# NAG Library Routine Document

## 1Purpose

c06fqf computes the discrete Fourier transforms of $m$ Hermitian sequences, each containing $n$ complex data values. This routine is designed to be particularly efficient on vector processors.

## 2Specification

Fortran Interface
 Subroutine c06fqf ( m, n, x, init, trig, work,
 Integer, Intent (In) :: m, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(m*n), trig(2*n) Real (Kind=nag_wp), Intent (Out) :: work(m*n) Character (1), Intent (In) :: init
#include <nagmk26.h>
 void c06fqf_ (const Integer *m, const Integer *n, double x[], const char *init, double trig[], double work[], Integer *ifail, const Charlen length_init)

## 3Description

Given $m$ Hermitian sequences of $n$ complex data values ${z}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06fqf simultaneously calculates the Fourier transforms of all the sequences defined by
 $x^kp = 1n ∑ j=0 n-1 zjp × exp -i 2πjk n , k= 0, 1, …, n-1 ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
The transformed values are purely real (see also the C06 Chapter Introduction).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
 $x^kp = 1n ∑ j=0 n-1 zjp × exp +i 2πjkn .$
To compute this form, this routine should be preceded by forming the complex conjugates of the ${\stackrel{^}{z}}_{k}^{p}$; that is $x\left(\mathit{k}\right)=-x\left(\mathit{k}\right)$, for $\mathit{k}=\left(n/2+1\right)×m+1,\dots ,m×n$.
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 $2$, $3$, $4$, $5$ and $6$. This routine is designed to be particularly efficient on vector processors, and it becomes especially fast as $m$, the number of transforms to be computed in parallel, increases.

## 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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of data values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{x}\left({\mathbf{m}}×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the data must be stored in x as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}-1\right)$; each of the $m$ sequences is stored in a row of the array in Hermitian form. If the $n$ data values ${z}_{j}^{p}$ are written as ${x}_{j}^{p}+i{y}_{j}^{p}$, then for $0\le j\le n/2$, ${x}_{j}^{p}$ is contained in ${\mathbf{x}}\left(p,j\right)$, and for $1\le j\le \left(n-1\right)/2$, ${y}_{j}^{p}$ is contained in ${\mathbf{x}}\left(p,n-j\right)$. (See also Section 2.1.2 in the C06 Chapter Introduction.)
On exit: the components of the $m$ discrete Fourier transforms, stored as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}-1\right)$. Each of the $m$ transforms is stored as a row of the array, overwriting the corresponding original sequence. If the $n$ components of the discrete Fourier transform are denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{p}$, for $\mathit{k}=0,1,\dots ,n-1$, the $mn$ elements of the array x contain the values
 $x^01 , x^02 ,…, x^0m , x^11 , x^12 ,…, x^1m ,…, x^ n-1 1 , x^ n-1 2 ,…, x^ n-1 m .$
4:     $\mathbf{init}$ – Character(1)Input
On entry: indicates whether trigonometric coefficients are to be calculated.
${\mathbf{init}}=\text{'I'}$
Calculate the required trigonometric coefficients for the given value of $n$, and store in the array trig.
${\mathbf{init}}=\text{'S'}$ or $\text{'R'}$
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of c06fpf or c06fqf. The routine performs a simple check that the current value of $n$ is consistent with the values stored in trig.
Constraint: ${\mathbf{init}}=\text{'I'}$, $\text{'S'}$ or $\text{'R'}$.
5:     $\mathbf{trig}\left(2×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{init}}=\text{'S'}$ or $\text{'R'}$, trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.
On exit: contains the required coefficients (computed by the routine if ${\mathbf{init}}=\text{'I'}$).
6:     $\mathbf{work}\left({\mathbf{m}}×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{init}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{init}}=\text{'I'}$, $\text{'S'}$ or $\text{'R'}$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{init}}=〈\mathit{\text{value}}〉$ but n and trig array incompatible.
${\mathbf{ifail}}=-99$
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

c06fqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fqf 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 c06fqf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06fqf 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 which are assumed to be Hermitian sequences of complex data stored in Hermitian form. The sequences are expanded into full complex form and printed. The discrete Fourier transforms are then computed (using c06fqf) and printed out. Inverse transforms are then calculated by conjugating and calling c06fpf showing that the original sequences are restored.

### 10.1Program Text

Program Text (c06fqfe.f90)

### 10.2Program Data

Program Data (c06fqfe.d)

### 10.3Program Results

Program Results (c06fqfe.r)