# NAG Library Function Document

## 1Purpose

nag_dwt (c09cac) computes the one-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wfilt (c09aac) must be called first to set up the DWT options.

## 2Specification

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

## 3Description

nag_dwt (c09cac) computes the one-dimensional DWT 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 and downsampling by two 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. To reduce distortion effects at the ends of the data array, several end extension methods are commonly used. Those provided are: periodic or circular convolution end extension, half-point symmetric end extension, whole-point symmetric end extension or zero end extension. The number ${n}_{c}$, of coefficients ${C}_{a}$ or ${C}_{d}$ is returned by the initialization function nag_wfilt (c09aac).
Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

## 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).
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
Either the initialization function has not been called first or array icomm has been corrupted.
Either the initialization function was called with ${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$ 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_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_dwt (c09cac) is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (c09cace.c)

### 10.2Program Data

Program Data (c09cace.d)

### 10.3Program Results

Program Results (c09cace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017