Integer, Intent (In)	::	ndim, 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 nagmk26.h

void	c06pjf_ ( const char direct, const Integer ndim, const Integer nd[], const Integer n, Complex x[], Complex work[], const Integer lwork, Integer *ifail, const Charlen length_direct)

3

Description

c06pjf computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values

z_{j_{1} j_{2} \dots j_{m}}

, where

j_{1} = 0, 1, \dots, n_{1} - 1, j_{2} = 0, 1, \dots, n_{2} - 1

, and so on. Thus the individual dimensions are

n_{1}, n_{2}, \dots, n_{m}

, and the total number of data values is

n = n_{1} \times n_{2} \times \dots \times n_{m}

The discrete Fourier transform is here defined (e.g., for

m = 2

) by:

{\hat{z}}_{k_{1}, k_{2}} = \frac{1}{\sqrt{n}} \sum_{j_{1} = 0}^{n_{1} - 1} \sum_{j_{2} = 0}^{n_{2} - 1} z_{j_{1} j_{2}} \times \exp (\pm 2 π i (\frac{j_{1} k_{1}}{n_{1}} + \frac{j_{2} k_{2}}{n_{2}})),

where

k_{1} = 0, 1, \dots, n_{1} - 1

and

k_{2} = 0, 1, \dots, n_{2} - 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.

The extension to higher dimensions is obvious. (Note the scale factor of

\frac{1}{\sqrt{n}}

in this definition.)

A call of c06pjf 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 array using column-major storage ordering of multidimensional data (i.e., with the first subscript

j_{1}

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: $ndim$ – IntegerInput: On entry: $m$ , the number of dimensions (or variables) in the multivariate data.

Constraint: $ndim \geq 1$ .
3: $nd (ndim)$ – Integer arrayInput: On entry: the elements of nd must contain the dimensions of the ndim variables; that is, $nd (i)$ must contain the dimension of the $i$ th variable.

Constraint: $nd (i) \geq 1$ , for $i = 1, 2, \dots, ndim$ .
4: $n$ – IntegerInput: On entry: $n$ , the total number of data values.

Constraint: n must equal the product of the first ndim elements of the array nd.
5: $x (n)$ – Complex (Kind=nag_wp) arrayInput/Output: On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, $z_{j_{1} j_{2} \dots j_{m}}$ is stored in $x (1 + j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3} + \dots)$ .

On exit: the corresponding elements of the computed transform.
6: $work (lwork)$ – Complex (Kind=nag_wp) arrayWorkspace: The workspace requirements as documented for c06pjf 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.
7: $lwork$ – IntegerInput: On entry: the dimension of the array work as declared in the (sub)program from which c06pjf is called.

Suggested value: $lwork \geq n + 3 \times \max (nd (i)) + 15$ , where $i = 1, 2, \dots, ndim$ .
8: $ifail$ – 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).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 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,

ndim < 1

$ifail = 2$

On entry,

direct \neq'F'

'B'

$ifail = 3$

On entry,

at least one of the first ndim elements of nd is less than

1

$ifail = 4$

On entry,

n does not equal the product of the first ndim elements of nd.

$ifail = 5$

On entry,

lwork is too small. The minimum amount of workspace required is returned in

work (1)

$ifail = 7$: An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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

c06pjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06pjf 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 \times \log (n)

, but also depends on the factorization of the individual dimensions

nd (i)

. c06pjf is faster if the only prime factors are

2

3

5

; and fastest of all if they are powers of

2

10

Example

This example reads in a bivariate sequence of complex data values and prints the two-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.

NAG Library Routine Document

c06pjf (fft_complex_multid)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results