c09dcf computes the one-dimensional multi-level maximal overlap discrete wavelet transform (MODWT). The initialization routine c09aaf must be called first to set up the MODWT options.

2 Specification

Fortran Interface

Subroutine c09dcf (

n, x, keepa, lenc, c, nwl, na, icomm, ifail)

Integer, Intent (In)	::	n, lenc, nwl
Integer, Intent (Inout)	::	icomm(100), ifail
Integer, Intent (Out)	::	na
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	c(lenc)
Character (1), Intent (In)	::	keepa

C Header Interface

#include <nag.h>

void	c09dcf_ (const Integer n, const double x[], const char keepa, const Integer lenc, double c[], const Integer nwl, Integer na, Integer icomm[], Integer ifail, const Charlen length_keepa)

The routine may be called by the names c09dcf or nagf_wav_dim1_mxolap_multi_fwd.

3 Description

c09dcf computes the multi-level MODWT for a data set,

x_{i}

, for

i = 1, 2, \dots, n

, in one dimension. For a chosen number of levels,

n_{l}

, with

n_{l} \leq l_{\max}

, where

l_{\max}

is returned by the initialization routine c09aaf in nwlmax, the transform is returned as a set of coefficients for the different levels stored in a single array. Periodic reflection is currently the only available end extension method to reduce the edge effects caused by finite data sets.

The argument keepa can be set to retain both approximation and detail coefficients at each level resulting in

n_{l} \times (n_{a} + n_{d})

coefficients being returned in the output array, c, where

n_{a}

is the number of approximation coefficients and

n_{d}

is the number of detail coefficients. Otherwise, only the detail coefficients are stored for each level along with the approximation coefficients for the final level, in which case the length of the output array, c, is

n_{a} + n_{l} \times n_{d}

. In the present implementation, for simplicity,

n_{a}

and

n_{d}

are chosen to be equal by adding zero padding to the wavelet filters where necessary.

4 References

Percival D B and Walden A T (2000) Wavelet Methods for Time Series Analysis Cambridge University Press

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: $keepa$ – Character(1) Input

On entry: determines whether the approximation coefficients are stored in array c for every level of the computed transform or else only for the final level. In both cases, the detail coefficients are stored in c for every level computed.

$keepa ='A'$: Retain approximation coefficients for all levels computed.
$keepa ='F'$: Retain approximation coefficients for only the final level computed.

Constraint:

keepa ='A'

'F'

4: $lenc$ – Integer Input

On entry: the dimension of the array c as declared in the (sub)program from which c09dcf is called. c must be large enough to contain the number of wavelet coefficients.

keepa ='F'

, the total number of coefficients,

n_{c}

, is returned in nwc by the call to the initialization routine c09aaf and corresponds to the MODWT being continued for the maximum number of levels possible for the given data set. When the number of levels,

n_{l}

, is chosen to be less than the maximum, then the number of stored coefficients is correspondingly smaller and lenc can be reduced by noting that

n_{d}

detail coefficients are stored at each level, with the storage increased at the final level to contain the

n_{a}

approximation coefficients.

keepa ='A'

n_{d}

detail coefficients and

n_{a}

approximation coefficients are stored for each level computed, requiring

lenc \geq n_{l} \times (n_{a} + n_{d}) = 2 \times n_{l} \times n_{a}

, since the numbers of stored approximation and detail coefficients are equal. The number of approximation (or detail) coefficients at each level,

n_{a}

, is returned in na.

Constraints:

if $keepa ='F'$ , $lenc \geq (n_{l} + 1) \times n_{a}$ ;
if $keepa ='A'$ , $lenc \geq 2 \times n_{l} \times n_{a}$ .

5: $c (lenc)$ – Real (Kind=nag_wp) array Output

On exit: the coefficients of a multi-level wavelet transform of the dataset.

The coefficients are stored in c as follows:

keepa ='F'

$c (1 : n_{a})$: Contains the level $n_{l}$ approximation coefficients;
$c (n_{a} + (i - 1) \times n_{d} + 1 : n_{a} + i \times n_{d})$: Contains the level $(n_{l} - i + 1)$ detail coefficients, for $i = 1, 2, \dots, n_{l}$ ;

keepa ='A'

$c ((i - 1) \times n_{a} + 1 : i \times n_{a})$: Contains the level $(n_{l} - i + 1)$ approximation coefficients, for $i = 1, 2, \dots, n_{l}$ ;
$c (n_{l} \times n_{a} + (i - 1) \times n_{d} + 1 : n_{l} \times n_{a} + i \times n_{d})$: Contains the level i detail coefficients, for $i = 1, 2, \dots, n_{l}$ ;

The values

n_{a}

and

n_{d}

denote the numbers of approximation and detail coefficients respectively, which are equal and returned in na.

6: $nwl$ – Integer Input

On entry: the number of levels,

n_{l}

, in the multi-level resolution to be performed.

Constraint:

1 \leq nwl \leq l_{\max}

, where

l_{\max}

is the value returned in nwlmax (the maximum number of levels) by the call to the initialization routine c09aaf.

7: $na$ – Integer Output

On exit: na contains the number of approximation coefficients,

n_{a}

, at each level which is equal to the number of detail coefficients,

n_{d}

. With periodic end extension (

mode ='P'

in c09aaf) this is the same as the length, n, of the data array, x.

8: $icomm (100)$ – 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.

9: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

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

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 $⟨ value ⟩$ .

$ifail = 2$: On entry, $keepa = ⟨ value ⟩$ was an illegal value.

$ifail = 4$: On entry, lenc is set too small: $lenc = ⟨ value ⟩$ .
Constraint: $lenc \geq ⟨ value ⟩$ .

$ifail = 6$: On entry, $nwl = ⟨ value ⟩$ .
Constraint: $nwl \geq 1$ .

On entry, nwl is larger than the maximum number of levels returned by the initialization function: $nwl = ⟨ value ⟩$ , maximum = $⟨ value ⟩$ .

$ifail = 8$: On entry, the initialization routine c09aaf has not been called first or it has not been called with $wtrans ='U'$ , or the communication array icomm has become 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

c09dcf is not threaded in any implementation.

9 Further Comments

The wavelet coefficients at each level can be extracted from the output array c using the information contained in na on exit.

10 Example

A set of data values (

n = 64

) is decomposed using the MODWT over two levels and then the inverse (c09ddf) is applied to restore the original data set.

c09dc: FL CL CPP AD

NAG FL Interfacec09dcf (dim1_​mxolap_​multi_​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
c09dcf (dim1_mxolap_multi_fwd)