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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 nc${n}_{c}$ 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$n$, nag_wav_1d_sngl_inv (c09cb) will reconstruct the data array yi${y}_{i}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$, from which the coefficients were derived.

None.

## Parameters

### Compulsory Input Parameters

1:     ca(lenc) – double array
lenc, the dimension of the array, must satisfy the constraint lencnc${\mathbf{lenc}}\ge {n}_{c}$, where nc${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
The nc${n}_{c}$ approximation coefficients, Ca${C}_{a}$. These will normally be the result of some transformation on the coefficients computed by nag_wav_1d_sngl_fwd (c09ca).
2:     cd(lenc) – double array
lenc, the dimension of the array, must satisfy the constraint lencnc${\mathbf{lenc}}\ge {n}_{c}$, where nc${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
The nc${n}_{c}$ detail coefficients, Cd${C}_{d}$. These will normally be the result of some transformation on the coefficients computed by nag_wav_1d_sngl_fwd (c09ca).
3:     n – int64int32nag_int scalar
n$n$, 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:     icomm(100$100$) – 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:     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 as declared in the (sub)program from which nag_wav_1d_sngl_inv (c09cb) is called.
Constraint: lencnc${\mathbf{lenc}}\ge {n}_{c}$, where nc${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).

None.

### Output Parameters

1:     y(n) – double array
The reconstructed data based on approximation and detail coefficients Ca${C}_{a}$ and Cd${C}_{d}$ and the transform options supplied to the initialization function nag_wav_1d_init (c09aa).
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, lenc < nc${\mathbf{lenc}}<{n}_{c}$, where nc${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
ifail = 4${\mathbf{ifail}}=4$
On entry, n is inconsistent with the value passed to the initialization function nag_wav_1d_init (c09aa).
ifail = 6${\mathbf{ifail}}=6$
On entry, the initialization function nag_wav_1d_init (c09aa) has not been called first or it has been called with wtrans = 'M'${\mathbf{wtrans}}=\text{'M'}$, or the communication array icomm has become corrupted.

## 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

function nag_wav_1d_sngl_inv_example
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] = nag_wav_1d_init(wavnam, wtrans, mode, n);

if ifail == int64(0)
% Compute the transform
[ca, cd, icomm, ifail] = nag_wav_1d_sngl_fwd(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] = nag_wav_1d_sngl_inv(ca, cd, n, icomm);

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

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

function c09cb_example
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

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