NAG Library Routine Document
C06HDF
1 Purpose
C06HDF computes the discrete quarterwave Fourier cosine transforms of $m$ sequences of real data values. This routine is designed to be particularly efficient on vector processors.
2 Specification
INTEGER 
M, N, IFAIL 
REAL (KIND=nag_wp) 
X(M*N), TRIG(2*N), WORK(M*N) 
CHARACTER(1) 
DIRECT, INIT 

3 Description
Given
$m$ sequences of
$n$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$, C06HDF simultaneously calculates the quarterwave Fourier cosine transforms of all the sequences defined by
or its inverse
for
$k=0,1,\dots ,n1$ and
$p=1,2,\dots ,m$.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
A call of C06HDF 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)). (See the
C06 Chapter Introduction.)
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
$2$,
$3$,
$4$,
$5$ 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.
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: DIRECT – CHARACTER(1)Input
On entry: if the forward transform as defined in
Section 3 is to be computed, then
DIRECT must be set equal to 'F'.
If the backward transform is to be computed then
DIRECT must be set equal to 'B'.
Constraint:
${\mathbf{DIRECT}}=\text{'F'}$ or $\text{'B'}$.
 2: M – INTEGERInput
On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{M}}\ge 1$.
 3: N – INTEGERInput
On entry: $n$, the number of real values in each sequence.
Constraint:
${\mathbf{N}}\ge 1$.
 4: X(${\mathbf{M}}\times {\mathbf{N}}$) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the data must be stored in
X as if in a twodimensional array of dimension
$\left(1:{\mathbf{M}},0:{\mathbf{N}}1\right)$; 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
$\mathit{j}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$, then the
$mn$ elements of the array
X must contain the values
On exit: the
$m$ quarterwave cosine transforms stored as if in a twodimensional 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 quarterwave cosine transform are denoted by
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for
$\mathit{k}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$, then the
$mn$ elements of the array
X contain the values
 5: 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 C06HAF, C06HBF, C06HCF or C06HDF. 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 $\text{'R'}$.
 6: TRIG($2\times {\mathbf{N}}$) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if
${\mathbf{INIT}}=\text{'S'}$ or
$\text{'R'}$,
TRIG must contain the required trigonometric coefficients calculated in a previous call of the routine. Otherwise
TRIG need not be set.
On exit: contains the required coefficients (computed by the routine if ${\mathbf{INIT}}=\text{'I'}$).
 7: WORK(${\mathbf{M}}\times {\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
 8: 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,  ${\mathbf{N}}<1$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{INIT}}\ne \text{'I'}$, $\text{'S'}$ or $\text{'R'}$. 
 ${\mathbf{IFAIL}}=4$

Not used at this Mark.
 ${\mathbf{IFAIL}}=5$

On entry,  ${\mathbf{INIT}}=\text{'S'}$ or $\text{'R'}$, but the array TRIG and the current value of N are inconsistent. 
 ${\mathbf{IFAIL}}=6$

On entry,  ${\mathbf{DIRECT}}\ne \text{'F'}$ or $\text{'B'}$. 
 ${\mathbf{IFAIL}}=7$

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 C06HDF is approximately proportional to $nm\mathrm{log}n$, but also depends on the factors of $n$. C06HDF 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 quarterwave cosine transforms as computed by C06HDF 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.
9.1 Program Text
Program Text (c06hdfe.f90)
9.2 Program Data
Program Data (c06hdfe.d)
9.3 Program Results
Program Results (c06hdfe.r)