NAG Library Routine Document

C06PYF

+− 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

C06PYF computes the three-dimensional discrete Fourier transform of a trivariate sequence of real data values.

2 Specification

SUBROUTINE C06PYF (

N1, N2, N3, X, Y, IFAIL)

INTEGER	N1, N2, N3, IFAIL
REAL (KIND=nag_wp)	X(N1N2N3)
COMPLEX (KIND=nag_wp)	Y((N1/2+1)N2N3)

3 Description

C06PYF computes the three-dimensional discrete Fourier transform of a trivariate sequence of real data values

x_{j_{1} j_{2} j_{3}}

, for

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

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

and

j_{3} = 0, 1, \dots, n_{3} - 1

The discrete Fourier transform is here defined by

{\hat{z}}_{k_{1} k_{2} k_{3}} = \frac{1}{\sqrt{n_{1} n_{2} n_{3}}} \sum_{j_{1} = 0}^{n_{1} - 1} \sum_{j_{2} = 0}^{n_{2} - 1} \sum_{j_{3} = 0}^{n_{3} - 1} x_{j_{1} j_{2} j_{3}} \times \exp (- 2 π i (\frac{j_{1} k_{1}}{n_{1}} + \frac{j_{2} k_{2}}{n_{2}} + \frac{j_{3} k_{3}}{n_{3}})),

where

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

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

and

k_{3} = 0, 1, \dots, n_{3} - 1

. (Note the scale factor of

\frac{1}{\sqrt{n_{1} n_{2} n_{3}}}

in this definition.)

The transformed values

{\hat{z}}_{k_{1} k_{2} k_{3}}

are complex. Because of conjugate symmetry (i.e.,

{\hat{z}}_{k_{1} k_{2} k_{3}}

is the complex conjugate of

{\hat{z}}_{(n_{1} - k_{1}) k_{2} k_{3}}

), only slightly more than half of the Fourier coefficients need to be stored in the output.

A call of C06PYF followed by a call of C06PZF will restore the original data.

This routine calls C06PQF and C06PRF to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).

4 References

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

5 Parameters

1: N1 – INTEGERInput: On entry: $n_{1}$ , the first dimension of the transform.
Constraint: $N1 \geq 1$ .
2: N2 – INTEGERInput: On entry: $n_{2}$ , the second dimension of the transform.
Constraint: $N2 \geq 1$ .
3: N3 – INTEGERInput: On entry: $n_{3}$ , the third dimension of the transform.
Constraint: $N3 \geq 1$ .
4: X( $N1 \times N2 \times N3$ ) – REAL (KIND=nag_wp) arrayInput: On entry: the real input dataset $x$ , where $x_{j_{1} j_{2} j_{3}}$ is stored in $X (j_{3} \times n_{1} n_{2} + j_{2} \times n_{1} + j_{1} + 1)$ , for $j_{1} = 0, 1, \dots, n_{1} - 1$ , $j_{2} = 0, 1, \dots, n_{2} - 1$ and $j_{3} = 0, 1, \dots, n_{3} - 1$ . That is, if X is regarded as a three-dimensional array of dimension $(0 : N1 - 1, 0 : N2 - 1, 0 : N3 - 1)$ , then $X (j_{1}, j_{2}, j_{3})$ must contain $x_{j_{1} j_{2} j_{3}}$ .
5: Y( $(N1 / 2 + 1) \times N2 \times N3$ ) – COMPLEX (KIND=nag_wp) arrayOutput: On exit: the complex output dataset $\hat{z}$ , where ${\hat{z}}_{k_{1} k_{2} k_{3}}$ is stored in $Y (k_{3} \times (n_{1} / 2 + 1) n_{2} + k_{2} \times (n_{1} / 2 + 1) + k_{1} + 1)$ , for $k_{1} = 0, 1, \dots, n_{1} / 2$ , $k_{2} = 0, 1, \dots, n_{2} - 1$ and $k_{3} = 0, 1, \dots, n_{3} - 1$ . That is, if Y is regarded as a three-dimensional array of dimension $(0 : N1 / 2, 0 : N2 - 1, 0 : N3 - 1)$ , then $Y (k_{1}, k_{2}, k_{3})$ contains ${\hat{z}}_{k_{1} k_{2} k_{3}}$ . Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
6: 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, $N1 = ⟨value⟩$ .
Constraint: $N1 \geq 1$ .

$IFAIL = 2$: On entry, $N2 = ⟨value⟩$ .
Constraint: $N2 \geq 1$ .

$IFAIL = 3$: On entry, $N3 = ⟨value⟩$ .
Constraint: $N3 \geq 1$ .

$IFAIL = 4$: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

$IFAIL = - 999$: Dynamic memory allocation failed.

7 Accuracy

Some indication of accuracy can be obtained by performing a forward transform using C06PYF and a backward transform using C06PZF, and comparing the results with the original sequence (in exact arithmetic they would be identical).

8 Further Comments

The time taken by C06PYF is approximately proportional to

n_{1} n_{2} n_{3} \log (n_{1} n_{2} n_{3})

, but also depends on the factors of

n_{1}

n_{2}

and

n_{3}

. C06PYF is fastest if the only prime factors of

n_{1}

n_{2}

and

n_{3}

are

2

3

and

5

, and is particularly slow if one of the dimensions is a large prime, or has large prime factors.

Workspace is internally allocated by C06PYF. The total size of these arrays is approximately proportional to

n_{1} n_{2} n_{3}

9 Example

This example reads in a trivariate sequence of real data values and prints their discrete Fourier transforms as computed by C06PYF. Inverse transforms are then calculated by calling C06PZF showing that the original sequences are restored.

NAG Library Routine DocumentC06PYF