NAG Library Function Document

nag_fft_multid_single (c06pfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_fft_multid_single (c06pfc) computes the discrete Fourier transform of one variable in a multivariate sequence of complex data values.

2 Specification

#include <nag.h>

#include <nagc06.h>

void	nag_fft_multid_single (Nag_TransformDirection direct, Integer ndim, Integer l, const Integer nd[], Integer n, Complex x[], NagError *fail)

3 Description

nag_fft_multid_single (c06pfc) computes the discrete Fourier transform of one variable (the

l

th say) in a multivariate 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 function computes

n / n_{l}

one-dimensional transforms defined by

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

where

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

\frac{1}{\sqrt{n_{l}}}

in this definition.)

A call of nag_fft_multid_single (c06pfc) with

direct = Nag_ForwardTransform

followed by a call with

direct = Nag_BackwardTransform

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

j_{1}

varying most rapidly).

This function 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 – Nag_TransformDirectionInput

On entry: if the forward transform as defined in Section 3 is to be computed, then direct must be set equal to Nag_ForwardTransform.

If the backward transform is to be computed then direct must be set equal to Nag_BackwardTransform.

Constraint:

direct = Nag_ForwardTransform

Nag_BackwardTransform

2: ndim – IntegerInput

On entry:

m

, the number of dimensions (or variables) in the multivariate data.

Constraint:

ndim \geq 1

3: l – IntegerInput

On entry:

l

, the index of the variable (or dimension) on which the discrete Fourier transform is to be performed.

Constraint:

1 \leq l \leq ndim

4: nd[ndim] – const IntegerInput

On entry: the elements of nd must contain the dimensions of the ndim variables; that is,

nd [i - 1]

must contain the dimension of the

i

th variable.

Constraints:

$nd [i - 1] \geq 1$ , for $i = 1, 2, \dots, ndim$ ;
$nd [l - 1]$ must have less than $31$ prime factors (counting repetitions).

5: 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.

6: x[n] – ComplexInput/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 [j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3} + \dots]

On exit: the corresponding elements of the computed transform.

7: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: $nd [l - 1]$ must have less than $31$ prime factors: $nd [l - 1] = 〈value〉$ .; On entry, $l = 〈value〉$ .
Constraint: $l \geq 1$ and $l \leq ndim$ .; On entry, $ndim = 〈value〉$ .
Constraint: $ndim \geq 1$ .
NE_INT_2: n must equal the product of the dimensions held in array nd: $n = 〈value〉$ , product of nd elements is $〈value〉$ .; On entry $nd [i - 1] = 〈value〉$ and $i = 〈value〉$ .
Constraint: $nd [i - 1] \geq 1$ .
NE_INTERNAL_ERROR: 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.

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_{l}

, but also depends on the factorization of

n_{l}

. nag_fft_multid_single (c06pfc) is faster if the only prime factors of

n_{l}

are

2

3

5

; and fastest of all if

n_{l}

is a power of

2

9 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.

NAG Library Function Documentnag_fft_multid_single (c06pfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_fft_multid_single (c06pfc)