NAG C Library Function Document

nag_fft_3d (c06pxc)

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 j1 j2 j3 , for j1=0,1,,n1-1, j2=0,1,,n2-1 and j3=0,1,,n3-1.
The discrete Fourier transform is here defined by
z^ k1 k2 k3 = 1 n1 n2 n3 j1=0 n1-1 j2=0 n2-1 j3=0 n3-1 z j1 j2 j3 × exp ±2πi j1 k1 n1 + j2 k2 n2 + j3 k3 n3 ,  
where k1=0,1,,n1-1 , k2=0,1,,n2-1  and k3=0,1,,n3-1 .
(Note the scale factor of 1 n1 n2 n3  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, 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:     n1 IntegerInput
On entry: n1, the first dimension of the transform.
Constraint: n11.
3:     n2 IntegerInput
On entry: n2, the second dimension of the transform.
Constraint: n21.
4:     n3 IntegerInput
On entry: n3, the third dimension of the transform.
Constraint: n31.
5:     x[ n1×n2×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 j1 j2 j3  is stored in x[ j1 + n1 j2 + n1 n2 j3 ].
On exit: the corresponding elements of the computed transform.
6:     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.
NE_INT
On entry, n1=value.
Constraint: n11.
On entry, n2=value.
Constraint: n21.
On entry, n3=value.
Constraint: n31.
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 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_3d (c06pxc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_fft_3d (c06pxc) 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 n1 n2 n3 × log n1 n2 n3 , but also depends on the factorization of the individual dimensions n1 , n2  and n3 . nag_fft_3d (c06pxc) is faster if the only prime factors are 2, 3 or 5; and fastest of all if they are powers of 2.

10
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.

10.1
Program Text

Program Text (c06pxce.c)

10.2
Program Data

Program Data (c06pxce.d)

10.3
Program Results

Program Results (c06pxce.r)