# NAG CL Interfacec06pzc (fft_​hermitian_​3d)

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## 1Purpose

c06pzc computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values.

## 2Specification

 #include
 void c06pzc (Integer n1, Integer n2, Integer n3, const Complex y[], double x[], NagError *fail)
The function may be called by the names: c06pzc or nag_sum_fft_hermitian_3d.

## 3Description

c06pzc computes the three-dimensional inverse discrete Fourier transform of a trivariate Hermitian sequence of complex data values ${z}_{{j}_{1}{j}_{2}{j}_{3}}$, for ${j}_{1}=0,1,\dots ,{n}_{1}-1$, ${j}_{2}=0,1,\dots ,{n}_{2}-1$ and ${j}_{3}=0,1,\dots ,{n}_{3}-1$.
The discrete Fourier transform is here defined by
 $x^ k1 k2 k3 = 1 n1 n2 n3 ∑ j1=0 n1-1 ∑ j2=0 n2-1 ∑ j3=0 n3-1 z j1 j2 j3 × exp(2πi( j1 k1 n1 + j2 k2 n2 + j3 k3 n3 )) ,$
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.)
Because the input data satisfies conjugate symmetry (i.e., ${z}_{{j}_{1}{j}_{2}{j}_{3}}$ is the complex conjugate of ${z}_{\left({n}_{1}-{j}_{1}\right)\left({n}_{2}-{j}_{2}\right)\left({n}_{3}-{j}_{3}\right)}$), the transformed values ${\stackrel{^}{x}}_{{k}_{1}{k}_{2}{k}_{3}}$ are real.
A call of c06pyc followed by a call of c06pzc will restore the original data.
This function performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).

## 4References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

## 5Arguments

1: $\mathbf{n1}$Integer Input
On entry: ${n}_{1}$, the first dimension of the transform.
Constraint: ${\mathbf{n1}}\ge 1$.
2: $\mathbf{n2}$Integer Input
On entry: ${n}_{2}$, the second dimension of the transform.
Constraint: ${\mathbf{n2}}\ge 1$.
3: $\mathbf{n3}$Integer Input
On entry: ${n}_{3}$, the third dimension of the transform.
Constraint: ${\mathbf{n3}}\ge 1$.
4: $\mathbf{y}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array y must be at least $\left({\mathbf{n1}}/2+1\right)×{\mathbf{n2}}×{\mathbf{n3}}$.
On entry: the Hermitian sequence of complex input dataset $z$, where ${z}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in ${\mathbf{y}}\left[{j}_{3}×\left({n}_{1}/2+1\right){n}_{2}+{j}_{2}×\left({n}_{1}/2+1\right)+{j}_{1}\right]$, for ${j}_{1}=0,1,\dots ,{n}_{1}/2$, ${j}_{2}=0,1,\dots ,{n}_{2}-1$ and ${j}_{3}=0,1,\dots ,{n}_{3}-1$.
5: $\mathbf{x}\left[{\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}\right]$double Output
On exit: the real output dataset $\stackrel{^}{x}$, where ${\stackrel{^}{x}}_{{k}_{1}{k}_{2}{k}_{3}}$ is stored in ${\mathbf{x}}\left[{k}_{3}×{n}_{1}{n}_{2}+{k}_{2}×{n}_{1}+{k}_{1}\right]$, for ${k}_{1}=0,1,\dots ,{n}_{1}-1$, ${k}_{2}=0,1,\dots ,{n}_{2}-1$ and ${k}_{3}=0,1,\dots ,{n}_{3}-1$.
6: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n1}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n1}}\ge 1$.
On entry, ${\mathbf{n2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n2}}\ge 1$.
On entry, ${\mathbf{n3}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n3}}\ge 1$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

Some indication of accuracy can be obtained by performing a forward transform using c06pyc and a backward transform using c06pzc, and comparing the results with the original sequence (in exact arithmetic they would be identical).

## 8Parallelism and Performance

c06pzc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pzc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by c06pzc is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}\mathrm{log}\left({n}_{1}{n}_{2}{n}_{3}\right)$, but also depends on the factors of ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$. c06pzc is fastest if the only prime factors of ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$ are $2$, $3$ and $5$, and is particularly slow if one of the dimensions is a large prime, or has large prime factors.
Workspace is internally allocated by c06pzc. The total size of these arrays is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}$.