# NAG CL Interfacec09cdc (dim1_​multi_​inv)

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## 1Purpose

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 c09ccc from an original set of data. The initialization function c09aac must be called first to set up the DWT options.

## 2Specification

 #include
 void c09cdc (Integer nwlinv, Integer lenc, const double c[], Integer n, double y[], const Integer icomm[], NagError *fail)
The function may be called by the names: c09cdc, nag_wav_dim1_multi_inv or nag_imldwt.

## 3Description

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

None.

## 5Arguments

1: $\mathbf{nwlinv}$Integer Input
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 c09ccc. The data will be reconstructed to level $\left({\mathbf{nwl}}-{\mathbf{nwlinv}}\right)$, where level $0$ is the original input dataset provided to c09ccc.
Constraint: $1\le {\mathbf{nwlinv}}\le {n}_{\mathrm{fwd}}$, where ${n}_{\mathrm{fwd}}$ is the value used in a preceding call to c09ccc.
2: $\mathbf{lenc}$Integer Input
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 c09ccc.
3: $\mathbf{c}\left[{\mathbf{lenc}}\right]$const double Input
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 c09ccc. See c09ccc for details.
4: $\mathbf{n}$Integer Input
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 c09ccc, this must be the same as argument n used in the call to 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 c09ccc.
5: $\mathbf{y}\left[{\mathbf{n}}\right]$double Output
On exit: the dataset reconstructed from the multi-level wavelet transform coefficients and the transformation options supplied to the initialization function c09aac.
6: $\mathbf{icomm}\left[100\right]$const Integer Communication Array
On entry: contains details of the discrete wavelet transform and the problem dimension for the forward transform previously computed by c09ccc.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

c09cdc is not threaded in any implementation.