NAG FL Interface
c06rgf (fft_qtrsine)
1
Purpose
c06rgf computes the discrete quarterwave Fourier sine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
idir, m, n 
Integer, Intent (Inout) 
:: 
ifail 
Real (Kind=nag_wp), Intent (Inout) 
:: 
x(n,m) 

C Header Interface
#include <nag.h>
void 
c06rgf_ (const Integer *idir, const Integer *m, const Integer *n, double x[], Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
c06rgf_ (const Integer &idir, const Integer &m, const Integer &n, double x[], Integer &ifail) 
}

The routine may be called by the names c06rgf or nagf_sum_fft_qtrsine.
3
Description
Given
$m$ sequences of
$n$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=1,2,\dots ,n$ and
$\mathit{p}=1,2,\dots ,m$,
c06rgf simultaneously calculates the quarterwave Fourier sine transforms of all the sequences defined by
or its inverse
where
$k=1,2,\dots ,n$ and
$p=1,2,\dots ,m$.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
A call of c06rgf with ${\mathbf{idir}}=1$ followed by a call with ${\mathbf{idir}}=1$ will restore the original data.
The two transforms are also known as typeIII DST and typeII DST, respectively.
The transform calculated by this routine can be used to solve Poisson's equation when the solution is specified at the left boundary, and the derivative of 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 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
Arguments

1:
$\mathbf{idir}$ – Integer
Input

On entry: indicates the transform, as defined in
Section 3, to be computed.
 ${\mathbf{idir}}=1$
 Forward transform.
 ${\mathbf{idir}}=1$
 Inverse transform.
Constraint:
${\mathbf{idir}}=1$ or $1$.

2:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.

3:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of real values in each sequence.
Constraint:
${\mathbf{n}}\ge 1$.

4:
$\mathbf{x}\left({\mathbf{n}},{\mathbf{m}}\right)$ – Real (Kind=nag_wp) array
Input/Output

On entry: the data values of the $\mathit{p}$th sequence to be transformed, denoted by
${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left(j,p\right)$.
On exit: the $n$ components of the $\mathit{p}$th quarterwave sine transform, denoted by
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left(k,p\right)$, overwriting the corresponding original values.

5:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$0$ is recommended.
When the value $\mathbf{1}$ 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
${\mathbf{ifail}}=0$ 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, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 1$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{idir}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{idir}}=1$ or $1$.
 ${\mathbf{ifail}}=4$

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.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
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).
8
Parallelism and Performance
c06rgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 implementationspecific information.
The time taken by c06rgf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06rgf 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. Workspace of order $\mathit{O}\left(n\right)$ is internally allocated by this routine.
10
Example
This example reads in sequences of real data values and prints their quarterwave sine transforms as computed by c06rgf with ${\mathbf{idir}}=1$. It then calls the routine again with ${\mathbf{idir}}=1$ and prints the results which may be compared with the original data.
10.1
Program Text
10.2
Program Data
10.3
Program Results