NAG Library Function Document
nag_sum_fft_real_2d (c06pvc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values.
||nag_sum_fft_real_2d (Integer m,
const double x,
nag_sum_fft_real_2d (c06pvc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values , for and .
The discrete Fourier transform is here defined by
. (Note the scale factor of
in this definition.)
The transformed values are complex. Because of conjugate symmetry (i.e., is the complex conjugate of ), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of nag_sum_fft_real_2d (c06pvc) followed by a call of nag_sum_fft_hermitian_2d (c06pwc)
will restore the original data.
This function performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974)
and Temperton (1983)
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350
m – IntegerInput
On entry: , the first dimension of the transform.
n – IntegerInput
On entry: , the second dimension of the transform.
x – const doubleInput
On entry: the real input dataset , where
is stored in , for and .
y – ComplexOutput
On exit: the complex output dataset , where
is stored in , for and . Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
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, .
On entry, .
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 forward transform using nag_sum_fft_real_2d (c06pvc) and a backward transform using nag_sum_fft_hermitian_2d (c06pwc)
, and comparing the results with the original sequence (in exact arithmetic they would be identical).
8 Parallelism and Performance
nag_sum_fft_real_2d (c06pvc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_real_2d (c06pvc) 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 by nag_sum_fft_real_2d (c06pvc) is approximately proportional to , but also depends on the factors of and . nag_sum_fft_real_2d (c06pvc) is fastest if the only prime factors of and are , and , and is particularly slow if or is a large prime, or has large prime factors.
Workspace is internally allocated by nag_sum_fft_real_2d (c06pvc). The total size of these arrays is approximately proportional to .
This example reads in a bivariate sequence of real data values and prints their discrete Fourier transforms as computed by nag_sum_fft_real_2d (c06pvc). Inverse transforms are then calculated by calling nag_sum_fft_hermitian_2d (c06pwc)
showing that the original sequences are restored.
10.1 Program Text
Program Text (c06pvce.c)
10.2 Program Data
Program Data (c06pvce.d)
10.3 Program Results
Program Results (c06pvce.r)