NAG Library Routine Document

c09fcf  (dim3_multi_fwd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{m}$ – IntegerInput

On entry: the number of rows of each two-dimensional frame.

Constraint: this must be the same as the value m passed to the initialization routine c09acf.

2:     $\mathbf{n}$ – IntegerInput

On entry: the number of columns of each two-dimensional frame.

Constraint: this must be the same as the value n passed to the initialization routine c09acf.

3:     $\mathbf{fr}$ – IntegerInput

On entry: the number of two-dimensional frames.

Constraint: this must be the same as the value fr passed to the initialization routine c09acf.

4:     $\mathbf{a}\left(\mathbf{lda},\mathbf{sda},\mathbf{fr}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the $m$ by $n$ by $\mathit{fr}$ three-dimensional input data $A$, where with $A_{ijk}$ stored in $\mathbf{a}\left(i,j,k\right)$.

5:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which c09fcf is called.

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

6:     $\mathbf{sda}$ – IntegerInput

On entry: the second dimension of the array a as declared in the (sub)program from which c09fcf is called.

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

7:     $\mathbf{lenc}$ – IntegerInput

On entry: the dimension of the array c as declared in the (sub)program from which c09fcf is called.

Constraint: $\mathbf{lenc}\ge n_{\mathrm{ct}}$, where $n_{\mathrm{ct}}$ is the total number of wavelet coefficients that correspond to a transform with nwl levels.

8:     $\mathbf{c}\left(\mathbf{lenc}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the coefficients of the discrete wavelet transform. If you need to access or modify the approximation coefficients or any specific set of detail coefficients then the use of c09fyf or c09fzf is recommended. For completeness the following description provides details of precisely how the coefficients are stored in c but this information should only be required in rare cases.

Let $q\left(\mathit{i}\right)$ denote the number of coefficients of each type at level $\mathit{i}$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$, such that $q\left(i\right)=\mathbf{dwtlvm}\left(n_{\mathrm{fwd}}-i+1\right)\times \mathbf{dwtlvn}\left(n_{\mathrm{fwd}}-i+1\right)\times \mathbf{dwtlvfr}\left(n_{\mathrm{fwd}}-i+1\right)$. Then, letting $k_1=q\left(n_{\mathrm{fwd}}\right)$ and $k_{\mathit{j}+1}=k_{\mathit{j}}+q\left(n_{\mathrm{fwd}}-⌈\mathit{j}/7⌉+1\right)$, for $\mathit{j}=1,2,\dots ,7n_{\mathrm{fwd}}$, the coefficients are stored in c as follows:

$\mathbf{c}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,k_1$

Contains the level $n_{\mathrm{fwd}}$ approximation coefficients, $a_{n_{\mathrm{fwd}}}$. Note that for computational efficiency reasons these coefficients are stored as $\mathbf{dwtlvm}\left(1\right)\times \mathbf{dwtlvn}\left(1\right)\times \mathbf{dwtlvfr}\left(1\right)$ in c.

$\mathbf{c}\left(\mathit{i}\right)$, for $\mathit{i}=k_j+1,\dots ,k_{j+1}$

Contains the level $n_{\mathrm{fwd}}-⌈j/7⌉+1$ detail coefficients. These are:

• LLH coefficients if $j\mathrm{ mod }7=1$;
• LHL coefficients if $j\mathrm{ mod }7=2$;
• LHH coefficients if $j\mathrm{ mod }7=3$;
• HLL coefficients if $j\mathrm{ mod }7=4$;
• HLH coefficients if $j\mathrm{ mod }7=5$;
• HHL coefficients if $j\mathrm{ mod }7=6$;
• HHH coefficients if $j\mathrm{ mod }7=0$,

for $j=1,\dots ,7n_{\mathrm{fwd}}$. See Section 2.1 in the C09 Chapter Introduction for a description of how these coefficients are produced.

Note that for computational efficiency reasons these coefficients are stored as $\mathbf{dwtlvfr}\left(⌈j/7⌉\right)\times \mathbf{dwtlvm}\left(⌈j/7⌉\right)\times \mathbf{dwtlvn}\left(⌈j/7⌉\right)$ in c.

9:     $\mathbf{nwl}$ – IntegerInput

On entry: the number of levels, $n_{\mathrm{fwd}}$, in the multi-level resolution to be performed.

Constraint: $1\le \mathbf{nwl}\le l_{\max }$, where $l_{\max }$ is the value returned in nwlmax (the maximum number of levels) by the call to the initialization routine c09acf.

10:   $\mathbf{dwtlvm}\left(\mathbf{nwl}\right)$ – Integer arrayOutput

On exit: the number of coefficients in the first dimension for each coefficient type at each level. $\mathbf{dwtlvm}\left(\mathit{i}\right)$ contains the number of coefficients in the first dimension (for each coefficient type computed) at the ($n_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$.

11:   $\mathbf{dwtlvn}\left(\mathbf{nwl}\right)$ – Integer arrayOutput

On exit: the number of coefficients in the second dimension for each coefficient type at each level. $\mathbf{dwtlvn}\left(\mathit{i}\right)$ contains the number of coefficients in the second dimension (for each coefficient type computed) at the ($n_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$.

12:   $\mathbf{dwtlvfr}\left(\mathbf{nwl}\right)$ – Integer arrayOutput

On exit: the number of coefficients in the third dimension for each coefficient type at each level. $\mathbf{dwtlvfr}\left(\mathit{i}\right)$ contains the number of coefficients in the third dimension (for each coefficient type computed) at the ($n_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$.

13:   $\mathbf{icomm}\left(260\right)$ – Integer arrayCommunication Array

On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine c09acf.

On exit: contains additional information on the computed transform.

14:   $\mathbf{ifail}$ – 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, $\mathbf{fr}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{fr}=〈\mathit{\text{value}}〉$, the value of fr on initialization (see c09acf).

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}=〈\mathit{\text{value}}〉$, the value of m on initialization (see c09acf).

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}=〈\mathit{\text{value}}〉$, the value of n on initialization (see c09acf).

$\mathbf{ifail}=2$

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

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

$\mathbf{ifail}=3$

On entry, $\mathbf{lenc}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lenc}\ge 〈\mathit{\text{value}}〉$, the total number of coefficents to be generated.

$\mathbf{ifail}=5$

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

On entry, $\mathbf{nwl}=〈\mathit{\text{value}}〉$ and $\mathbf{nwlmax}=〈\mathit{\text{value}}〉$ in c09acf.
Constraint: $\mathbf{nwl}\le \mathbf{nwlmax}$ in c09acf.

$\mathbf{ifail}=6$

Either the communication array icomm has been corrupted or there has not been a prior call to the initialization routine c09acf.

The initialization routine was called with $\mathbf{wtrans}=\text{'S'}$.

$\mathbf{ifail}=-99$

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 (c09fcfe.f90)

10.2

Program Data

Program Data (c09fcfe.d)

10.3

Program Results

Program Results (c09fcfe.r)