C09 Chapter Contents
C09 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC09CDF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

C09CDF computes the inverse one-dimensional multi-level discrete wavelet transform (DWT). This routine reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by C09CCF from an original set of data. The initialization routine C09AAF must be called first to set up the DWT options.

## 2  Specification

 SUBROUTINE C09CDF ( NWL, LENC, C, N, Y, ICOMM, IFAIL)
 INTEGER NWL, LENC, N, ICOMM(100), IFAIL REAL (KIND=nag_wp) C(LENC), Y(N)

## 3  Description

C09CDF performs the inverse operation of C09CCF. That is, given a set of wavelet coefficients, computed by C09CCF using a DWT as set up by the initialization routine C09AAF, on a real data array of length $n$, C09CDF 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 C09CCF. This results in a partial reconstruction.

None.

## 5  Parameters

1:     NWL – INTEGERInput
On entry: the number, ${n}_{l}$, of levels to be used in the inverse multi-level transform. The number of levels must less than or equal to the number used in the computation of the wavelet coefficients.
Constraint: $1\le {\mathbf{NWL}}\le {n}_{\mathrm{fwd}}$, where ${n}_{\mathrm{fwd}}$ is the value used in a preceding call to C09CCF.
2:     LENC – INTEGERInput
On entry: the dimension of the array C as declared in the (sub)program from which C09CDF is called.
Constraint: ${\mathbf{LENC}}\ge {n}_{c}$, where ${n}_{c}$ is the total number of coefficients that correspond to a transform with ${n}_{\mathrm{fwd}}$ levels and is unchanged from the preceding call to C09CCF.
3:     C(LENC) – REAL (KIND=nag_wp) arrayInput
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:
${\mathbf{C}}\left(\mathit{i}\right)$, 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 C09CCF. See C09CCF for details.
4:     N – INTEGERInput
On entry: $n$, the length of the data array, $y$, to be reconstructed. For a full reconstruction from ${n}_{\mathrm{fwd}}$ levels, this is the same as parameter N in the preceding call to C09CCF. For a partial reconstruction of ${n}_{l}<{n}_{\mathrm{fwd}}$ levels, this will be equal to ${\mathbf{DWTLEV}}\left({n}_{l}+2\right)$ as returned from C09CCF.
5:     Y(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the dataset reconstructed from the multi-level wavelet transform coefficients and the transformation options supplied to the initialization routine C09AAF.
6:     ICOMM($100$) – INTEGER arrayCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension for the forward transform previously computed by C09CCF.
7:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{NWL}}<1$, or ${\mathbf{NWL}}>\text{}$ the number of levels used in the computation of the wavelet coefficients by a call to C09CCF.
${\mathbf{IFAIL}}=2$
On entry, LENC is too small. LENC must be at least the number of wavelet coefficients required for a transform operating on NWL levels. If ${\mathbf{NWL}}={l}_{\mathrm{max}}$, the maximum number of levels as returned in NWL by the initial call to C09AAF, then LENC must be at least ${n}_{c}$, the value returned in NWC by the same call to C09AAF.
${\mathbf{IFAIL}}=4$
On entry, N is too small for the required level of reconstruction.
${\mathbf{IFAIL}}=6$
On entry, the initialization routine C09AAF has not been called first or it has been called with ${\mathbf{WTRANS}}=\text{'S'}$, or the communication array ICOMM has become corrupted.

## 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.