NAG FL Interface
c06pff (fft_​complex_​multid_​1d)

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1 Purpose

c06pff computes the discrete Fourier transform of one variable in a multivariate sequence of complex data values.

2 Specification

Fortran Interface
Subroutine c06pff ( direct, ndim, l, nd, n, x, work, lwork, ifail)
Integer, Intent (In) :: ndim, l, nd(ndim), n, lwork
Integer, Intent (Inout) :: ifail
Complex (Kind=nag_wp), Intent (Inout) :: x(n)
Complex (Kind=nag_wp), Intent (Out) :: work(lwork)
Character (1), Intent (In) :: direct
C Header Interface
#include <nag.h>
void  c06pff_ (const char *direct, const Integer *ndim, const Integer *l, const Integer nd[], const Integer *n, Complex x[], Complex work[], const Integer *lwork, Integer *ifail, const Charlen length_direct)
The routine may be called by the names c06pff or nagf_sum_fft_complex_multid_1d.

3 Description

c06pff computes the discrete Fourier transform of one variable (the lth say) in a multivariate sequence of complex data values z j1 j2 jm , where j1=0,1,,n1-1 ,   j2=0,1,,n2-1 , and so on. Thus the individual dimensions are n1, n2, , nm , and the total number of data values is n = n1 × n2 ×× nm .
The routine computes n/nl one-dimensional transforms defined by
z^ j1 kl jm = 1nl jl=0 nl-1 z j1 jl jm × exp(± 2 π i jl kl nl ) ,  
where kl = 0 , 1 ,, nl-1 . The plus or minus sign in the argument of the exponential terms in the above definition determine the direction of the transform: a minus sign defines the forward direction and a plus sign defines the backward direction.
(Note the scale factor of 1nl in this definition.)
A call of c06pff with direct='F' followed by a call with direct='B' will restore the original data.
The data values must be supplied in a one-dimensional complex array using column-major storage ordering of multidimensional data (i.e., with the first subscript j1 varying most rapidly).
This routine calls c06prf to perform one-dimensional discrete Fourier transforms. Hence, 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).

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: ndim Integer Input
On entry: m, the number of dimensions (or variables) in the multivariate data.
Constraint: ndim1.
3: l Integer Input
On entry: l, the index of the variable (or dimension) on which the discrete Fourier transform is to be performed.
Constraint: 1 l ndim.
4: nd(ndim) Integer array Input
On entry: the elements of nd must contain the dimensions of the ndim variables; that is, nd(i) must contain the dimension of the ith variable.
Constraint: nd(i)1, for i=1,2,,ndim.
5: n Integer Input
On entry: n, the total number of data values.
Constraint: n must equal the product of the first ndim elements of the array nd.
6: x(n) Complex (Kind=nag_wp) array Input/Output
On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, z j1 j2 jm is stored in x( 1 + j1 + n1 j2 + n1 n2 j3 + ).
On exit: the corresponding elements of the computed transform.
7: work(lwork) Complex (Kind=nag_wp) array Workspace
The workspace requirements as documented for c06pff may be an overestimate in some implementations.
On exit: the real part of work(1) contains the minimum workspace required for the current value of n with this implementation.
8: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which c06pff is called.
Suggested value: lwork n + nd(l) + 15 .
9: 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, ndim=value.
Constraint: ndim1.
On entry, l=value.
Constraint: 1lndim.
On entry, direct=value.
Constraint: direct='F' or 'B'.
On entry, nd(value)=value.
Constraint: nd(i)1, for all i.
On entry, n=value, product of nd elements is value.
Constraint: n must equal the product of the dimensions held in array nd.
On entry, lwork=value.
Constraint: lwork must be at least value.
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.
c06pff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pff 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 is approximately proportional to n×lognl , but also depends on the factorization of nl . c06pff is faster if the only prime factors of nl are 2, 3 or 5; and fastest of all if nl is a power of 2.

10 Example

This example reads in a bivariate sequence of complex data values and prints the discrete Fourier transform of the second variable. It then performs an inverse transform and prints the sequence so obtained, which may be compared with the original data values.

10.1 Program Text

Program Text (c06pffe.f90)

10.2 Program Data

Program Data (c06pffe.d)

10.3 Program Results

Program Results (c06pffe.r)