# NAG FL Interfacec09caf (dim1_​sngl_​fwd)

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## 1Purpose

c09caf computes the one-dimensional discrete wavelet transform (DWT) at a single level. The initialization routine c09aaf must be called first to set up the DWT options.

## 2Specification

Fortran Interface
 Subroutine c09caf ( n, x, lenc, ca, cd,
 Integer, Intent (In) :: n, lenc Integer, Intent (Inout) :: icomm(100), ifail Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: ca(lenc), cd(lenc)
#include <nag.h>
 void c09caf_ (const Integer *n, const double x[], const Integer *lenc, double ca[], double cd[], Integer icomm[], Integer *ifail)
The routine may be called by the names c09caf or nagf_wav_dim1_sngl_fwd.

## 3Description

c09caf 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 routine c09aaf.
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 routine c09aaf.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array 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 as declared in the (sub)program from which c09caf is called. 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 routine c09aaf.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the value returned in nwc by the call to the initialization routine c09aaf.
4: $\mathbf{ca}\left({\mathbf{lenc}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{ca}}\left(i\right)$ contains the $i$th approximation coefficient, ${C}_{a}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{c}$.
5: $\mathbf{cd}\left({\mathbf{lenc}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{cd}}\left(\mathit{i}\right)$ 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 array Communication Array
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine c09aaf.
On exit: contains additional information on the computed transform.
7: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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, n is inconsistent with the value passed to the initialization routine: ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$, n should be $⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=3$
On entry, array dimension lenc not large enough: ${\mathbf{lenc}}=⟨\mathit{\text{value}}⟩$ but must be at least $⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=6$
Either the initialization routine has not been called first or array icomm has been corrupted.
Either the initialization routine was called with ${\mathbf{wtrans}}=\text{'M'}$ or array icomm has been corrupted.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL 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

c09caf is not threaded in any implementation.

None.

## 10Example

This example computes the one-dimensional discrete wavelet decomposition for $8$ values using the Daubechies wavelet, ${\mathbf{wavnam}}=\text{'DB4'}$, with zero end extension.

### 10.1Program Text

Program Text (c09cafe.f90)

### 10.2Program Data

Program Data (c09cafe.d)

### 10.3Program Results

Program Results (c09cafe.r)