NAG Library Function Document
nag_fft_multid_full (c06pjc) computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.
||nag_fft_multid_full (Nag_TransformDirection direct,
const Integer nd,
nag_fft_multid_full (c06pjc) computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values , where , and so on. Thus the individual dimensions are , and the total number of data values is .
The discrete Fourier transform is here defined (e.g., for
. 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
The extension to higher dimensions is obvious. (Note the scale factor of in this definition.)
A call of nag_fft_multid_full (c06pjc) with followed by a call with 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 varying most rapidly).
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)
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
direct – Nag_TransformDirectionInput
: if the forward transform as defined in Section 3
is to be computed, then direct
must be set equal to
If the backward transform is to be computed then direct
must be set equal to
ndim – IntegerInput
On entry: , the number of dimensions (or variables) in the multivariate data.
nd[ndim] – const IntegerInput
: the elements of nd
must contain the dimensions of the ndim
variables; that is,
must contain the dimension of the
, for .
n – IntegerInput
On entry: , the total number of data values.
must equal the product of the first ndim
elements of the array nd
x[n] – ComplexInput/Output
: the complex data values. Data values are stored in x
using column-major ordering for storing multidimensional arrays; that is,
is stored in
On exit: the corresponding elements of the computed transform.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
must equal the product of the dimensions held in array nd
, product of nd
On entry and .
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
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 Users' Note
for your implementation for any additional implementation-specific information.
The time taken is approximately proportional to , but also depends on the factorization of the individual dimensions . nag_fft_multid_full (c06pjc) is faster if the only prime factors are , or ; and fastest of all if they are powers of .
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.
10.1 Program Text
Program Text (c06pjce.c)
10.2 Program Data
Program Data (c06pjce.d)
10.3 Program Results
Program Results (c06pjce.r)