# NAG Library Routine Document

## 1Purpose

c06pqf computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored column-wise in a complex storage format.

## 2Specification

Fortran Interface
 Subroutine c06pqf ( n, m, x, work,
 Integer, Intent (In) :: n, m Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x((n+2)*m), work(*) Character (1), Intent (In) :: direct
#include nagmk26.h
 void c06pqf_ (const char *direct, const Integer *n, const Integer *m, double x[], double work[], Integer *ifail, const Charlen length_direct)

## 3Description

Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06pqf simultaneously calculates the Fourier transforms of all the sequences defined by
 $z^kp = 1n ∑ j=0 n-1 xjp × exp -i 2πjk n , k0,1,…,n-1 ​ and ​ p=1,2,…,m .$
The transformed values ${\stackrel{^}{z}}_{k}^{p}$ are complex, but for each value of $p$ the ${\stackrel{^}{z}}_{k}^{p}$ form a Hermitian sequence (i.e., ${\stackrel{^}{z}}_{n-k}^{p}$ is the complex conjugate of ${\stackrel{^}{z}}_{k}^{p}$), so they are completely determined by $mn$ real numbers (since ${\stackrel{^}{z}}_{0}^{p}$ is real, as is ${\stackrel{^}{z}}_{n/2}^{p}$ for $n$ even).
Alternatively, given $m$ Hermitian sequences of $n$ complex data values ${z}_{j}^{p}$, this routine simultaneously calculates their inverse (backward) discrete Fourier transforms defined by
 $x^kp = 1n ∑ j=0 n-1 zjp × exp i 2πjk n , k=0,1,…,n-1 ​ and ​ p=1,2,…,m .$
The transformed values ${\stackrel{^}{x}}_{k}^{p}$ are real.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in the above definition.)
A call of c06pqf 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 coding is provided for the factors $2$, $3$, $4$ and $5$.

## 4References

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

## 5Arguments

1:     $\mathbf{direct}$ – Character(1)Input
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.
If the backward transform is to be computed, direct must be set equal to 'B'.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of real or complex values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
4:     $\mathbf{x}\left(\left({\mathbf{n}}+2\right)×{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the data must be stored in x as if in a two-dimensional array of dimension $\left(0:{\mathbf{n}}+1,1:{\mathbf{m}}\right)$; each of the $m$ sequences is stored in a column of the array. In other words, if the data values of the $p$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{p}$, for $\mathit{j}=0,1,\dots ,n-1$:
• if ${\mathbf{direct}}=\text{'F'}$, ${\mathbf{x}}\left(\left(\mathit{p}-1\right)×\left({\mathbf{n}}+2\right)+\mathit{j}\right)$ must contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$;
• if ${\mathbf{direct}}=\text{'B'}$, ${\mathbf{x}}\left(\left(\mathit{p}-1\right)×\left({\mathbf{n}}+2\right)+2×\mathit{k}\right)$ and ${\mathbf{x}}\left(\left(\mathit{p}-1\right)×\left({\mathbf{n}}+2\right)+2×\mathit{k}+1\right)$ must contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$. (Note that for the sequence ${\stackrel{^}{z}}_{k}^{p}$ to be Hermitian, the imaginary part of ${\stackrel{^}{z}}_{0}^{p}$, and of ${\stackrel{^}{z}}_{n/2}^{p}$ for $n$ even, must be zero.)
On exit:
• if ${\mathbf{direct}}=\text{'F'}$ and x is declared with bounds $\left(0:{\mathbf{n}}+1,1:{\mathbf{m}}\right)$ then ${\mathbf{x}}\left(2×\mathit{k},\mathit{p}\right)$ and ${\mathbf{x}}\left(2×\mathit{k}+1,\mathit{p}\right)$ will contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$;
• if ${\mathbf{direct}}=\text{'B'}$ and x is declared with bounds $\left(0:{\mathbf{n}}+1,1:{\mathbf{m}}\right)$ then ${\mathbf{x}}\left(\mathit{j},\mathit{p}\right)$ will contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$.
5:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $\left({\mathbf{m}}+2\right)×{\mathbf{n}}+15$.
The workspace requirements as documented for c06pqf may be an overestimate in some implementations.
On exit: ${\mathbf{work}}\left(1\right)$ contains the minimum workspace required for the current values of m and n with this implementation.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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 $\text{'B'}$.
${\mathbf{ifail}}=4$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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).

## 8Parallelism and Performance

c06pqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pqf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by c06pqf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06pqf 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.

## 10Example

This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by c06pqf with ${\mathbf{direct}}=\text{'F'}$), after expanding them from complex Hermitian form into a full complex sequences.
Inverse transforms are then calculated by calling c06pqf with ${\mathbf{direct}}=\text{'B'}$ showing that the original sequences are restored.

### 10.1Program Text

Program Text (c06pqfe.f90)

### 10.2Program Data

Program Data (c06pqfe.d)

### 10.3Program Results

Program Results (c06pqfe.r)

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