c09 Chapter Contents
c09 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_imlmodwt (c09ddc)

## 1  Purpose

nag_imlmodwt (c09ddc) computes the inverse one-dimensional multi-level maximal overlap discrete wavelet transform (MODWT). This function reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by nag_mlmodwt (c09dcc) from an original set of data. The initialization function nag_wfilt (c09aac) must be called first to set up the MODWT options.

## 2  Specification

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

## 3  Description

nag_imlmodwt (c09ddc) performs the inverse operation of nag_mlmodwt (c09dcc). That is, given a set of wavelet coefficients computed by nag_mlmodwt (c09dcc) using a MODWT as set up by the initialization function nag_wfilt (c09aac) on a real array of length $n$, nag_imlmodwt (c09ddc) will reconstruct the data array ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, from which the coefficients were derived.
Percival D B and Walden A T (2000) Wavelet Methods for Time Series Analysis Cambridge University Press

## 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_mlmodwt (c09dcc). The data will be reconstructed to level $\left({\mathbf{nwl}}-{\mathbf{nwlinv}}\right)$, where level $0$ is the original input dataset provided to nag_mlmodwt (c09dcc).
Constraint: $1\le {\mathbf{nwlinv}}\le {n}_{\mathrm{fwd}}$, where ${n}_{\mathrm{fwd}}$ is the value used in a preceding call to nag_mlmodwt (c09dcc).
2:    $\mathbf{keepa}$Nag_WaveletCoefficientsInput
On entry: determines whether the approximation coefficients are stored in array c for every level of the computed transform or else only for the final level. In both cases, the detail coefficients are stored in c for every level computed.
${\mathbf{keepa}}=\mathrm{Nag_StoreAll}$
Retain approximation coefficients for all levels computed.
${\mathbf{keepa}}=\mathrm{Nag_StoreFinal}$
Retain approximation coefficients for only the final level computed.
Constraint: ${\mathbf{keepa}}=\mathrm{Nag_StoreAll}$ or $\mathrm{Nag_StoreFinal}$.
3:    $\mathbf{lenc}$IntegerInput
On entry: the dimension of the array c.
Constraints:
• if ${\mathbf{keepa}}=\mathrm{Nag_StoreFinal}$, ${\mathbf{lenc}}\ge \left({n}_{l}+1\right)×{n}_{a}$;
• if ${\mathbf{keepa}}=\mathrm{Nag_StoreAll}$, ${\mathbf{lenc}}\ge 2×{n}_{l}×{n}_{a}$, where ${n}_{a}$ is the number of approximation or detail coefficients at each level and is unchanged from the preceding call to nag_mlmodwt (c09dcc).
4:    $\mathbf{c}\left[{\mathbf{lenc}}\right]$const doubleInput
On entry: the coefficients of a multi-level wavelet transform of the dataset.
The coefficients are stored in c as follows:
If ${\mathbf{keepa}}=\mathrm{Nag_StoreFinal}$,
${\mathbf{C}}\left(1:{n}_{a}\right)$
Contains the level ${n}_{l}$ approximation coefficients;
${\mathbf{C}}\left({n}_{a}+\left(i-1\right)×{n}_{d}+1:{n}_{a}+i×{n}_{d}\right)$
Contains the level $\left({n}_{l}-\mathit{i}+1\right)$ detail coefficients, for $\mathit{i}=1,2,\dots ,{n}_{l}$;
If ${\mathbf{keepa}}=\mathrm{Nag_StoreAll}$,
${\mathbf{C}}\left(\left(i-1\right)×{n}_{a}+1:i×{n}_{a}\right)$
Contains the level $\left({n}_{l}-\mathit{i}+1\right)$ approximation coefficients, for $\mathit{i}=1,2,\dots ,{n}_{l}$;
${\mathbf{C}}\left({n}_{l}×{n}_{a}+\left(i-1\right)×{n}_{d}+1:{n}_{l}×{n}_{a}+i×{n}_{d}\right)$
Contains the level $\mathit{i}$ detail coefficients, for $\mathit{i}=1,2,\dots ,{n}_{l}$.
The values ${n}_{a}$ and ${n}_{d}$ denote the numbers of approximation and detail coefficients respectively, which are equal. This number is returned as output in na from a preceding call to nag_mlmodwt (c09dcc). See nag_mlmodwt (c09dcc) for details.
5:    $\mathbf{n}$IntegerInput
On entry: $n$, the length of the data array, $y$, to be reconstructed.
Constraint: This must be the same as the value n passed to the initialization function nag_wfilt (c09aac).
6:    $\mathbf{y}\left[{\mathbf{n}}\right]$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).
7:    $\mathbf{icomm}\left[100\right]$const IntegerCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension for the forward transform previously computed by nag_mlmodwt (c09dcc).
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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
On entry, n is inconsistent with the value passed to the initialization function: ${\mathbf{n}}=〈\mathit{\text{value}}〉$, n should be $〈\mathit{\text{value}}〉$.
On entry, the initialization function nag_wfilt (c09aac) has not been called first or it has not been called with ${\mathbf{wtrans}}=\mathrm{Nag_MODWTMulti}$, or the communication array icomm has become corrupted.
NE_INT
On entry, ${\mathbf{nwlinv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nwlinv}}\ge 1$.
NE_INT_2
On entry, nwlinv is larger than the number of levels computed by the preceding call to nag_mlmodwt (c09dcc): ${\mathbf{nwlinv}}=〈\mathit{\text{value}}〉$, expected $〈\mathit{\text{value}}〉$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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.

## 8  Parallelism and Performance

nag_imlmodwt (c09ddc) is not threaded in any implementation.