of the transform.
Constraint: $n1 \geq 1$ .
3: n2 – IntegerInput: On entry: $n_{2}$ , the second dimension of the transform.
Constraint: $n2 \geq 1$ .
4: n3 – IntegerInput: On entry: $n_{3}$ , the third dimension of the transform.
Constraint: $n3 \geq 1$ .
5: x[ $n1 \times n2 \times n3$ ] – ComplexInput/Output: On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, $z_{j_{1} j_{2} j_{3}}$ is stored in $x [j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3}]$ .

On exit: the corresponding elements of the computed transform.
6: 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.
NE_INT: On entry, $n1 = 〈value〉$ .
Constraint: n1 must have less than $31$ prime factors.; On entry, $n1 = 〈value〉$ .
Constraint: $n1 \geq 1$ .; On entry, $n2 = 〈value〉$ .
Constraint: n2 must have less than $31$ prime factors.; On entry, $n2 = 〈value〉$ .
Constraint: $n2 \geq 1$ .; On entry, $n3 = 〈value〉$ .
Constraint: n3 must have less than $31$ prime factors.; On entry, $n3 = 〈value〉$ .
Constraint: $n3 \geq 1$ .
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 Further Comments

The time taken is approximately proportional to

n_{1} n_{2} n_{3} \times \log (n_{1} n_{2} n_{3})

, but also depends on the factorization of the individual dimensions

n_{1}

n_{2}

and

n_{3}

. nag_fft_3d (c06pxc) is faster if the only prime factors are

2

3

5

; and fastest of all if they are powers of

2

9 Example

This example reads in a trivariate sequence of complex data values and prints the three-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_3d (c06pxc)