naginterfaces.library.sum.fft_​complex_​3d

naginterfaces.library.sum.fft_complex_3d(direct, n1, n2, n3, x)

fft_complex_3d computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).

For full information please refer to the NAG Library document for c06px

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06pxf.html

Parameters

directstr, length 1

If the forward transform as defined in Notes 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’.

n1int

$n_1$, the first dimension of the transform.

n2int

$n_2$, the second dimension of the transform.

n3int

$n_3$, the third dimension of the transform.

xcomplex, array-like, shape $(n1\times n2\times n3)$

The complex data values. Data values are stored in $x$ using column-major ordering for storing multidimensional arrays; that is, $z_{j_1{j}_2{j}_3}$ is stored in $x[1+j_1+n_1{j}_2+n_1{n}_2{j}_3]$.

Returns

xcomplex, ndarray, shape $(n1\times n2\times n3)$

The corresponding elements of the computed transform.

Raises

NagValueError

(errno $1$)

On entry, $n1=⟨value⟩$.

Constraint: $n1\ge 1$.

(errno $2$)

On entry, $n2=⟨value⟩$.

Constraint: $n2\ge 1$.

(errno $3$)

On entry, $n3=⟨value⟩$.

Constraint: $n3\ge 1$.

(errno $4$)

On entry, $direct=⟨value⟩$.

Constraint: $direct=\text{'F'}$ or $\text{'B'}$.

(errno $8$)

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_3d computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values $z_{j_1{j}_2{j}_3}$, for ${\text{j}}_3=0,1,\dots ,n_3-1$, for ${\text{j}}_2=0,1,\dots ,n_2-1$, for ${\text{j}}_1=0,1,\dots ,n_1-1$.

The discrete Fourier transform is here defined by

$${\hat{z}}_{k_1{k}_2{k}_3}=\frac{1}{\sqrt{n_1{n}_2{n}_3}}\sum _{j_1=0}^{n_1-1}\sum _{j_2=0}^{n_2-1}\sum _{j_3=0}^{n_3-1}{z}_{j_1{j}_2{j}_3}\times exp\left(\pm 2\pi i(\frac{j_1{k}_1}{n_1}+\frac{j_2{k}_2}{n_2}+\frac{j_3{k}_3}{n_3})\right)\text{,}$$

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 minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.

A call of fft_complex_3d with $direct=\text{'F'}$ followed by a call with $direct=\text{'B'}$ will restore the original data.

This function performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

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