NAG Library Routine Document
C06RBF
1 Purpose
C06RBF computes the discrete Fourier cosine transforms of $m$ sequences of real data values.
2 Specification
INTEGER 
M, N, IFAIL 
REAL (KIND=nag_wp) 
X(M*(N+3)), WORK(*) 

3 Description
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
(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 selfsorting algorithm, described in
Temperton (1983), together with pre and postprocessing stages described in
Swarztrauber (1982). Special coding is provided for the factors
$2$,
$3$,
$4$ and
$5$.
4 References
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 mixedradix real Fourier transforms J. Comput. Phys. 52 340–350
5 Parameters
 1: M – INTEGERInput
On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{M}}\ge 1$.
 2: 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: X(${\mathbf{M}}\times \left({\mathbf{N}}+3\right)$) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the data must be stored in
X as if in a twodimensional 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
$\mathit{p}=1,2,\dots ,m$, then the first
$m\left(n+1\right)$ elements of the array
X must contain the values
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 twodimensional 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
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for
$\mathit{k}=0,1,\dots ,n$ and
$\mathit{p}=1,2,\dots ,m$, then the
$m\left(n+3\right)$ elements of the array
X contain the values
 4: WORK($*$) – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array
WORK
must be at least
${\mathbf{M}}\times {\mathbf{N}}+2\times {\mathbf{N}}+15$.
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: 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,  ${\mathbf{N}}<1$. 
 ${\mathbf{IFAIL}}=3$

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).
The time taken by C06RBF is approximately proportional to $nm\mathrm{log}n$, 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.
9 Example
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.
9.1 Program Text
Program Text (c06rbfe.f90)
9.2 Program Data
Program Data (c06rbfe.d)
9.3 Program Results
Program Results (c06rbfe.r)