NAG Library Routine Document

c06prf (fft_complex_1d_multi_row)


c06prf computes the discrete Fourier transforms of m sequences, each containing n complex data values.


Fortran Interface
Subroutine c06prf ( direct, m, n, x, work, ifail)
Integer, Intent (In):: m, n
Integer, Intent (Inout):: ifail
Complex (Kind=nag_wp), Intent (Inout):: x(m*n), work(*)
Character (1), Intent (In):: direct
C Header Interface
#include <nagmk26.h>
void  c06prf_ (const char *direct, const Integer *m, const Integer *n, Complex x[], Complex work[], Integer *ifail, const Charlen length_direct)


Given m sequences of n complex data values zjp , for j=0,1,,n-1 and p=1,2,,m, c06prf simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
z^kp = 1n j=0 n-1 zjp × exp ±i 2πjk n ,   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 c06prf 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, 4 and 5.


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


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 – IntegerInput
On entry: m, the number of sequences to be transformed.
Constraint: m1.
3:     n – IntegerInput
On entry: n, the number of complex values in each sequence.
Constraint: n1.
4:     x m×n – Complex (Kind=nag_wp) arrayInput/Output
On entry: the complex data must be stored in x as if in a two-dimensional array of dimension 1:m,0:n-1; each of the m sequences is stored in a row of each array. In other words, if the elements of the pth sequence to be transformed are denoted by zjp, for j=0,1,,n-1, xj×m+p must contain zjp.
On exit: is overwritten by the complex transforms.
5:     work* – Complex (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least m×n + 2×n + 15.
The workspace requirements as documented for c06prf may be an overestimate in some implementations.
On exit: the real part of work1 contains the minimum workspace required for the current values of m and n with this implementation.
6:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 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 argument, the recommended value is 0. 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).

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 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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).

Parallelism and Performance

c06prf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06prf 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.

Further Comments

The time taken by c06prf is approximately proportional to nm logn, but also depends on the factors of n. c06prf 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.


This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by c06prf with direct='F'). Inverse transforms are then calculated using c06prf with direct='B' and printed out, showing that the original sequences are restored.

Program Text

Program Text (c06prfe.f90)

Program Data

Program Data (c06prfe.d)

Program Results

Program Results (c06prfe.r)