C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06PRF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

C06PRF computes the discrete Fourier transforms of $m$ sequences, each containing $n$ complex data values.

## 2  Specification

 SUBROUTINE C06PRF ( DIRECT, M, N, X, WORK, IFAIL)
 INTEGER M, N, IFAIL COMPLEX (KIND=nag_wp) X(M*N), WORK(*) CHARACTER(1) DIRECT

## 3  Description

Given $m$ sequences of $n$ complex data values ${z}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, C06PRF simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
 $z^kp = 1n ∑ j=0 n-1 zjp × exp ±i 2πjk n , k= 0, 1, …, n-1 ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of C06PRF with ${\mathbf{DIRECT}}=\text{'F'}$ followed by a call with ${\mathbf{DIRECT}}=\text{'B'}$ will restore the original data.
The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors $2$, $3$, $4$ and $5$.

## 4  References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

## 5  Parameters

1:     DIRECT – CHARACTER(1)Input
On entry: if the forward transform as defined in Section 3 is to be computed, then DIRECT must be set equal to 'F'.
If the backward transform is to be computed then DIRECT must be set equal to 'B'.
Constraint: ${\mathbf{DIRECT}}=\text{'F'}$ or $\text{'B'}$.
2:     M – INTEGERInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{M}}\ge 1$.
3:     N – INTEGERInput
On entry: $n$, the number of complex values in each sequence.
Constraint: ${\mathbf{N}}\ge 1$.
4:     X(${\mathbf{M}}×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayInput/Output
On entry: the complex data must be stored in X as if in a two-dimensional array of dimension $\left(1:{\mathbf{M}},0:{\mathbf{N}}-1\right)$; each of the $m$ sequences is stored in a row of each array. In other words, if the elements of the $p$th sequence to be transformed are denoted by ${z}_{\mathit{j}}^{p}$, for $\mathit{j}=0,1,\dots ,n-1$, then ${\mathbf{X}}\left(j×{\mathbf{M}}+p\right)$ must contain ${z}_{j}^{p}$.
On exit: is overwritten by the complex transforms.
5:     WORK($*$) – COMPLEX (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least ${\mathbf{M}}×{\mathbf{N}}+2×{\mathbf{N}}+15$.
The workspace requirements as documented for C06PRF may be an overestimate in some implementations.
On exit: the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum workspace required for the current values of M and N with this implementation.
6:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{M}}<1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{N}}<1$.
${\mathbf{IFAIL}}=3$
 On entry, ${\mathbf{DIRECT}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{IFAIL}}=4$
 On entry, N has more than $30$ prime factors.
${\mathbf{IFAIL}}=5$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

## 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  Further Comments

The time taken by C06PRF is approximately proportional to $nm\mathrm{log}n$, but also depends on the factors of $n$. C06PRF 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.

## 9  Example

This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by C06PRF with ${\mathbf{DIRECT}}=\text{'F'}$). Inverse transforms are then calculated using C06PRF with ${\mathbf{DIRECT}}=\text{'B'}$ and printed out, showing that the original sequences are restored.

### 9.1  Program Text

Program Text (c06prfe.f90)

### 9.2  Program Data

Program Data (c06prfe.d)

### 9.3  Program Results

Program Results (c06prfe.r)