# NAG Library Routine Document

## 1Purpose

c06rdf computes the discrete quarter-wave Fourier cosine transforms of $m$ sequences of real data values.

## 2Specification

Fortran Interface
 Subroutine c06rdf ( m, n, x, work,
 Integer, Intent (In) :: m, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(m*(n+2)), work(*) Character (1), Intent (In) :: direct
#include nagmk26.h
 void c06rdf_ (const char *direct, const Integer *m, const Integer *n, 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 $\mathit{p}=1,2,\dots ,m$, c06rdf simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
 $x^kp = 1n 12 x0p + ∑ j=1 n-1 xjp × cos j 2k-1 π2n , if ​ direct='F' ,$
or its inverse
 $xkp = 2n ∑ j= 0 n- 1 x^ j p × cos 2j- 1 k π2n , if ​ direct='B' ,$
where $k=0,1,\dots ,n-1$ and $p=1,2,\dots ,m$.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
A call of c06rdf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The transform calculated by this routine can be used to solve Poisson's equation when the derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (see Swarztrauber (1977)).
The routine 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 $5$.

## 4References

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

## 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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of real values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{x}\left({\mathbf{m}}×\left({\mathbf{n}}+2\right)\right)$ – 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 other words, if the data values of the $\mathit{p}$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, the first $mn$ elements of the array x must contain the values
 $x01 , x02 ,…, x0m , x11 , x12 ,…, x1m ,…, x n-1 1 , x n-1 2 ,…, x n-1 m .$
The $\left(n+1\right)$th and $\left(n+2\right)$th elements of each row ${x}_{n}^{\mathit{p}},{x}_{n+1}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$, are required as workspace. These $2m$ elements may contain arbitrary values as they are set to zero by the routine.
On exit: the $m$ quarter-wave cosine 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 in a row of the array, overwriting the corresponding original sequence. If the $n$ components of the $\mathit{p}$th quarter-wave cosine transform are denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, the $m\left(n+2\right)$ elements of the array x contain the values
5:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least ${\mathbf{m}}×{\mathbf{n}}+2×{\mathbf{n}}+2×{\mathbf{m}}+15$.
The workspace requirements as documented for c06rdf 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

c06rdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06rdf 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 c06rdf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06rdf 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 quarter-wave cosine transforms as computed by c06rdf with ${\mathbf{direct}}=\text{'F'}$. It then calls the routine again with ${\mathbf{direct}}=\text{'B'}$ and prints the results which may be compared with the original data.

### 10.1Program Text

Program Text (c06rdfe.f90)

### 10.2Program Data

Program Data (c06rdfe.d)

### 10.3Program Results

Program Results (c06rdfe.r)

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