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.

2 Specification

Fortran Interface

Subroutine c09caf (

n, x, lenc, ca, cd, icomm, ifail)

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)

C Header Interface

#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.

3 Description

c09caf computes the one-dimensional DWT of a given input data array,

x_{i}

, for

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}

C_{d}

is returned by the initialization routine c09aaf.

4 References

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

5 Arguments

1: $n$ – 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: $x (n)$ – Real (Kind=nag_wp) array Input: On entry: x contains the input dataset $x_{i}$ , for $i = 1, 2, \dots, n$ .
3: $lenc$ – 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: $lenc \geq n_{c}$ , where $n_{c}$ is the value returned in nwc by the call to the initialization routine c09aaf.
4: $ca (lenc)$ – Real (Kind=nag_wp) array Output: On exit: $ca (i)$ contains the $i$ th approximation coefficient, $C_{a} (i)$ , for $i = 1, 2, \dots, n_{c}$ .
5: $cd (lenc)$ – Real (Kind=nag_wp) array Output: On exit: $cd (i)$ contains the $i$ th detail coefficient, $C_{d} (i)$ , for $i = 1, 2, \dots, n_{c}$ .
6: $icomm (100)$ – 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: $ifail$ – 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 $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, n is inconsistent with the value passed to the initialization routine: $n = ⟨ value ⟩$ , n should be $⟨ value ⟩$ .

$ifail = 3$: On entry, array dimension lenc not large enough: $lenc = ⟨ value ⟩$ but must be at least $⟨ value ⟩$ .

$ifail = 6$: Either the initialization routine has not been called first or array icomm has been corrupted.

Either the initialization routine was called with $wtrans ='M'$ or array icomm has been corrupted.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

c09caf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example computes the one-dimensional discrete wavelet decomposition for

8

values using the Daubechies wavelet,

wavnam ='DB4'

, with zero end extension.

c09ca: FL CL CPP AD

NAG FL Interfacec09caf (dim1_​sngl_​fwd)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
c09caf (dim1_sngl_fwd)