c09 Chapter Contents
c09 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_imldwt (c09cdc)

## 1  Purpose

nag_imldwt (c09cdc) 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_mldwt (c09ccc) from an original set of data. The initialization function nag_wfilt (c09aac) must be called first to set up the DWT options.

## 2  Specification

 #include #include
 void nag_imldwt (Integer nwlinv, Integer lenc, const double c[], Integer n, double y[], const Integer icomm[], NagError *fail)

## 3  Description

nag_imldwt (c09cdc) performs the inverse operation of nag_mldwt (c09ccc). That is, given a set of wavelet coefficients, computed up to level ${n}_{\mathrm{fwd}}$ by nag_mldwt (c09ccc) using a DWT as set up by the initialization function nag_wfilt (c09aac), on a real data array of length $n$, nag_imldwt (c09cdc) 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_mldwt (c09ccc). This results in a partial reconstruction.

None.

## 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 nag_mldwt (c09ccc). The data will be reconstructed to level $\left({\mathbf{nwl}}-{\mathbf{nwlinv}}\right)$, where level $0$ is the original input dataset provided to nag_mldwt (c09ccc).
Constraint: $1\le {\mathbf{nwlinv}}\le {n}_{\mathrm{fwd}}$, where ${n}_{\mathrm{fwd}}$ is the value used in a preceding call to nag_mldwt (c09ccc).
2:    $\mathbf{lenc}$IntegerInput
On entry: 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_mldwt (c09ccc).
3:    $\mathbf{c}\left[{\mathbf{lenc}}\right]$const doubleInput
On entry: 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:
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{k}_{1}$
Contains the level ${n}_{\mathrm{fwd}}$ approximation coefficients, ${a}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}={k}_{1}+1,\dots ,{k}_{2}$
Contains the level ${n}_{\mathrm{fwd}}$ detail coefficients ${d}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left[\mathit{i}-1\right]$, 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_mldwt (c09ccc). See nag_mldwt (c09ccc) for details.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the length of the data array, $y$, to be reconstructed. For a full reconstruction of nwl levels, where nwl is as supplied to nag_mldwt (c09ccc), this must be the same as argument n used in the call to nag_mldwt (c09ccc). For a partial reconstruction of ${\mathbf{nwlinv}}<{\mathbf{nwl}}$, this must be equal to ${\mathbf{dwtlev}}\left[{\mathbf{nwlinv}}+1\right]$, as returned from nag_mldwt (c09ccc).
5:    $\mathbf{y}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the dataset reconstructed from the multi-level wavelet transform coefficients and the transformation options supplied to the initialization function nag_wfilt (c09aac).
6:    $\mathbf{icomm}\left[100\right]$const IntegerCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension for the forward transform previously computed by nag_mldwt (c09ccc).
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_DIM_LEN
On entry, lenc is set too small: ${\mathbf{lenc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lenc}}\ge 〈\mathit{\text{value}}〉$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INITIALIZATION
Either the initialization function has not been called first or array icomm has been corrupted.
Either the initialization function was called with ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$ or array icomm has been corrupted.
On entry, n is inconsistent with the value passed to the initialization function: ${\mathbf{n}}=〈\mathit{\text{value}}〉$, n should be $〈\mathit{\text{value}}〉$.
NE_INT_2
On entry, ${\mathbf{nwlinv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nwlinv}}\ge 1$.
On entry, nwlinv is larger than the number of levels computed by the preceding call to nag_mldwt (c09ccc): ${\mathbf{nwlinv}}=〈\mathit{\text{value}}〉$, expected $\text{}=〈\mathit{\text{value}}〉$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  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.

Not applicable.