NAG Library Routine Document

c06rdf (fft_real_qtrcosine_simple)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{direct}$ – Character(1)Input

On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.

If the backward transform is to be computed, direct must be set equal to 'B'.

Constraint: $\mathbf{direct}=\text{'F'}$ or $\text{'B'}$.

2:     $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of sequences to be transformed.

Constraint: $\mathbf{m}\ge 1$.

3:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of real values in each sequence.

Constraint: $\mathbf{n}\ge 1$.

4:     $\mathbf{x}\left(\mathbf{m}\times \left(\mathbf{n}+2\right)\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the data must be stored in x as if in a two-dimensional array of dimension $\left(1:\mathbf{m},0:\mathbf{n}+1\right)$; each of the $m$ sequences is stored in a row of the array. In other words, if the data values of the $\mathit{p}$th sequence to be transformed are denoted by $x_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, the first $mn$ elements of the array x must contain the values

$$x_0^1,x_0^2,\dots ,x_0^m,x_1^1,x_1^2,\dots ,x_1^m,\dots ,x_{n-1}^1,x_{n-1}^2,\dots ,x_{n-1}^m\text{.}$$

The $\left(n+1\right)$th and $\left(n+2\right)$th elements of each row $x_n^{\mathit{p}},x_{n+1}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$, are required as workspace. These $2m$ elements may contain arbitrary values as they are set to zero by the routine.

On exit: the $m$ quarter-wave cosine transforms stored as if in a two-dimensional array of dimension $\left(1:\mathbf{m},0:\mathbf{n}+1\right)$. Each of the $m$ transforms is stored in a row of the array, overwriting the corresponding original sequence. If the $n$ components of the $\mathit{p}$th quarter-wave cosine transform are denoted by ${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, the $m\left(n+2\right)$ elements of the array x contain the values

$${\hat{x}}_0^1,{\hat{x}}_0^2,\dots ,{\hat{x}}_0^m,{\hat{x}}_1^1,{\hat{x}}_1^2,\dots ,{\hat{x}}_1^m,\dots ,{\hat{x}}_{n-1}^1,{\hat{x}}_{n-1}^2,\dots ,{\hat{x}}_{n-1}^m,0,0,\dots ,0\left(2m\text{ times}\right)\text{.}$$

5:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace

Note: the dimension of the array work must be at least $\mathbf{m}\times \mathbf{n}+2\times \mathbf{n}+2\times \mathbf{m}+15$.

The workspace requirements as documented for c06rdf may be an overestimate in some implementations.

On exit: $\mathbf{work}\left(1\right)$ contains the minimum workspace required for the current values of m and n with this implementation.

6:     $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 1$.

$\mathbf{ifail}=2$

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

$\mathbf{ifail}=3$

On entry, $\mathbf{direct}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{direct}=\text{'F'}$ or $\text{'B'}$.

$\mathbf{ifail}=4$

An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (c06rdfe.f90)

10.2

Program Data

Program Data (c06rdfe.d)

10.3

Program Results

Program Results (c06rdfe.r)