NAG Library Routine Document

c06fpf (withdraw_fft_real_1d_multi_rfmt)


    1  Purpose
    7  Accuracy


c06fpf computes the discrete Fourier transforms of m sequences, each containing n real data values. This routine is designed to be particularly efficient on vector processors.


Fortran Interface
Subroutine c06fpf ( m, n, x, init, trig, work, ifail)
Integer, Intent (In):: m, n
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (Inout):: x(m*n), trig(2*n)
Real (Kind=nag_wp), Intent (Out):: work(m*n)
Character (1), Intent (In):: init
C Header Interface
#include nagmk26.h
void  c06fpf_ (const Integer *m, const Integer *n, double x[], const char *init, double trig[], double work[], Integer *ifail, const Charlen length_init)


Given m sequences of n real data values xjp , for j=0,1,,n-1 and p=1,2,,m, c06fpf simultaneously calculates the Fourier transforms of all the sequences defined by
z^ k p = 1n j=0 n-1 xjp × exp -i 2πjkn ,   k= 0, 1, , n-1 ​ and ​ p= 1,2,,m .  
(Note the scale factor 1n  in this definition.)
The transformed values z^kp  are complex, but for each value of p the z^kp  form a Hermitian sequence (i.e., z^n-kp  is the complex conjugate of z^kp ), so they are completely determined by mn  real numbers (see also the C06 Chapter Introduction).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term:
z^kp = 1n j=0 n-1 xjp × exp +i 2πjkn .  
To compute this form, this routine should be followed by forming the complex conjugates of the z^kp ; that is xk=-xk, for k=n/2+1×m+1,,m×n.
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 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.


Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350


1:     m – IntegerInput
On entry: m, the number of sequences to be transformed.
Constraint: m1.
2:     n – IntegerInput
On entry: n, the number of real values in each sequence.
Constraint: n1.
3:     x m×n – Real (Kind=nag_wp) arrayInput/Output
On entry: the 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 the array. In other words, if the data values of the pth sequence to be transformed are denoted by xjp, for j=0,1,,n-1, the mn elements of the array x must contain the values
x01 , x02 ,, x0m , x11 , x12 ,, x1m ,, x n-1 1 , x n-1 2 ,, x n-1 m .  
On exit: the m discrete Fourier transforms stored as if in a two-dimensional array of dimension 1:m,0:n-1. Each of the m transforms is stored in a row of the array in Hermitian form, overwriting the corresponding original sequence. If the n components of the discrete Fourier transform z^ k p  are written as akp + i bkp, then for 0 k n/2, akp is contained in xpk, and for 1 k n-1 / 2, bkp is contained in xpn-k. (See also Section 2.1.2 in the C06 Chapter Introduction.)
4:     init – Character(1)Input
On entry: indicates whether trigonometric coefficients are to be calculated.
Calculate the required trigonometric coefficients for the given value of n, and store in the array trig.
init='S' or 'R'
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of c06fpf or c06fqf. The routine performs a simple check that the current value of n is consistent with the values stored in trig.
Constraint: init='I', 'S' or 'R'.
5:     trig 2×n – Real (Kind=nag_wp) arrayInput/Output
On entry: if init='S' or 'R', trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.
On exit: contains the required coefficients (computed by the routine if init='I').
6:     work m×n – Real (Kind=nag_wp) arrayWorkspace
7:     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<1.
On entry,n<1.
On entry,init'I', 'S' or 'R'.
Not used at this Mark.
On entry,init='S' or 'R', but the array trig and the current value of n are inconsistent.
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
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

c06fpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fpf 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 c06fpf is approximately proportional to nm logn, but also depends on the factors of n. c06fpf 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 real data values and prints their discrete Fourier transforms (as computed by c06fpf). The Fourier transforms are expanded into full complex form using and printed. Inverse transforms are then calculated by conjugating and calling c06fqf showing that the original sequences are restored.

Program Text

Program Text (c06fpfe.f90)

Program Data

Program Data (c06fpfe.d)

Program Results

Program Results (c06fpfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017