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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_hermitian_1d_multi_rfmt (c06fq)

## Purpose

nag_sum_fft_hermitian_1d_multi_rfmt (c06fq) computes the discrete Fourier transforms of m$m$ Hermitian sequences, each containing n$n$ complex data values. This function is designed to be particularly efficient on vector processors.

## Syntax

[x, trig, ifail] = c06fq(m, n, x, init, trig)
[x, trig, ifail] = nag_sum_fft_hermitian_1d_multi_rfmt(m, n, x, init, trig)

## Description

Given m$m$ Hermitian 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_hermitian_1d_multi_rfmt (c06fq) simultaneously calculates the Fourier transforms of all the sequences defined by
 n − 1 x̂kp = 1/(sqrt(n)) ∑ zjp × exp( − i(2πjk)/n),  k = 0,1, … ,n − 1​ and ​p = 1,2, … ,m. j = 0
$x^kp = 1n ∑ j=0 n-1 zjp × exp( -i 2πjk n ) , k= 0, 1, …, n-1 ​ and ​ p= 1, 2, …, m .$
(Note the scale factor 1/(sqrt(n)) $\frac{1}{\sqrt{n}}$ in this definition.)
The transformed values are purely real (see also the C06 Chapter Introduction).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
 n − 1 x̂kp = 1/(sqrt(n)) ∑ zjp × exp( + i(2πjk)/n). j = 0
$x^kp = 1n ∑ j=0 n-1 zjp × exp( +i 2πjkn ) .$
To compute this form, this function should be preceded by forming the complex conjugates of the kp ${\stackrel{^}{z}}_{k}^{p}$; that is x(k) = x(k)$x\left(\mathit{k}\right)=-x\left(\mathit{k}\right)$, for k = (n / 2 + 1) × m + 1,,m × n$\mathit{k}=\left(n/2+1\right)×m+1,\dots ,m×n$.
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$2$, 3$3$, 4$4$, 5$5$ and 6$6$. This function is designed to be particularly efficient on vector processors, and it becomes especially fast as m$m$, the number of transforms to be computed in parallel, increases.

## References

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

## Parameters

### Compulsory Input Parameters

1:     m – int64int32nag_int scalar
m$m$, the number of sequences to be transformed.
Constraint: m1${\mathbf{m}}\ge 1$.
2:     n – int64int32nag_int scalar
n$n$, the number of data values in each sequence.
Constraint: n1${\mathbf{n}}\ge 1$.
3:     x( m × n ${\mathbf{m}}×{\mathbf{n}}$) – double array
The data must be stored in x as if in a two-dimensional array of dimension (1 : m,0 : n1)$\left(1:{\mathbf{m}},0:{\mathbf{n}}-1\right)$; each of the m$m$ sequences is stored in a row of the array in Hermitian form. If the n$n$ data values zjp${z}_{j}^{p}$ are written as xjp + i yjp${x}_{j}^{p}+i{y}_{j}^{p}$, then for 0 j n / 2$0\le j\le n/2$, xjp${x}_{j}^{p}$ is contained in x(p,j)${\mathbf{x}}\left(p,j\right)$, and for 1 j (n1) / 2$1\le j\le \left(n-1\right)/2$, yjp${y}_{j}^{p}$ is contained in x(p,nj)${\mathbf{x}}\left(p,n-j\right)$. (See also Section [Real transforms] in the C06 Chapter Introduction.)
4:     init – string (length ≥ 1)
Indicates whether trigonometric coefficients are to be calculated.
init = 'I'${\mathbf{init}}=\text{'I'}$
Calculate the required trigonometric coefficients for the given value of n$n$, and store in the array trig.
init = 'S'${\mathbf{init}}=\text{'S'}$ or 'R'$\text{'R'}$
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of nag_sum_fft_real_1d_multi_rfmt (c06fp), nag_sum_fft_hermitian_1d_multi_rfmt (c06fq) or nag_sum_fft_complex_1d_multi_rfmt (c06fr). The function performs a simple check that the current value of n$n$ is consistent with the values stored in trig.
Constraint: init = 'I'${\mathbf{init}}=\text{'I'}$, 'S'$\text{'S'}$ or 'R'$\text{'R'}$.
5:     trig( 2 × n $2×{\mathbf{n}}$) – double array
If init = 'S'${\mathbf{init}}=\text{'S'}$ or 'R'$\text{'R'}$, trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.

