NAG Library Function Document
nag_sum_fft_cosine (c06rfc)
1
Purpose
nag_sum_fft_cosine (c06rfc) computes the discrete Fourier cosine transforms of sequences of real data values. The elements of each sequence and its transform are stored contiguously.
2
Specification
#include <nag.h> |
#include <nagc06.h> |
void |
nag_sum_fft_cosine (Integer m,
Integer n,
double x[],
NagError *fail) |
|
3
Description
Given
sequences of
real data values
, for
and
,
nag_sum_fft_cosine (c06rfc) simultaneously calculates the Fourier cosine transforms of all the sequences defined by
(Note the scale factor
in this definition.)
This transform is also known as type-I DCT.
Since the Fourier cosine transform defined above is its own inverse, two consecutive calls of nag_sum_fft_cosine (c06rfc) will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at both left and right boundaries (see
Swarztrauber (1977)).
The function uses a variant of the fast Fourier transform (FFT) algorithm (see
Brigham (1974)) known as the Stockham self-sorting algorithm, described in
Temperton (1983), together with pre- and post-processing stages described in
Swarztrauber (1982). Special coding is provided for the factors
,
,
and
.
4
References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350
5
Arguments
- 1:
– IntegerInput
-
On entry: , the number of sequences to be transformed.
Constraint:
.
- 2:
– IntegerInput
-
On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is .
Constraint:
.
- 3:
– doubleInput/Output
-
On entry: the data sequences to be transformed. The data values of the th sequence to be transformed, denoted by
, for and , must be stored in .
On exit: the Fourier cosine transforms, overwriting the corresponding original sequences. The components of the th Fourier cosine transform, denoted by
, for and , are stored in .
- 4:
– 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: .
- 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 subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).
8
Parallelism and Performance
nag_sum_fft_cosine (c06rfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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_cosine (c06rfc) is approximately proportional to , but also depends on the factors of . nag_sum_fft_cosine (c06rfc) is fastest if the only prime factors of are , and , and is particularly slow if is a large prime, or has large prime factors.
This function internally allocates a workspace of order double values.
10
Example
This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by nag_sum_fft_cosine (c06rfc)). It then calls nag_sum_fft_cosine (c06rfc) again and prints the results which may be compared with the original sequence.
10.1
Program Text
Program Text (c06rfce.c)
10.2
Program Data
Program Data (c06rfce.d)
10.3
Program Results
Program Results (c06rfce.r)