NAG Library Function Document

nag_sum_fft_sine (c06rec)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_sum_fft_sine (c06rec) computes the discrete Fourier sine transforms of m 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_sine (Integer m, Integer n, double x[], NagError *fail)

3
Description

Given m sequences of n-1 real data values xjp , for j=1,2,,n-1 and p=1,2,,m, nag_sum_fft_sine (c06rec) simultaneously calculates the Fourier sine transforms of all the sequences defined by
x^ k p = 2n j=1 n-1 xjp × sin jk πn ,   k= 1, 2, , n-1 ​ and ​ p= 1, 2, , m .  
(Note the scale factor 2n  in this definition.)
This transform is also known as type-I DST.
Since the Fourier sine transform defined above is its own inverse, two consecutive calls of nag_sum_fft_sine (c06rec) will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when 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 2, 3, 4 and 5.

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:     m IntegerInput
On entry: m, the number of sequences to be transformed.
Constraint: m1.
2:     n IntegerInput
On entry: one more than the number of real values in each sequence, i.e., the number of values in each sequence is n-1.
Constraint: n1.
3:     x[n-1×m] doubleInput/Output
On entry: the pth sequence to be transformed, denoted by xjp, for j=1,2,,n-1 and p=1,2,,m, must be stored in x[p-1×n-1+j-1].
On exit: the m Fourier sine transforms, overwriting the corresponding original sequences. The n-1 components of the pth Fourier sine transform, denoted by x^kp, for k=1,2,,n-1 and p=1,2,,m, are stored in x[p-1×n-1+k-1].
4:     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, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
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_sine (c06rec) 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.

9
Further Comments

The time taken by nag_sum_fft_sine (c06rec) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_sine (c06rec) is fastest if the only prime factors of n are 2, 3 and 5, and is particularly slow if n is a large prime, or has large prime factors. Workspace is internally allocated by this function. The total amount of memory allocated is On double values.

10
Example

This example reads in sequences of real data values and prints their Fourier sine transforms (as computed by nag_sum_fft_sine (c06rec)). It then calls nag_sum_fft_sine (c06rec) again and prints the results which may be compared with the original sequence.

10.1
Program Text

Program Text (c06rece.c)

10.2
Program Data

Program Data (c06rece.d)

10.3
Program Results

Program Results (c06rece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017