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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_wav_1d_multi_inv (c09cd)

## Purpose

nag_wav_1d_multi_inv (c09cd) computes the inverse one-dimensional multi-level discrete wavelet transform (DWT). This function reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by nag_wav_1d_multi_fwd (c09cc) from an original set of data. The initialization function nag_wav_1d_init (c09aa) must be called first to set up the DWT options.

## Syntax

[y, ifail] = c09cd(nwlinv, c, n, icomm, 'lenc', lenc)
[y, ifail] = nag_wav_1d_multi_inv(nwlinv, c, n, icomm, 'lenc', lenc)

## Description

nag_wav_1d_multi_inv (c09cd) performs the inverse operation of nag_wav_1d_multi_fwd (c09cc). That is, given a set of wavelet coefficients, computed up to level ${n}_{\mathrm{fwd}}$ by nag_wav_1d_multi_fwd (c09cc) using a DWT as set up by the initialization function nag_wav_1d_init (c09aa), on a real data array of length $n$, nag_wav_1d_multi_inv (c09cd) will reconstruct the data array ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, from which the coefficients were derived. If the original input dataset is level $0$, then it is possible to terminate reconstruction at a higher level by specifying fewer than the number of levels used in the call to nag_wav_1d_multi_fwd (c09cc). This results in a partial reconstruction.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{nwlinv}$int64int32nag_int scalar
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 nag_wav_1d_multi_fwd (c09cc). The data will be reconstructed to level $\left({\mathbf{nwl}}-{\mathbf{nwlinv}}\right)$, where level $0$ is the original input dataset provided to nag_wav_1d_multi_fwd (c09cc).
Constraint: $1\le {\mathbf{nwlinv}}\le {\mathbf{nwl}}$, where nwl is the value used in a preceding call to nag_wav_1d_multi_fwd (c09cc).
2:     $\mathrm{c}\left({\mathbf{lenc}}\right)$ – double array
The coefficients of a multi-level wavelet transform of the dataset.
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 ${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{\mathrm{fwd}}-\mathit{j}+1\right)$, for $\mathit{j}=1,2,\dots ,{n}_{\mathrm{fwd}}$, 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 $\mathit{i}={k}_{1}+1,\dots ,{k}_{2}$
Contains the level ${n}_{\mathrm{fwd}}$ detail coefficients ${d}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left(\mathit{i}\right)$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level ${n}_{\mathrm{fwd}}-\mathit{j}+1$ detail coefficients, for $\mathit{j}=2,3,\dots ,{n}_{\mathrm{fwd}}$.
The values $q\left(\mathit{i}\right)$, for $\mathit{i}={n}_{\mathrm{fwd}},{n}_{\mathrm{fwd}}-1,\dots ,1$, are contained in dwtlev which is produced as output by a preceding call to nag_wav_1d_multi_fwd (c09cc). See nag_wav_1d_multi_fwd (c09cc) for details.
3:     $\mathrm{n}$int64int32nag_int scalar
$n$, the length of the data array, $y$, to be reconstructed. For a full reconstruction of nwl levels, where nwl is as supplied to nag_wav_1d_multi_fwd (c09cc), this must be the same as argument n used in the call to nag_wav_1d_multi_fwd (c09cc). For a partial reconstruction of ${\mathbf{nwlinv}}<{\mathbf{nwl}}$, this must be equal to ${\mathbf{dwtlev}}\left({\mathbf{nwlinv}}+2\right)$, as returned from nag_wav_1d_multi_fwd (c09cc).
4:     $\mathrm{icomm}\left(100\right)$int64int32nag_int array
Contains details of the discrete wavelet transform and the problem dimension for the forward transform previously computed by nag_wav_1d_multi_fwd (c09cc).

### Optional Input Parameters

1:     $\mathrm{lenc}$int64int32nag_int scalar
Default: the dimension of the array c.
The dimension of the array c.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the total number of coefficients that correspond to a transform with nwlinv levels and is unchanged from the preceding call to nag_wav_1d_multi_fwd (c09cc).

### Output Parameters

1:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The dataset reconstructed from the multi-level wavelet transform coefficients and the transformation options supplied to the initialization function nag_wav_1d_init (c09aa).
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{nwlinv}}\ge 1$.
On entry, nwlinv is larger than the number of levels computed by the preceding call to nag_wav_1d_multi_fwd (c09cc).
${\mathbf{ifail}}=2$
lenc is too small.
${\mathbf{ifail}}=4$
On entry, n is inconsistent with the value passed to the initialization function.
${\mathbf{ifail}}=6$
Either the initialization function has not been called first or array icomm has been corrupted.
Either the initialization function was called with ${\mathbf{wtrans}}=\text{'S'}$ or array icomm has been corrupted.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The accuracy of the wavelet transform depends only on the floating-point operations used in the convolution and downsampling and should thus be close to machine precision.

None.

