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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sum_fft_real_qtrcosine (c06hd)

Purpose

nag_sum_fft_real_qtrcosine (c06hd) computes the discrete quarter-wave Fourier cosine transforms of m$m$ sequences of real data values. This function is designed to be particularly efficient on vector processors.
Note: this function is scheduled to be withdrawn, please see c06hd in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[x, trig, ifail] = c06hd(direct, m, n, x, init, trig)
[x, trig, ifail] = nag_sum_fft_real_qtrcosine(direct, m, n, x, init, trig)

Description

Given m$m$ sequences of n$n$ real data values xjp ${x}_{\mathit{j}}^{\mathit{p}}$, 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_real_qtrcosine (c06hd) simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
kp = 1/(sqrt(n))
 ( n − 1 ) (1/2)x0p + ∑ xjp × cos(j(2k − 1)π/(2n)) j = 1
,   if ​ direct = 'F' ,
$x^ k p = 1n { 12 x0p + ∑ j=1 n-1 xjp × cos( j (2k-1) π 2n ) } , if ​ direct='F' ,$
or its inverse
 n − 1 xkp = 2/(sqrt(n)) ∑ x̂jp × cos((2j − 1)kπ/(2n)),   if ​direct = 'B', j = 0
$xkp = 2n ∑ j= 0 n- 1 x^jp × cos( (2j- 1) k π2n ) , if ​ direct='B' ,$
for k = 0,1,,n1 $k=0,1,\dots ,n-1$ and p = 1,2,,m $p=1,2,\dots ,m$.
(Note the scale factor 1/(sqrt(n)) $\frac{1}{\sqrt{n}}$ in this definition.)
A call of nag_sum_fft_real_qtrcosine (c06hd) with direct = 'F'${\mathbf{direct}}=\text{'F'}$ followed by a call with direct = 'B'${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (see Swarztrauber (1977)). (See the C06 Chapter Introduction.)
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). 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
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

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:     m – int64int32nag_int scalar
m$m$, the number of sequences to be transformed.
Constraint: m1${\mathbf{m}}\ge 1$.
3:     n – int64int32nag_int scalar
n$n$, the number of real values in each sequence.
Constraint: n1${\mathbf{n}}\ge 1$.
4:     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 other words, if the data values of the p$\mathit{p}$th sequence to be transformed are denoted by xjp${x}_{\mathit{j}}^{\mathit{p}}$, for j = 0,1,,n1$\mathit{j}=0,1,\dots ,n-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, then the mn$mn$ elements of the array x must contain the values
 x01 , x02 , … , x0m , x11 , x12 , … , x1m , … , xn − 11 , xn − 12 , … , xn − 1m . $x01 , x02 ,…, x0m , x11 , x12 ,…, x1m ,…, x n-1 1 , x n-1 2 ,…, x n-1 m .$
5:     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_sine (c06ha), nag_sum_fft_real_cosine (c06hb), nag_sum_fft_real_qtrsine (c06hc) or nag_sum_fft_real_qtrcosine (c06hd). 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'$\text{'R'}$.
6:     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 calculated in a previous call of the function. Otherwise trig need not be set.

None.

work

Output Parameters

1:     x( m × n ${\mathbf{m}}×{\mathbf{n}}$) – double array
The m$m$ quarter-wave cosine 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 in a row of the array, overwriting the corresponding original sequence. If the n$n$ components of the p$\mathit{p}$th quarter-wave cosine transform are denoted by kp${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for k = 0,1,,n1$\mathit{k}=0,1,\dots ,n-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, 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${\mathbf{ifail}}=6$
 On entry, direct ≠ 'F'${\mathbf{direct}}\ne \text{'F'}$ or 'B'$\text{'B'}$.
ifail = 7${\mathbf{ifail}}=7$
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_real_qtrcosine (c06hd) 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_real_qtrcosine (c06hd) 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_real_qtrcosine_example
direct = 'Forward';
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 = zeros(12,1);
[xOut, trigOut, ifail] = nag_sum_fft_real_qtrcosine(direct, m, n, x, init, trig)
```
```

xOut =

0.7257
0.7479
0.6713
-0.2216
-0.6172
-0.1363
0.1011
0.4112
-0.0064
0.2355
0.0791
-0.0285
-0.1406
0.1331
0.4758
-0.2282
-0.0906
0.1475

trigOut =

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

ifail =

0

```
```function c06hd_example
direct = 'Forward';
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 = zeros(12,1);
[xOut, trigOut, ifail] = c06hd(direct, m, n, x, init, trig)
```
```

xOut =

0.7257
0.7479
0.6713
-0.2216
-0.6172
-0.1363
0.1011
0.4112
-0.0064
0.2355
0.0791
-0.0285
-0.1406
0.1331
0.4758
-0.2282
-0.0906
0.1475

trigOut =

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

ifail =

0

```