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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_wav_1d_sngl_inv (c09cb)

## Purpose

nag_wav_1d_sngl_inv (c09cb) computes the inverse one-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wav_1d_init (c09aa) must be called first to set up the DWT options.

## Syntax

[y, ifail] = c09cb(ca, cd, n, icomm, 'lenc', lenc)
[y, ifail] = nag_wav_1d_sngl_inv(ca, cd, n, icomm, 'lenc', lenc)

## Description

nag_wav_1d_sngl_inv (c09cb) performs the inverse operation of nag_wav_1d_sngl_fwd (c09ca). That is, given sets of ${n}_{c}$ approximation coefficients and detail coefficients, computed by nag_wav_1d_sngl_fwd (c09ca) 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_sngl_inv (c09cb) will reconstruct the data array ${y}_{i}$, for $\mathit{i}=1,2,\dots ,n$, from which the coefficients were derived.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{ca}\left({\mathbf{lenc}}\right)$ – double array
The ${n}_{c}$ approximation coefficients, ${C}_{a}$. These will normally be the result of some transformation on the coefficients computed by nag_wav_1d_sngl_fwd (c09ca).
2:     $\mathrm{cd}\left({\mathbf{lenc}}\right)$ – double array
The ${n}_{c}$ detail coefficients, ${C}_{d}$. These will normally be the result of some transformation on the coefficients computed by nag_wav_1d_sngl_fwd (c09ca).
3:     $\mathrm{n}$int64int32nag_int scalar
$n$, the length of the original data array from which the wavelet coefficients were computed by nag_wav_1d_sngl_fwd (c09ca) and the length of the data array y that is to be reconstructed by this function.
Constraint: This must be the same as the value n passed to the initialization function nag_wav_1d_init (c09aa).
4:     $\mathrm{icomm}\left(100\right)$int64int32nag_int array
Contains details of the discrete wavelet transform and the problem dimension and, possibly, additional information on the previously computed forward transform.

### Optional Input Parameters

1:     $\mathrm{lenc}$int64int32nag_int scalar
Default: the dimension of the arrays ca, cd. (An error is raised if these dimensions are not equal.)
The dimension of the arrays ca and cd.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).

### Output Parameters

1:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The reconstructed data based on approximation and detail coefficients ${C}_{a}$ and ${C}_{d}$ and the transform 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:
${\mathbf{ifail}}=1$
On entry, array dimension lenc not large enough.
${\mathbf{ifail}}=4$
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 ${\mathbf{wtrans}}=\text{'M'}$ 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_sngl_fwd (c09ca).
```function c09cb_example

fprintf('c09cb example results\n\n');

n = int64(8);
wavnam = 'DB4';
mode = 'zero';
wtrans = 'Single Level';
x = [1; 3; 5; 7; 6; 4; 5; 2];
fprintf('\n Input Data:\n');
fprintf('%8.4f ', x);
fprintf('\n');

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

if ifail == int64(0)
% Compute the transform
[ca, cd, icomm, ifail] = c09ca(x, nwc, icomm);

if ifail == int64(0)
fprintf(' Approximation coefficients CA :\n');
fprintf('%8.4f ', ca);
fprintf('\n');
fprintf(' Detail coefficients        CD :\n');
fprintf('%8.4f ', cd);
fprintf('\n');

% Reconstruct original data
[y, ifail] = c09cb(ca, cd, n, icomm);

if ifail == int64(0)
fprintf(' Reconstruction       Y : \n');
fprintf('%8.4f ', y);
fprintf('\n');
end
end
end

```
```c09cb example results

Input Data:
1.0000   3.0000   5.0000   7.0000   6.0000   4.0000   5.0000   2.0000
Approximation coefficients CA :
0.0011  -0.0043  -0.0174   4.4778   8.9557   7.3401   2.5816
Detail coefficients        CD :
0.0237   0.0410  -0.5966   1.7763  -0.7517   0.3332  -0.1188
Reconstruction       Y :
1.0000   3.0000   5.0000   7.0000   6.0000   4.0000   5.0000   2.0000
```