NAG Library Function Document
nag_sum_fft_hermitian_3d (c06pzc)
1
Purpose
nag_sum_fft_hermitian_3d (c06pzc) computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values.
2
Specification
#include <nag.h> |
#include <nagc06.h> |
void |
nag_sum_fft_hermitian_3d (Integer n1,
Integer n2,
Integer n3,
const Complex y[],
double x[],
NagError *fail) |
|
3
Description
nag_sum_fft_hermitian_3d (c06pzc) computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values , for , and .
The discrete Fourier transform is here defined by
where
,
and
. (Note the scale factor of
in this definition.)
Because the input data satisfies conjugate symmetry (i.e., is the complex conjugate of ), the transformed values are real.
A call of
nag_sum_fft_real_3d (c06pyc) followed by a call of
nag_sum_fft_hermitian_3d (c06pzc) 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).
4
References
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
5
Arguments
- 1:
– IntegerInput
-
On entry: , the first dimension of the transform.
Constraint:
.
- 2:
– IntegerInput
-
On entry: , the second dimension of the transform.
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the third dimension of the transform.
Constraint:
.
- 4:
– const ComplexInput
-
Note: the dimension,
dim, of the array
y
must be at least
.
On entry: the Hermitian sequence of complex input dataset , where
is stored in , for , and .
- 5:
– doubleOutput
-
On exit: the real output dataset , where
is stored in , for , and .
- 6:
– 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 had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- 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 forward transform using
nag_sum_fft_real_3d (c06pyc) and a backward transform using
nag_sum_fft_hermitian_3d (c06pzc), and comparing the results with the original sequence (in exact arithmetic they would be identical).
8
Parallelism and Performance
nag_sum_fft_hermitian_3d (c06pzc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_hermitian_3d (c06pzc) 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.
The time taken by nag_sum_fft_hermitian_3d (c06pzc) is approximately proportional to , but also depends on the factors of , and . nag_sum_fft_hermitian_3d (c06pzc) is fastest if the only prime factors of , and are , and , and is particularly slow if one of the dimensions is a large prime, or has large prime factors.
Workspace is internally allocated by nag_sum_fft_hermitian_3d (c06pzc). The total size of these arrays is approximately proportional to .
10
Example
See
Section 10 in
nag_sum_fft_real_3d (c06pyc).