# NAG C Library Function Document

## 1Purpose

nag_modwt (c09dac) computes the one-dimensional maximal overlap discrete wavelet transform (MODWT) at a single level. The initialization function nag_wfilt (c09aac) must be called first to set up the MODWT options.

## 2Specification

 #include #include
 void nag_modwt (Integer n, const double x[], Integer lenc, double ca[], double cd[], Integer icomm[], NagError *fail)

## 3Description

nag_modwt (c09dac) computes the one-dimensional MODWT of a given input data array, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, at a single level. For a chosen wavelet filter pair, the output coefficients are obtained by applying convolution to the input, $x$. The approximation (or smooth) coefficients, ${C}_{a}$, are produced by the low pass filter and the detail coefficients, ${C}_{d}$, by the high pass filter. Periodic (circular) convolution is available as an end extension method for application to finite data sets. The number ${n}_{c}$, of coefficients ${C}_{a}$ or ${C}_{d}$ is returned by the initialization function nag_wfilt (c09aac).
Percival D B and Walden A T (2000) Wavelet Methods for Time Series Analysis Cambridge University Press

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: the number of elements, $n$, in the data array $x$.
Constraint: this must be the same as the value n passed to the initialization function nag_wfilt (c09aac).
2:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: x contains the input dataset ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
3:    $\mathbf{lenc}$IntegerInput
On entry: the dimension of the arrays ca and cd. This must be at least the number, ${n}_{c}$, of approximation coefficients, ${C}_{a}$, and detail coefficients, ${C}_{d}$, of the discrete wavelet transform as returned in nwc by the call to the initialization function nag_wfilt (c09aac). Note that ${n}_{c}=n$ for periodic end extension, but this is not the case for other end extension methods which will be available in future releases.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the value returned in nwc by the call to the initialization function nag_wfilt (c09aac).
4:    $\mathbf{ca}\left[{\mathbf{lenc}}\right]$doubleOutput
On exit: ${\mathbf{ca}}\left[i-1\right]$ contains the $i$th approximation coefficient, ${C}_{a}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{c}$.
5:    $\mathbf{cd}\left[{\mathbf{lenc}}\right]$doubleOutput
On exit: ${\mathbf{cd}}\left[\mathit{i}-1\right]$ contains the $\mathit{i}$th detail coefficient, ${C}_{d}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{c}$.
6:    $\mathbf{icomm}\left[100\right]$IntegerCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wfilt (c09aac).
On exit: contains additional information on the computed transform.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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, array dimension lenc not large enough: ${\mathbf{lenc}}=〈\mathit{\text{value}}〉$ but must be at least $〈\mathit{\text{value}}〉$.
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 ${\mathbf{wtrans}}=\mathrm{Nag_MODWTSingle}$, or the communication array icomm has become corrupted.
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.

## 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

nag_modwt (c09dac) is not threaded in any implementation.

None.

## 10Example

This example computes the one-dimensional maximal overlap discrete wavelet decomposition for $8$ values using the Daubechies wavelet, ${\mathbf{wavnam}}=\mathrm{Nag_Daubechies4}$.

### 10.1Program Text

Program Text (c09dace.c)

### 10.2Program Data

Program Data (c09dace.d)

### 10.3Program Results

Program Results (c09dace.r)