# NAG CL Interfacec06pac (fft_​realherm_​1d)

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## 1Purpose

c06pac calculates the discrete Fourier transform of a sequence of $n$ real data values or of a Hermitian sequence of $n$ complex data values stored in compact form in a double array.

## 2Specification

 #include
 void c06pac (Nag_TransformDirection direct, double x[], Integer n, NagError *fail)
The function may be called by the names: c06pac or nag_sum_fft_realherm_1d.

## 3Description

Given a sequence of $n$ real data values ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, c06pac calculates their discrete Fourier transform (in the forward direction) defined by
 $z^k = 1n ∑ j=0 n-1 xj × exp(-i 2πjk n ) , k= 0, 1, …, n-1 .$
The transformed values ${\stackrel{^}{z}}_{k}$ are complex, but they form a Hermitian sequence (i.e., ${\stackrel{^}{z}}_{n-k}$ is the complex conjugate of ${\stackrel{^}{z}}_{k}$), so they are completely determined by $n$ real numbers (since ${\stackrel{^}{z}}_{0}$ is real, as is ${\stackrel{^}{z}}_{n/2}$ for $n$ even).
Alternatively, given a Hermitian sequence of $n$ complex data values ${z}_{j}$, this function calculates their inverse (backward) discrete Fourier transform defined by
 $x^k = 1n ∑ j=0 n-1 zj × exp(i 2πjk n ) , k= 0, 1, …, n-1 .$
The transformed values ${\stackrel{^}{x}}_{k}$ are real.
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in the above definitions.)
A call of c06pac with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ will restore the original data.
c06pac 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).
The same functionality is available using the forward and backward transform function pair: c06pvc and c06pwc on setting ${\mathbf{n}}=1$. This pair use a different storage solution; real data is stored in a double array, while Hermitian data (the first unconjugated half) is stored in a Complex array.

## 4References

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

## 5Arguments

1: $\mathbf{direct}$Nag_TransformDirection Input
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to $\mathrm{Nag_ForwardTransform}$.
If the backward transform is to be computed, direct must be set equal to $\mathrm{Nag_BackwardTransform}$.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2: $\mathbf{x}\left[{\mathbf{n}}+2\right]$double Input/Output
On entry:
• if ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$, ${\mathbf{x}}\left[\mathit{j}\right]$ must contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$;
• if ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$, ${\mathbf{x}}\left[2×\mathit{k}\right]$ and ${\mathbf{x}}\left[2×\mathit{k}+1\right]$ must contain the real and imaginary parts respectively of ${z}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$. (Note that for the sequence ${z}_{k}$ to be Hermitian, the imaginary part of ${z}_{0}$, and of ${z}_{n/2}$ for $n$ even, must be zero.)
On exit:
• if ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$, ${\mathbf{x}}\left[2×\mathit{k}\right]$ and ${\mathbf{x}}\left[2×\mathit{k}+1\right]$ will contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$;
• if ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$, ${\mathbf{x}}\left[\mathit{j}\right]$ will contain ${\stackrel{^}{x}}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of data values.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{fail}$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, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
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 subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

## 8Parallelism and Performance

c06pac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pac 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 is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. c06pac is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$. This function internally allocates a workspace of $3n+100$ double values.

## 10Example

This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by c06pac with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using c06pac with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$, and prints the sequence so obtained alongside the original data values.

### 10.1Program Text

Program Text (c06pace.c)

### 10.2Program Data

Program Data (c06pace.d)

### 10.3Program Results

Program Results (c06pace.r)