NAG Library Routine Document

c09edf (dim2_multi_inv)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{nwlinv}$ – IntegerInput

On entry: the number of levels to be used in the inverse multi-level transform. The number of levels must be less than or equal to $n_{\mathrm{fwd}}$, which has the value of argument nwl as used in the computation of the wavelet coefficients using c09ecf. The data will be reconstructed to level $\left(\mathbf{nwl}-\mathbf{nwlinv}\right)$, where level $0$ is the original input dataset provided to c09ecf.

Constraint: $1\le \mathbf{nwlinv}\le \mathbf{nwl}$, where nwl is the value used in a preceding call to c09ecf.

2:     $\mathbf{lenc}$ – IntegerInput

On entry: the dimension of the array c as declared in the (sub)program from which c09edf is called.

Constraint: $\mathbf{lenc}\ge n_{\mathrm{ct}}$, where $n_{\mathrm{ct}}$ is the total number of coefficients that correspond to a transform with nwlinv levels and is unchanged from the preceding call to c09ecf.

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

On entry: the coefficients of a multi-level wavelet transform of the original matrix, $A$, which may have been filtered or otherwise manipulated.

Let $q\left(\mathit{i}\right)$ be the number of coefficients (of each type) at level $\mathit{i}$, for $\mathit{i}=n_{\mathrm{fwd}},n_{\mathrm{fwd}}-1,\dots ,1$. Then, setting $k_1=q\left(n_{\mathrm{fwd}}\right)$ and $k_{j+1}=k_j+q\left(n_{\mathrm{fwd}}-⌈j/3⌉+1\right)$, for $j=1,2,\dots ,3n_{\mathrm{fwd}}$, 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, $a_{n_{\mathrm{fwd}}}$.

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

Contains the level $n_{\mathrm{fwd}}-⌈j/3⌉+1$ vertical, horizontal and diagonal coefficients. These are:

• vertical coefficients if $j\mathrm{ mod }3=1$;
• horizontal coefficients if $j\mathrm{ mod }3=2$;
• diagonal coefficients if $j\mathrm{ mod }3=0$,

for $j=1,\dots ,3n_{\mathrm{fwd}}$.

Note that the coefficients in c may be extracted according to level and type into two-dimensional arrays using c09eyf, and inserted using c09ezf.

4:     $\mathbf{m}$ – IntegerInput

On entry: the number of elements, $m$, in the first dimension of the reconstructed matrix $B$. For a full reconstruction of nwl levels, where nwl is as supplied to c09ecf, this must be the same as argument m used in the call to c09ecf. For a partial reconstruction of $\mathbf{nwlinv}<\mathbf{nwl}$ levels, this must be equal to $\mathbf{dwtlvm}\left(\mathbf{nwlinv}+1\right)$, as returned from c09ecf.

5:     $\mathbf{n}$ – IntegerInput

On entry: the number of elements, $n$, in the second dimension of the reconstructed matrix $B$. For a full reconstruction of nwl levels, where nwl is as supplied to c09fcf, this must be the same as argument n used in the call to c09ecf. For a partial reconstruction of $\mathbf{nwlinv}<\mathbf{nwl}$, this must be equal to $\mathbf{dwtlvn}\left(\mathbf{nwlinv}+1\right)$, as returned from c09ecf.

6:     $\mathbf{b}\left(\mathbf{ldb},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the $m$ by $n$ reconstructed matrix, $B$, based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization routine c09abf.

7:     $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which c09edf is called.

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

8:     $\mathbf{icomm}\left(180\right)$ – 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 c09abf.

9:     $\mathbf{ifail}$ – 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, $\mathbf{nwlinv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nwlinv}\ge 1$.

On entry, $\mathbf{nwlinv}=〈\mathit{\text{value}}〉$ and $n_{\mathrm{fwd}}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nwlinv}\le n_{\mathrm{fwd}}$.

$\mathbf{ifail}=2$

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

$\mathbf{ifail}=3$

On entry, $\mathbf{lenc}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lenc}\ge 〈\mathit{\text{value}}〉$, the total number of coefficients generated by the preceding call to c09ecf.

$\mathbf{ifail}=4$

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 〈\mathit{\text{value}}〉$, the number of coefficients in the first dimension at the required level of reconstruction.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 〈\mathit{\text{value}}〉$, the number of coefficients in the second dimension at the required level of reconstruction.

$\mathbf{ifail}=6$

Either the initialization routine has not been called first or icomm has been corrupted.

Either the initialization routine was called with $\mathbf{wtrans}=\text{'S'}$ or icomm has been corrupted.

$\mathbf{ifail}=-99$

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