NAG FL Interface
c06rcf (fft_​real_​qtrsine_​simple)

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1 Purpose

c06rcf computes the discrete quarter-wave Fourier sine transforms of m sequences of real data values.

2 Specification

Fortran Interface
Subroutine c06rcf ( direct, m, n, x, work, ifail)
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
C Header Interface
#include <nag.h>
void  c06rcf_ (const char *direct, const Integer *m, const Integer *n, double x[], double work[], Integer *ifail, const Charlen length_direct)
The routine may be called by the names c06rcf or nagf_sum_fft_real_qtrsine_simple.

3 Description

Given m sequences of n real data values xjp , for j=1,2,,n and p=1,2,,m, c06rcf simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by
x^kp = 1n ( j=1 n-1 xjp×sin(j(2k-1)π2n)+12 (-1) k-1 xnp) ,   if ​ direct='F' ,  
or its inverse
xkp = 2n j=1 n x^ j p × sin((2j-1)kπ2n) ,   if ​ direct='B' ,  
where k=1,2,,n and p=1,2,,m.
(Note the scale factor 1n in this definition.)
A call of c06rcf with direct='F' followed by a call with direct='B' will restore the original data.
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 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.

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 mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5 Arguments

1: 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: direct='F' or 'B'.
2: m Integer Input
On entry: m, the number of sequences to be transformed.
Constraint: m1.
3: n Integer Input
On entry: n, the number of real values in each sequence.
Constraint: n1.
4: x( m×(n+2) ) Real (Kind=nag_wp) array Input/Output
On entry: the data must be stored in x as if in a two-dimensional array of dimension (1:m,1:n+2); each of the m sequences is stored in a row of the array. In other words, if the data values of the pth sequence to be transformed are denoted by xjp, for j=1,2,,n and p=1,2,,m, the first mn elements of the array x must contain the values
x11 , x12 ,, x1m , x21 , x22 ,, x2m ,, xn1 , xn2 ,, xnm .  
The (n+1)th and (n+2)th elements of each row x n+1 p , x n+2 p , for p=1,2,,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 sine transforms stored as if in a two-dimensional array of dimension (1:m,1:n+2). Each of the m transforms is stored in a row of the array, overwriting the corresponding original sequence. If the n components of the pth quarter-wave sine transform are denoted by x^kp, for k=1,2,,n and p=1,2,,m, the m(n+2) elements of the array x contain the values
x^11 , x^12 ,, x^1m , x^21 , x^22 ,, x^2m ,, x^n1 , x^n2 ,, x^nm , 0 , 0 ,, 0  (2m times) .  
5: work(*) Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least m×n + 2×n + 2×m + 15.
The workspace requirements as documented for c06rcf may be an overestimate in some implementations.
On exit: work(1) contains the minimum workspace required for the current values of m and n with this implementation.
6: 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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
On entry, direct=value.
Constraint: direct='F' or 'B'.
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.
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.
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.
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

Background information to multithreading can be found in the Multithreading documentation.
c06rcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06rcf 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.

9 Further Comments

The time taken by c06rcf is approximately proportional to nm log(n), but also depends on the factors of n. c06rcf 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.

10 Example

This example reads in sequences of real data values and prints their quarter-wave sine transforms as computed by c06rcf with direct='F'. It then calls the routine again with direct='B' and prints the results which may be compared with the original data.

10.1 Program Text

Program Text (c06rcfe.f90)

10.2 Program Data

Program Data (c06rcfe.d)

10.3 Program Results

Program Results (c06rcfe.r)