naginterfaces.library.sum.fft_complex_3d_sep¶

naginterfaces.library.sum.fft_complex_3d_sep(n1, n2, n3, x, y)[source]¶

fft_complex_3d_sep computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values. This function is designed to be particularly efficient on vector processors.

For full information please refer to the NAG Library document for c06fx

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06fxf.html

Parameters

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.
xfloat, array-like, shape $(n 1 \times n 2 \times n 3)$: The real and imaginary parts of the complex data values must be stored in arrays $x$ and $y$ respectively. If $x$ and $y$ are regarded as three-dimensional arrays of dimension $(0 : n 1 - 1, 0 : n 2 - 1, 0 : n 3 - 1)$ , $x [j_{1} - 1, j_{2} - 1, j_{3} - 1]$ and $y [j_{1} - 1, j_{2} - 1, j_{3} - 1]$ must contain the real and imaginary parts of $z_{j_{1} j_{2} j_{3}}$ .
yfloat, array-like, shape $(n 1 \times n 2 \times n 3)$: The real and imaginary parts of the complex data values must be stored in arrays $x$ and $y$ respectively. If $x$ and $y$ are regarded as three-dimensional arrays of dimension $(0 : n 1 - 1, 0 : n 2 - 1, 0 : n 3 - 1)$ , $x [j_{1} - 1, j_{2} - 1, j_{3} - 1]$ and $y [j_{1} - 1, j_{2} - 1, j_{3} - 1]$ must contain the real and imaginary parts of $z_{j_{1} j_{2} j_{3}}$ .

Returns

xfloat, ndarray, shape $(n 1 \times n 2 \times n 3)$: The real and imaginary parts respectively of the corresponding elements of the computed transform.
yfloat, ndarray, shape $(n 1 \times n 2 \times n 3)$: The real and imaginary parts respectively of the corresponding elements of the computed transform.

Raises

NagValueError

(errno $1$ )

On entry, $n 1 = ⟨ v a l u e ⟩$ .

Constraint: $n 1 \geq 1$ .

(errno $2$ )

On entry, $n 2 = ⟨ v a l u e ⟩$ .

Constraint: $n 2 \geq 1$ .

(errno $3$ )

On entry, $n 3 = ⟨ v a l u e ⟩$ .

Constraint: $n 3 \geq 1$ .

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

fft_complex_3d_sep computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values $z_{j_{1} j_{2} j_{3}}$ , for $j_{3} = 0, 1, \dots, n_{3} - 1$ , for $j_{2} = 0, 1, \dots, n_{2} - 1$ , for $j_{1} = 0, 1, \dots, n_{1} - 1$ .

The discrete Fourier transform is here defined by

{^z}_{k_{1} k_{2} k_{3}} = \frac{1}{\sqrt{n_{1} n_{2} n_{3}}} n_{1} - 1 \sum j_{1} = 0 n_{2} - 1 \sum j_{2} = 0 n_{3} - 1 \sum j_{3} = 0 z_{j_{1} j_{2} j_{3}} \times e x p (- 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 $k_{1} = 0, 1, \dots, n_{1} - 1$ , $k_{2} = 0, 1, \dots, n_{2} - 1$ , $k_{3} = 0, 1, \dots, n_{3} - 1$ .

(Note the scale factor of $\frac{1}{\sqrt{n_{1} n_{2} n_{3}}}$ in this definition.)

To compute the inverse discrete Fourier transform, defined with $e x p (+ 2 π i (\dots))$ in the above formula instead of $e x p (- 2 π i (\dots))$ , this function should be preceded and followed by forming the complex conjugates of the data values and the transform.

This function performs, for each dimension, multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)). It is designed to be particularly efficient on vector processors.

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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