naginterfaces.library.sum.fft_​complex_​multid_​1d

naginterfaces.library.sum.fft_complex_multid_1d(direct, l, nd, x)

fft_complex_multid_1d computes the discrete Fourier transform of one variable in a multivariate sequence of complex data values.

For full information please refer to the NAG Library document for c06pf

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06pff.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’.

lint

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

ndint, array-like, shape $\left(\text{ndim}\right)$

The elements of $nd$ must contain the dimensions of the $\text{ndim}$ variables; that is, $nd[i-1]$ must contain the dimension of the $i$th variable.

xcomplex, array-like, shape $\left(n\right)$

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

Returns

xcomplex, ndarray, shape $\left(n\right)$

The corresponding elements of the computed transform.

Raises

NagValueError

(errno $1$)

On entry, $\text{ndim}=⟨value⟩$.

Constraint: $\text{ndim}\ge 1$.

(errno $2$)

On entry, $l=⟨value⟩$.

Constraint: $1\le l\le \text{ndim}$.

(errno $3$)

On entry, $direct=⟨value⟩$.

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

(errno $4$)

On entry, $nd[⟨value⟩]=⟨value⟩$.

Constraint: $nd[i-1]\ge 1$, for all $i$.

(errno $5$)

On entry, $n=⟨value⟩$, product of $nd$ elements is $⟨value⟩$.

Constraint: $\text{n}$ must equal the product of the dimensions held in array $nd$.

(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_multid_1d computes the discrete Fourier transform of one variable (the $l$th say) in a multivariate sequence of complex data values $z_{j_1{j}_2\cdots j_m}$, where $j_1=0,1,\dots ,n_1-1\text{,}{j}_2=0,1,\dots ,n_2-1$, and so on. Thus the individual dimensions are $n_1,n_2,\dots ,n_m$, and the total number of data values is $n=n_1\times n_2\times \cdots \times n_m$.

The function computes $n/n_l$ one-dimensional transforms defined by

$${\hat{z}}_{j_1\dots k_l\dots j_m}=\frac{1}{\sqrt{n_l}}\sum _{j_l=0}^{n_l-1}{z}_{j_1\dots j_l\dots j_m}\times exp(\pm \frac{2\pi i{j}_l{k}_l}{n_l})\text{,}$$

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 fft_complex_multid_1d with $direct=\text{'F'}$ followed by a call with $direct=\text{'B'}$ 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).

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