c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_sum_fft_complex_1d_multi (c06psc)

## 1  Purpose

nag_sum_fft_complex_1d_multi (c06psc) computes the discrete Fourier transforms of $m$ sequences each containing $n$ complex data values.

## 2  Specification

 #include #include
 void nag_sum_fft_complex_1d_multi (Nag_TransformDirection direct, Integer n, Integer m, Complex x[], NagError *fail)

## 3  Description

Given $m$ sequences of $n$ complex data values ${z}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, nag_sum_fft_complex_1d_multi (c06psc) simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
 $z^kp=1n ∑j=0 n-1zjp×exp±i2πjkn , k=0,1,…,n-1​ and ​p=1,2,…,m.$
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of nag_sum_fft_complex_1d_multi (c06psc) 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 code is provided for the factors $2$, $3$ and $5$.

## 4  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

## 5  Arguments

1:    $\mathbf{direct}$Nag_TransformDirectionInput
On entry: if the forward transform as defined in Section 3 is to be computed, then direct must be set equal to $\mathrm{Nag_ForwardTransform}$.
If the backward transform is to be computed then direct must be set equal to $\mathrm{Nag_BackwardTransform}$.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of complex values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
4:    $\mathbf{x}\left[{\mathbf{n}}×{\mathbf{m}}\right]$ComplexInput/Output
On entry: the complex data values ${z}_{\mathit{j}}^{p}$ stored in ${\mathbf{x}}\left[\left(\mathit{p}-1\right)×{\mathbf{n}}+\mathit{j}\right]$, for $\mathit{j}=0,1,\dots ,{\mathbf{n}}-1$ and $\mathit{p}=1,2,\dots ,{\mathbf{m}}$.
On exit: is overwritten by the complex transforms.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
$〈\mathit{\text{value}}〉$ is an invalid value of direct.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7  Accuracy

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).

## 8  Parallelism and Performance

nag_sum_fft_complex_1d_multi (c06psc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_1d_multi (c06psc) 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 nag_sum_fft_complex_1d_multi (c06psc) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_complex_1d_multi (c06psc) 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. This function internally allocates a workspace of $nm+n+15$ Complex values.

## 10  Example

This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by nag_sum_fft_complex_1d_multi (c06psc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$). Inverse transforms are then calculated using nag_sum_fft_complex_1d_multi (c06psc) with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ and printed out, showing that the original sequences are restored.

### 10.1  Program Text

Program Text (c06psce.c)

### 10.2  Program Data

Program Data (c06psce.d)

### 10.3  Program Results

Program Results (c06psce.r)