NAG Library Routine Document

1Purpose

c06rbf computes the discrete Fourier cosine transforms of $m$ sequences of real data values.

2Specification

Fortran Interface
 Subroutine c06rbf ( m, n, x, work,
 Integer, Intent (In) :: m, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(m*(n+3)), work(*)
#include <nagmk26.h>
 void c06rbf_ (const Integer *m, const Integer *n, double x[], double work[], Integer *ifail)

3Description

Given $m$ sequences of $n+1$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, c06rbf simultaneously calculates the Fourier cosine transforms of all the sequences defined by
 $x^ k p = 2n 12 x0p + ∑ j=1 n-1 xjp × cos jk πn + 12 -1k xnp , k= 0, 1, …, n ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
Since the Fourier cosine transform is its own inverse, two consecutive calls of c06rbf 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 both left and right boundaries (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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathbf{n}$ – IntegerInput
On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is $n+1$.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{x}\left({\mathbf{m}}×\left({\mathbf{n}}+3\right)\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the data must be stored in x as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}+2\right)$; each of the $m$ sequences is stored in a row of the array. In other words, if the $\left(n+1\right)$ data values of the $\mathit{p}$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, the first $m\left(n+1\right)$ elements of the array x must contain the values
 $x01 , x02 ,…, x0m , x11 , x12 ,…, x1m ,…, xn1 , xn2 ,…, xnm .$
The $\left(n+2\right)$th and $\left(n+3\right)$th elements of each row ${x}_{n+2}^{\mathit{p}},{x}_{n+3}^{\mathit{p}}$, 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 $m$ Fourier cosine transforms stored as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}+2\right)$. Each of the $m$ transforms is stored in a row of the array, overwriting the corresponding original data. If the $\left(n+1\right)$ components of the $\mathit{p}$th Fourier cosine transform are denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, the $m\left(n+3\right)$ elements of the array x contain the values
4:     $\mathbf{work}\left(*\right)$ – 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 c06rbf 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.
5:     $\mathbf{ifail}$ – 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: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.
${\mathbf{ifail}}=-99$
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

c06rbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06rbf 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 c06rbf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06rbf 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 Fourier cosine transforms (as computed by c06rbf). It then calls the routine again and prints the results which may be compared with the original sequence.

10.1Program Text

Program Text (c06rbfe.f90)

10.2Program Data

Program Data (c06rbfe.d)

10.3Program Results

Program Results (c06rbfe.r)