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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_complex_1d_multi_col (c06ps)

## Purpose

nag_sum_fft_complex_1d_multi_col (c06ps) computes the discrete Fourier transforms of m$m$ sequences, stored as columns of an array, each containing n$n$ complex data values.

## Syntax

[x, ifail] = c06ps(direct, n, m, x)
[x, ifail] = nag_sum_fft_complex_1d_multi_col(direct, n, m, x)

## Description

Given m$m$ sequences of n$n$ complex data values zjp ${z}_{\mathit{j}}^{\mathit{p}}$, for j = 0,1,,n1$\mathit{j}=0,1,\dots ,n-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, nag_sum_fft_complex_1d_multi_col (c06ps) simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
 n − 1 ẑkp = 1/(sqrt(n)) ∑ zjp × exp( ± i(2πjk)/n),  k = 0,1, … ,n − 1​ and ​p = 1,2, … ,m. j = 0
$z^kp=1n ∑j=0 n-1zjp×exp(±i2πjkn) , k=0,1,…,n-1​ and ​p=1,2,…,m.$
(Note the scale factor 1/(sqrt(n)) $\frac{1}{\sqrt{n}}$ 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_col (c06ps) with direct = 'F'${\mathbf{direct}}=\text{'F'}$ followed by a call with direct = 'B'${\mathbf{direct}}=\text{'B'}$ 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$2$, 3$3$, 4$4$ and 5$5$.

## 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

## Parameters

### Compulsory Input Parameters

1:     direct – string (length ≥ 1)
If the forward transform as defined in Section [Description] is to be computed, then direct must be set equal to 'F'.
If the backward transform is to be computed then direct must be set equal to 'B'.
Constraint: direct = 'F'${\mathbf{direct}}=\text{'F'}$ or 'B'$\text{'B'}$.
2:     n – int64int32nag_int scalar
n$n$, the number of complex values in each sequence.
Constraint: n1${\mathbf{n}}\ge 1$.
3:     m – int64int32nag_int scalar
m$m$, the number of sequences to be transformed.
Constraint: m1${\mathbf{m}}\ge 1$.
4:     x( n × m ${\mathbf{n}}×{\mathbf{m}}$) – complex array
The complex data values zjp${z}_{\mathit{j}}^{p}$ stored in x((p1) × n + j + 1)${\mathbf{x}}\left(\left(\mathit{p}-1\right)×{\mathbf{n}}+\mathit{j}+1\right)$, for j = 0,1,,n1$\mathit{j}=0,1,\dots ,{\mathbf{n}}-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,{\mathbf{m}}$.

None.

work

### Output Parameters

1:     x( n × m ${\mathbf{n}}×{\mathbf{m}}$) – complex array
Stores the complex transforms.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, m < 1${\mathbf{m}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, n < 1${\mathbf{n}}<1$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, direct ≠ 'F'${\mathbf{direct}}\ne \text{'F'}$ or 'B'$\text{'B'}$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, n has more than 30$30$ prime factors.
ifail = 5${\mathbf{ifail}}=5$
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

## 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).

The time taken by nag_sum_fft_complex_1d_multi_col (c06ps) is approximately proportional to nm log(n)$nm\mathrm{log}\left(n\right)$, but also depends on the factors of n$n$. nag_sum_fft_complex_1d_multi_col (c06ps) is fastest if the only prime factors of n$n$ are 2$2$, 3$3$ and 5$5$, and is particularly slow if n$n$ is a large prime, or has large prime factors.

## Example

```function nag_sum_fft_complex_1d_multi_col_example
direct = 'F';
n = int64(6);
m = int64(3);
x = [ 0.3854 + 0.5417i;
0.6772 + 0.2983i;
0.1138 + 0.1181i;
0.6751 + 0.7255i;
0.6362 + 0.8638i;
0.1424 + 0.8723i;
0.9172 + 0.9089i;
0.0644 + 0.3118i;
0.6037 + 0.3465i;
0.643 + 0.6198i;
0.0428 + 0.2668i;
0.4815 + 0.1614i;
0.1156 + 0.6214i;
0.0685 + 0.8681i;
0.206 + 0.706i;
0.863 + 0.8652i;
0.6967 + 0.919i;
0.2792 + 0.3355i];
[xOut, ifail] = nag_sum_fft_complex_1d_multi_col(direct, n, m, x)
```
```

xOut =

1.0737 + 1.3961i
-0.5706 - 0.0409i
0.1733 - 0.2958i
-0.1467 - 0.1521i
0.0518 + 0.4517i
0.3625 - 0.0321i
1.1237 + 1.0677i
0.1728 + 0.0386i
0.4185 + 0.7481i
0.1530 + 0.1752i
0.3686 + 0.0565i
0.0101 + 0.1403i
0.9100 + 1.7617i
-0.3054 + 0.0624i
0.4079 - 0.0695i
-0.0785 + 0.0725i
-0.1193 + 0.1285i
-0.5314 - 0.4335i

ifail =

0

```
```function c06ps_example
direct = 'F';
n = int64(6);
m = int64(3);
x = [ 0.3854 + 0.5417i;
0.6772 + 0.2983i;
0.1138 + 0.1181i;
0.6751 + 0.7255i;
0.6362 + 0.8638i;
0.1424 + 0.8723i;
0.9172 + 0.9089i;
0.0644 + 0.3118i;
0.6037 + 0.3465i;
0.643 + 0.6198i;
0.0428 + 0.2668i;
0.4815 + 0.1614i;
0.1156 + 0.6214i;
0.0685 + 0.8681i;
0.206 + 0.706i;
0.863 + 0.8652i;
0.6967 + 0.919i;
0.2792 + 0.3355i];
[xOut, ifail] = c06ps(direct, n, m, x)
```
```

xOut =

1.0737 + 1.3961i
-0.5706 - 0.0409i
0.1733 - 0.2958i
-0.1467 - 0.1521i
0.0518 + 0.4517i
0.3625 - 0.0321i
1.1237 + 1.0677i
0.1728 + 0.0386i
0.4185 + 0.7481i
0.1530 + 0.1752i
0.3686 + 0.0565i
0.0101 + 0.1403i
0.9100 + 1.7617i
-0.3054 + 0.0624i
0.4079 - 0.0695i
-0.0785 + 0.0725i
-0.1193 + 0.1285i
-0.5314 - 0.4335i

ifail =

0

```