None.

work

### Output Parameters

1:     x( m × n ${\mathbf{m}}×{\mathbf{n}}$) – double array
The components of the m$m$ discrete Fourier transforms, stored as if in a two-dimensional array of dimension (1 : m,0 : n1) $\left(1:{\mathbf{m}},0:{\mathbf{n}}-1\right)$. Each of the m$m$ transforms is stored as a row of the array, overwriting the corresponding original sequence. If the n$n$ components of the discrete Fourier transform are denoted by kp ${\stackrel{^}{x}}_{\mathit{k}}^{p}$, for k = 0,1,,n1$\mathit{k}=0,1,\dots ,n-1$, then the mn $mn$ elements of the array x contain the values
 x̂01 , x̂02 , … , x̂0m , x̂11 , x̂12 , … , x̂1m , … , x̂n − 11 , x̂n − 12 , … , x̂n − 1m . $x^01 , x^02 ,…, x^0m , x^11 , x^12 ,…, x^1m ,…, x^ n-1 1 , x^ n-1 2 ,…, x^ n-1 m .$
2:     trig( 2 × n $2×{\mathbf{n}}$) – double array
Contains the required coefficients (computed by the function if init = 'I'${\mathbf{init}}=\text{'I'}$).
3:     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, init ≠ 'I'${\mathbf{init}}\ne \text{'I'}$, 'S'$\text{'S'}$ or 'R'$\text{'R'}$.
ifail = 4${\mathbf{ifail}}=4$
Not used at this Mark.
ifail = 5${\mathbf{ifail}}=5$
 On entry, init = 'S'${\mathbf{init}}=\text{'S'}$ or 'R'$\text{'R'}$, but the array trig and the current value of n are inconsistent.
ifail = 6${\mathbf{ifail}}=6$
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_hermitian_1d_multi_rfmt (c06fq) 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_hermitian_1d_multi_rfmt (c06fq) 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_hermitian_1d_multi_rfmt_example
m = int64(3);
n = int64(6);
x = [0.3854;
0.5417;
0.9172;
0.6772;
0.2983;
0.0644;
0.1138;
0.1181;
0.6037;
0.6751;
0.7255;
0.643;
0.6362;
0.8638;
0.0428;
0.1424;
0.8723;
0.4815];
init = 'Initial';
trig = [0;
0;
0;
0;
0;
0;
0;
0;
0;
0;
0;
0];
[xOut, trigOut, ifail] = nag_sum_fft_hermitian_1d_multi_rfmt(m, n, x, init, trig)
```
```

xOut =

1.0788
0.8573
1.1825
0.6623
1.2261
0.2625
-0.2391
0.3533
0.6744
-0.5783
-0.2222
0.5523
0.4592
0.3413
0.0540
-0.4388
-1.2291
-0.4790

trigOut =

1
1
1
1
1
6
0
0
0
0
0
6

ifail =

0

```
```function c06fq_example
m = int64(3);
n = int64(6);
x = [0.3854;
0.5417;
0.9172;
0.6772;
0.2983;
0.0644;
0.1138;
0.1181;
0.6037;
0.6751;
0.7255;
0.643;
0.6362;
0.8638;
0.0428;
0.1424;
0.8723;
0.4815];
init = 'Initial';
trig = [0;
0;
0;
0;
0;
0;
0;
0;
0;
0;
0;
0];
[xOut, trigOut, ifail] = c06fq(m, n, x, init, trig)
```
```

xOut =

1.0788
0.8573
1.1825
0.6623
1.2261
0.2625
-0.2391
0.3533
0.6744
-0.5783
-0.2222
0.5523
0.4592
0.3413
0.0540
-0.4388
-1.2291
-0.4790

trigOut =

1
1
1
1
1
6
0
0
0
0
0
6

ifail =

0

```