NAG FL Interface
c06psf (fft_​complex_​1d_​multi_​col)

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1 Purpose

c06psf computes the discrete Fourier transforms of m sequences, stored as columns of an array, each containing n complex data values.

2 Specification

Fortran Interface
Subroutine c06psf ( direct, n, m, x, work, ifail)
Integer, Intent (In) :: n, m
Integer, Intent (Inout) :: ifail
Complex (Kind=nag_wp), Intent (Inout) :: x(n*m), work(*)
Character (1), Intent (In) :: direct
C Header Interface
#include <nag.h>
void  c06psf_ (const char *direct, const Integer *n, const Integer *m, Complex x[], Complex work[], Integer *ifail, const Charlen length_direct)
The routine may be called by the names c06psf or nagf_sum_fft_complex_1d_multi_col.

3 Description

Given m sequences of n complex data values zjp , for j=0,1,,n-1 and p=1,2,,m, c06psf simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
z^kp=1n j=0 n-1zjp×exp(±i2πjkn) ,  k=0,1,,n-1​ and ​p=1,2,,m.  
(Note the scale factor 1n in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of c06psf with direct='F' followed by a call with direct='B' will restore the original data.
The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors 2, 3 and 5.

4 References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

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: n Integer Input
On entry: n, the number of complex values in each sequence.
Constraint: n1.
3: m Integer Input
On entry: m, the number of sequences to be transformed.
Constraint: m1.
4: x( n×m ) Complex (Kind=nag_wp) array Input/Output
On entry: the complex data values zjp stored in x((p-1)×n+j+1), for j=0,1,,n-1 and p=1,2,,m.
On exit: is overwritten by the complex transforms.
5: work(*) Complex (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least n×m + n + 15.
The workspace requirements as documented for c06psf may be an overestimate in some implementations.
On exit: the real part of 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:
ifail=1
On entry, m=value.
Constraint: m1.
ifail=2
On entry, n=value.
Constraint: n1.
ifail=3
On entry, direct=value.
Constraint: direct='F' or 'B'.
ifail=5
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.
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.
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.
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

c06psf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06psf 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 c06psf is approximately proportional to nm log(n), but also depends on the factors of n. c06psf 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 complex data values and prints their discrete Fourier transforms (as computed by c06psf with direct='F'). Inverse transforms are then calculated using c06psf with direct='B' and printed out, showing that the original sequences are restored.

10.1 Program Text

Program Text (c06psfe.f90)

10.2 Program Data

Program Data (c06psfe.d)

10.3 Program Results

Program Results (c06psfe.r)