NAG Library Routine Document

C06FJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C06FJF computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.

2 Specification

SUBROUTINE C06FJF (

NDIM, ND, N, X, Y, WORK, LWORK, IFAIL)

INTEGER	NDIM, ND(NDIM), N, LWORK, IFAIL
REAL (KIND=nag_wp)	X(N), Y(N), WORK(LWORK)

3 Description

C06FJF 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 (- 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

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

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

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

in this definition.)

To compute the inverse discrete Fourier transform, defined with

\exp (+ 2 π i (\dots))

in the above formula instead of

\exp (- 2 π i (\dots))

, this routine should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in

y

The data values must be supplied in a pair of one-dimensional arrays (real and imaginary parts separately), in accordance with the Fortran convention for storing multidimensional data (i.e., with the first subscript

j_{1}

varying most rapidly).

This routine calls C06FCF to perform one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974), and hence there are some restrictions on the values of the

n_{i}

(see Section 5).

4 References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

5 Parameters

1: NDIM – INTEGERInput: On entry: $m$ , the number of dimensions (or variables) in the multivariate data.
Constraint: $NDIM \geq 1$ .
2: ND(NDIM) – INTEGER arrayInput: On entry: $ND (i)$ must contain $n_{i}$ (the dimension of the $i$ th variable) , for $i = 1, 2, \dots, m$ . The largest prime factor of each $ND (i)$ must not exceed $19$ , and the total number of prime factors of $ND (i)$ , counting repetitions, must not exceed $20$ .
Constraint: $ND (i) \geq 1$ , for $i = 1, 2, \dots, NDIM$ .
3: N – INTEGERInput: On entry: $n$ , the total number of data values.
Constraint: $N = ND (1) \times ND (2) \times \dots \times ND (NDIM)$ .
4: X(N) – REAL (KIND=nag_wp) arrayInput/Output: On entry: $X (1 + j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3} + \dots)$ must contain the real part of the complex data value $z_{j_{1} j_{2} \dots j_{m}}$ , for $0 \leq j_{1} \leq n_{1} - 1, 0 \leq j_{2} \leq n_{2} - 1, \dots$ ; i.e., the values are stored in consecutive elements of the array according to the Fortran convention for storing multidimensional arrays.

On exit: the real parts of the corresponding elements of the computed transform.
5: Y(N) – REAL (KIND=nag_wp) arrayInput/Output: On entry: the imaginary parts of the complex data values, stored in the same way as the real parts in the array X.

On exit: the imaginary parts of the corresponding elements of the computed transform.
6: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace
7: LWORK – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which C06FJF is called.
Constraint: $LWORK \geq 3 \times \max \{ND (i)\}$ .
8: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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,

N \neq ND (1) \times ND (2) \times \dots \times ND (NDIM)

$IFAIL = 10 \times l + 1$: At least one of the prime factors of $ND (l)$ is greater than $19$ .

$IFAIL = 10 \times l + 2$: $ND (l)$ has more than $20$ prime factors.

$IFAIL = 10 \times l + 3$

On entry,

ND (l) < 1

$IFAIL = 10 \times l + 4$

On entry,

LWORK < 3 \times ND (l)

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 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)

. C06FJF is faster if the only prime factors are

2

3

5

; and fastest of all if they are powers of

2

9 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 DocumentC06FJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

C06FJF