# NAG CL Interfacec06pqc (fft_​realherm_​1d_​multi_​col)

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## 1Purpose

c06pqc computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored column-wise in a complex storage format.

## 2Specification

 #include
 void c06pqc (Nag_TransformDirection direct, Integer n, Integer m, double x[], NagError *fail)
The function may be called by the names: c06pqc or nag_sum_fft_realherm_1d_multi_col.

## 3Description

Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06pqc simultaneously calculates the Fourier transforms of all the sequences defined by
 $z^kp = 1n ∑ j=0 n-1 xjp × exp(-i 2πjk n ) , k=0,1,…,n-1 ​ and ​ p=1,2,…,m .$
The transformed values ${\stackrel{^}{z}}_{k}^{p}$ are complex, but for each value of $p$ the ${\stackrel{^}{z}}_{k}^{p}$ form a Hermitian sequence (i.e., ${\stackrel{^}{z}}_{n-k}^{p}$ is the complex conjugate of ${\stackrel{^}{z}}_{k}^{p}$), so they are completely determined by $mn$ real numbers (since ${\stackrel{^}{z}}_{0}^{p}$ is real, as is ${\stackrel{^}{z}}_{n/2}^{p}$ for $n$ even).
Alternatively, given $m$ Hermitian sequences of $n$ complex data values ${z}_{j}^{p}$, this function simultaneously calculates their inverse (backward) discrete Fourier transforms defined by
 $x^kp = 1n ∑ j=0 n-1 zjp × exp(i 2πjk n ) , k=0,1,…,n-1 ​ and ​ p=1,2,…,m .$
The transformed values ${\stackrel{^}{x}}_{k}^{p}$ are real.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in the above definition.)
A call of c06pqc with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ will restore the original data.
The 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). Special coding is provided for the factors $2$, $3$, $4$ and $5$.

## 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: $\mathbf{direct}$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: $\mathbf{n}$Integer Input
On entry: $n$, the number of real or complex values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{m}$Integer Input
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
4: $\mathbf{x}\left[\left({\mathbf{n}}+2\right)×{\mathbf{m}}\right]$double Input/Output
On entry: the $m$ real or Hermitian data sequences to be transformed.
• if ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$, the $m$ real data sequences, ${x}^{\mathit{p}}=\left({x}_{1}^{\mathit{p}},{x}_{2}^{\mathit{p}},\dots ,{x}_{n}^{\mathit{p}}\right)$, for $\mathit{p}=1,2,\dots ,m$, should be stored sequentially in x, with a stride of $n+2$ between sequences.
• if ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$, the $m$ Hermitian data sequences, ${\stackrel{^}{z}}^{\mathit{p}}=\left({\stackrel{^}{z}}_{1}^{\mathit{p}},{\stackrel{^}{z}}_{2}^{\mathit{p}},\dots ,{\stackrel{^}{z}}_{n/2+1}^{\mathit{p}}\right)=\left({\stackrel{^}{z}}_{1}^{\mathit{p}}\mathbf{.}\mathbf{re},{\stackrel{^}{z}}_{1}^{\mathit{p}}\mathbf{.}\mathbf{im},{\stackrel{^}{z}}_{2}^{\mathit{p}}\mathbf{.}\mathbf{re},{\stackrel{^}{z}}_{2}^{\mathit{p}}\mathbf{.}\mathbf{im},\dots ,{\stackrel{^}{z}}_{n/2+1}^{\mathit{p}}\mathbf{.}\mathbf{re},{\stackrel{^}{z}}_{n/2+1}^{\mathit{p}}\mathbf{.}\mathbf{im}\right)$, for $\mathit{p}=1,2,\dots ,m$, should be stored sequentially in x, with a stride of $n+2$ between sequences.
In other words:
• if ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$, ${\mathbf{x}}\left[\left(\mathit{p}-1\right)×\left({\mathbf{n}}+2\right)+\mathit{j}-1\right]$ must contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$;
• if ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$, ${\mathbf{x}}\left[\left(\mathit{p}-1\right)×\left({\mathbf{n}}+2\right)+\left(2×\mathit{k}-1\right)-1\right]$ and ${\mathbf{x}}\left[\left(\mathit{p}-1\right)×\left({\mathbf{n}}+2\right)+\left(2×\mathit{k}\right)-1\right]$ must contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n/2+1$ and $\mathit{p}=1,2,\dots ,m$. (Note that for the sequence ${\stackrel{^}{z}}_{k}^{p}$ to be Hermitian, the imaginary part of ${\stackrel{^}{z}}_{1}^{p}$, and of ${\stackrel{^}{z}}_{n/2+1}^{p}$ for $n$ even, must be zero.)
On exit:
• if ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ then the $m$ sequences, ${\stackrel{^}{z}}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$ stored as described on entry for ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$
• if ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ then the $m$ sequences, ${x}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$ stored as described on entry for ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$
5: $\mathbf{fail}$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, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 1$.
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

c06pqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pqc 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 c06pqc is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06pqc is fastest if the only prime factors of $n$ are $2$, $3$ and $5$, and is particularly slow if $n$ is a large prime, or has large prime factors.

## 10Example

This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by c06pqc with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$), after expanding them from complex Hermitian form into full complex sequences.
Inverse transforms are then calculated by calling c06pqc with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ showing that the original sequences are restored.

### 10.1Program Text

Program Text (c06pqce.c)

### 10.2Program Data

Program Data (c06pqce.d)

### 10.3Program Results

Program Results (c06pqce.r)