## Example

See Example in nag_wav_1d_multi_fwd (c09cc).
```function c09cd_example

fprintf('c09cd example results\n\n');

n = int64(64);
wavnam = 'DB4';
mode = 'zero';
wtrans = 'Multilevel';
x = [ 6.5271; 6.512; 6.5016; 6.5237; 6.4625;
6.3496; 6.4025; 6.4035; 6.4407; 6.4746;
6.5095; 6.6551; 6.61; 6.5969; 6.6083;
6.652; 6.7113; 6.7227; 6.7196; 6.7649;
6.7794; 6.8037; 6.8308; 6.7712; 6.7067;
6.769; 6.7068; 6.7024; 6.6463; 6.6098;
6.59; 6.596; 6.5457; 6.547; 6.5797;
6.5895; 6.6275; 6.6795; 6.6598; 6.6925;
6.6873; 6.7223; 6.7205; 6.6843; 6.703;
6.647; 6.6008; 6.6061; 6.6097; 6.6485;
6.6394; 6.6571; 6.6357; 6.6224; 6.6073;
6.6075; 6.6379; 6.6294; 6.5906; 6.6258;
6.6369; 6.6515; 6.6826; 6.7042];
fprintf('\n Input Data:\n');
for i=1:8:double(n)
fprintf('%8.4f ', x(i:i+8-1));
fprintf('\n');
end
fprintf('\n');

% Query wavelet filter dimensions
[nwl, nf, nwc, icomm, ifail] = c09aa(wavnam, wtrans, mode, n);

if ifail == int64(0)
% Perform Discrete Wavelet transform
[c, dwtlev, icomm, ifail] = c09cc(x, nwc, nwl, icomm);

if ifail == int64(0)
fprintf(' Length of wavelet filter :             %10d\n', nf);
fprintf(' Number of Levels :                     %10d\n\n', nwl);
fprintf(' Number of coefficients in each level :\n     ');
fprintf(' %8d', dwtlev);
fprintf('\n');
fprintf(' Total number of wavelet coefficients : %10d\n\n', nwc);
fprintf(' Wavelet coefficients C : \n');
for i=1:8:double(nwc)
if i+8-1 <= numel(c)
fprintf('%8.4f ', c(i:i+8-1));
else
fprintf('%8.4f ', c(i:numel(c)));
end
fprintf('\n');
end
fprintf('\n');

% Reconstruct original data
[y, ifail] = c09cd(nwl, c, n, icomm);

if ifail == int64(0)
fprintf('\n Reconstruction       Y : \n');
for i=1:8:double(n)
fprintf('%8.4f ', y(i:i+8-1));
fprintf('\n');
end
fprintf('\n');
end
end
end

```
```c09cd example results

Input Data:
6.5271   6.5120   6.5016   6.5237   6.4625   6.3496   6.4025   6.4035
6.4407   6.4746   6.5095   6.6551   6.6100   6.5969   6.6083   6.6520
6.7113   6.7227   6.7196   6.7649   6.7794   6.8037   6.8308   6.7712
6.7067   6.7690   6.7068   6.7024   6.6463   6.6098   6.5900   6.5960
6.5457   6.5470   6.5797   6.5895   6.6275   6.6795   6.6598   6.6925
6.6873   6.7223   6.7205   6.6843   6.7030   6.6470   6.6008   6.6061
6.6097   6.6485   6.6394   6.6571   6.6357   6.6224   6.6073   6.6075
6.6379   6.6294   6.5906   6.6258   6.6369   6.6515   6.6826   6.7042

Length of wavelet filter :                      8
Number of Levels :                              6

Number of coefficients in each level :
7        7        8       10       14       21       35
Total number of wavelet coefficients :        102

Wavelet coefficients C :
0.0000  -0.0227  -0.3446   2.7574 -10.1970  44.8800  15.9443   0.0010
-0.4881 -10.2673  11.3258  -1.7469   2.0785  -0.7334  -0.0054  -0.1402
-5.8980  -1.1527   5.5613   2.1352   0.3203  -0.4004   0.0010   0.5229
0.5055  -2.7274  -0.0911  -0.2806  -0.3669   2.9467  -0.3799  -0.1552
0.0218   0.0922   5.4626  -2.1620   0.5196  -0.0287  -0.0199   0.0920
-0.0134  -0.1298  -5.5168   2.3105  -0.5383  -0.0155   0.3057   0.6186
-1.5542   0.2682   0.1566   0.0030  -0.0152  -0.0589   0.0126   0.0063
0.0171  -0.0268   0.0077  -0.0189   0.0207   0.0104  -0.3207  -0.6062
1.6288  -0.2414  -0.0671   3.1657  -1.1462   0.2785   0.0523  -0.0030
-0.0270  -0.0442   0.0090   0.0171  -0.0230  -0.0015   0.0213  -0.0402
-0.0263  -0.0099   0.0021  -0.0250   0.0210  -0.0028  -0.0298  -0.0095
0.0034   0.0281  -0.0188  -0.0002  -0.0173  -0.0076  -0.0014   0.0184
-0.0318   0.0048   0.0047  -3.2555   1.1710  -0.2913

Reconstruction       Y :
6.5271   6.5120   6.5016   6.5237   6.4625   6.3496   6.4025   6.4035
6.4407   6.4746   6.5095   6.6551   6.6100   6.5969   6.6083   6.6520
6.7113   6.7227   6.7196   6.7649   6.7794   6.8037   6.8308   6.7712
6.7067   6.7690   6.7068   6.7024   6.6463   6.6098   6.5900   6.5960
6.5457   6.5470   6.5797   6.5895   6.6275   6.6795   6.6598   6.6925
6.6873   6.7223   6.7205   6.6843   6.7030   6.6470   6.6008   6.6061
6.6097   6.6485   6.6394   6.6571   6.6357   6.6224   6.6073   6.6075
6.6379   6.6294   6.5906   6.6258   6.6369   6.6515   6.6826   6.7042

```