naginterfaces.library.sum.fft_complex_multid¶

naginterfaces.library.sum.fft_complex_multid(direct, nd, x)[source]¶

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

For full information please refer to the NAG Library document for c06pj

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/c06/c06pjf.html

Parameters

directstr, length 1

If the forward transform as defined in Notes is to be computed, $d i r e c t$ must be set equal to ‘F’.

If the backward transform is to be computed, $d i r e c t$ must be set equal to ‘B’.

ndint, array-like, shape $(ndim)$

The elements of $n d$ must contain the dimensions of the $ndim$ variables; that is, $n d [i - 1]$ must contain the dimension of the $i$ th variable.

xcomplex, array-like, shape $(n)$

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 $x [1 + j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3} + \dots]$ .

Returns

xcomplex, ndarray, shape $(n)$: The corresponding elements of the computed transform.

Raises

NagValueError

(errno $1$ )

On entry, $ndim = ⟨ v a l u e ⟩$ .

Constraint: $ndim \geq 1$ .

(errno $2$ )

On entry, $d i r e c t = ⟨ v a l u e ⟩$ .

Constraint: $d i r e c t ='F'$ or $'B'$ .

(errno $3$ )

On entry, $n d [⟨ v a l u e ⟩] = ⟨ v a l u e ⟩$ .

Constraint: $n d [i - 1] \geq 1$ , for all $i$ .

(errno $4$ )

On entry, $n = ⟨ v a l u e ⟩$ , product of $n d$ elements is $⟨ v a l u e ⟩$ .

Constraint: $n$ must equal the product of the dimensions held in array $n d$ .

(errno $7$ )

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.

Notes

fft_complex_multid 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:

{^z}_{k_{1}, k_{2}} = \frac{1}{\sqrt{n}} n_{1} - 1 \sum j_{1} = 0 n_{2} - 1 \sum j_{2} = 0 z_{j_{1} j_{2}} \times e x p (\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 fft_complex_multid with $d i r e c t ='F'$ followed by a call with $d i r e c t ='B'$ 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 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).

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

NAG and Python

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naginterfaces.library.sum.fft_complex_multid¶

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naginterfaces.library.sum.fft_complex_multid¶