NAG CL Interface
c06fqc (withdraw_​fft_​hermitian_​1d_​multi_​rfmt)

Note: this function is deprecated and will be withdrawn at Mark 30.2. Replaced by c06pqc.
c06pqc provides a simpler interface for both forward and backward transforms.
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1 Purpose

c06fqc computes the discrete Fourier transforms of m Hermitian sequences, each containing n complex data values.

2 Specification

#include <nag.h>
void  c06fqc (Integer m, Integer n, double x[], const double trig[], NagError *fail)
The function may be called by the names: c06fqc, nag_sum_withdraw_fft_hermitian_1d_multi_rfmt or nag_fft_multiple_hermitian.

3 Description

Given m Hermitian sequences of n complex data values z j p , for j=0,1,,n - 1 and p=1,2,,m, this function simultaneously calculates the Fourier transforms of all the sequences defined by
x ^ k p = 1 n j=0 n-1 z j p exp(-2πijk/n) ,   for ​ k = 0 , 1 , , n - 1 ; p = 1 , 2 , , m .  
(Note the scale factor 1 / n in this definition.)
The transformed values are purely real.
The first call of c06fqc must be preceded by a call to c06gzc to initialize the array trig with trigonometric coefficients according to the value of n.
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
x ^ k p = 1 n j=0 n-1 z j p exp(+2πijk/n) .  
For that form, this function should be preceded by a call to c06gqc to form the complex conjugates of the z ^ j p .
The function uses a variant of the fast Fourier transform algorithm (Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is included for the factors 2, 3, 4, 5 and 6.

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: m Integer Input
On entry: the number of sequences to be transformed, m .
Constraint: m1 .
2: n Integer Input
On entry: the number of data values in each sequence, n .
Constraint: n1 .
3: x[m×n] double Input/Output
On entry: the m Hermitian sequences must be stored consecutively in x in Hermitian form. Sequence 1 should occupy the first n elements of x, sequence 2 the elements n to 2 n - 1 , so that in general sequence p occupies the array elements (p-1) n to pn - 1 . If the n data values z j p are written as x j p + iy j p , then for 0 j n / 2 , x j p should be in array element x[ (p-1) × n + j ] and for 1 j (n-1) / 2 , y j p should be in array element x[ (p-1) × n + n-j ] .
On exit: the components of the m discrete Fourier transforms, stored consecutively. Transform p occupies the elements (p-1) n to pn - 1 of x overwriting the corresponding original sequence; thus if the n components of the discrete Fourier transform are denoted by x ^ k p , for k=0,1,,n - 1, then the mn elements of the array x contain the values
x ^ 0 1 , x ^ 1 1 , , x ^ n-1 1 , x ^ 0 2 , x ^ 1 2 , , x ^ n-1 2 , , x ^ 0 m , x ^ 1 m , , x ^ n-1 m .  
4: trig[2×n] const double Input
On entry: trigonometric coefficients as returned by a call of c06gzc. c06fqc makes a simple check to ensure that trig has been initialized and that the initialization is compatible with the value of n
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
Value of n and trig array are incompatible or trig array not initialized.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.

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

Background information to multithreading can be found in the Multithreading documentation.
c06fqc is not threaded in any implementation.

9 Further Comments

The time taken is approximately proportional to nm log(n) , but also depends on the factors of n . The function 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.

10 Example

This program reads in sequences of real data values which are assumed to be Hermitian sequences of complex data stored in Hermitian form. The sequences are expanded into full complex form using c06gsc and printed. The discrete Fourier transforms are then computed (using c06fqc) and printed out. Inverse transforms are then calculated by calling c06fpc followed by c06gqc showing that the original sequences are restored.

10.1 Program Text

Program Text (c06fqce.c)

10.2 Program Data

Program Data (c06fqce.d)

10.3 Program Results

Program Results (c06fqce.r)