# NAG CL Interfacec09cac (dim1_​sngl_​fwd)

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## 1Purpose

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

## 2Specification

 #include
 void c09cac (Integer n, const double x[], Integer lenc, double ca[], double cd[], Integer icomm[], NagError *fail)
The function may be called by the names: c09cac, nag_wav_dim1_sngl_fwd or nag_dwt.

## 3Description

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 c09aac.
Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

## 5Arguments

1: $\mathbf{n}$Integer Input
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 c09aac.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: x contains the input dataset ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
3: $\mathbf{lenc}$Integer Input
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 c09aac.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the value returned in nwc by the call to the initialization function c09aac.
4: $\mathbf{ca}\left[{\mathbf{lenc}}\right]$double Output
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: $\mathbf{cd}\left[{\mathbf{lenc}}\right]$double Output
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: $\mathbf{icomm}\left[100\right]$Integer Communication Array
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function c09aac.
On exit: contains additional information on the computed transform.
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, array dimension lenc not large enough: ${\mathbf{lenc}}=⟨\mathit{\text{value}}⟩$ but must be at least $⟨\mathit{\text{value}}⟩$.
NE_BAD_PARAM
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 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

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)