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Discrete Orthogonal Polynomial Toolbox: DOPBox Version 1.8

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Discrete Orthogonal Polynomial Toolbox: DOPBox Version 1.8

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11 Apr 2013 (Updated )

A toolbox for discrete orthogonal polynomials and their applications to fitting, ODEs etc.

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Description

Diecrete Orthogonal Polynomials: Dopbox Version V1.8
March 2014
Introduction
------------
This a toolbox for Diecrete Orthogonal Polynomials called the DOPbox.
Discrete orthogonal polynomials have many applications, such as: in discrete
approximations; in the solution of ordinaty differential equations, in
particular boundary value problems and initial value problems. For the
generation of admissible functions etc.

Organization
------------
You will need to install all directories on you computer and set the matlab
path to include the directories and their sub-directories.

The library is organized in three main directories:

1) DOPbox: This directory contains the files central to the library.
2) SupportFns: These are supporting fnctions which make the generation of
   documentation simpler. They are used extensively in the examples and
   documentation provied.
3) Documentation: this directory contains matlab and -pdf files which
   document the use of the library functions.

Documentation
-------------
We recommend you start by looking at the GettingStarted.pdf documentation.

This contains an example of the use of each and every function in the

library.

None of the theory behind the library is explained in the documentation, the
reader is referred to the following publications, should they wish to study
the theoritical material:

This paper provied an introduction to the Gram polynomials

@inproceedings{
oleary2008b,
   Author = {O'Leary, P. and Harker, M.},
   Title = {An Algebraic Framework for Discrete Basis Functions in Computer

Vision},
   BookTitle = {2008 $6^{\textrm{th}}$ ICVGIP},
   Address= {Bhubaneswar, India},
   Publisher = {IEEE},
   Pages = {150-157},
   Year = {2008} }

DOI: 10.1109/ICVGIP.2008.107

This paper introduced the concept of local and global polynomial

approximations

@inproceedings{oleary2010C,
   Author = {O'Leary, P. and Harker, M.},
   Title = {Discrete Polynomial Moments and Savitzky-Golay Smoothing},
   BookTitle = {Waset Special Journal},
   Volume = {72},
   DOI = {},
   Pages = {439--443},
   Year = {2010}}

The PDF is available at

www.waset.org/journals/waset/v48/v48-85.pdf

This paper provies extenside theory and deviations for the application of

discrete
orthogonal polynomials to the solution of inverse boundary value problems.
The work is done the the bounds of an application in the monitoring of

structures.
We highly recommend reading this paper if more advanced applications of the

ideas are to be made.

@article{Oleary2012,
  author = {Paul O'Leary and Matthew Harker},
  title = {A Framework for the Evaluation of Inclinometer Data in the
               Measurement of Structures},
  journal = {IEEE T. Instrumentation and Measurement},
  volume = {61},
  number = {5},
  year = {2012},
  pages = {1237-1251},
  ee = {http://dx.doi.org/10.1109/TIM.2011.2180969}

http://dx.doi.org/10.1109/TIM.2011.2180969

Matthew harker and Paul O'Leary Marczh 2014

Changes
-------
Version V1.8

1) A code error in the function dopVal.m has been corrected

Version V1.7

1) A code error in dopDiffLocal was corrected. The function now works correctly with sparse matrices.

Version V1.6

1) A demonstration for constrained polynomials where the constraints are not at a node has been added.

2) A demonstration of a constraint located outside the range of the support has been added.

3) An example of using constrained basis functions as admissible functions
in a discrete Rayligh-Ritz solution to a Sturm-Liouville equation has been
added. This is an example where the constraints are located outside the range of the support.

4) The dopDiffLocal function has been modified to return a full differentiating matrix when the support length is equal to the number of points.

5) The rank of the differentiating matrix is tested and a warning is issued if the matrix is more than rank-1 deficient.

Acknowledgements

This file inspired Surface Reconstruction From Gradient Fields: Grad2 Surf Version 1.0.

Required Products MATLAB
MATLAB release MATLAB 7.14 (R2012a)
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Comments and Ratings (2)
05 Jun 2013 Arghad  
28 May 2013 Florian  
Updates
18 Apr 2013

A more efficient coding has been implemented for sparse matrices. This is particularly when doing local approximations to polynomials and derivatives, see the functions dopDiffLocal.m and dopApproxLocal.m.

22 May 2013

The numerical efficence of dopApproxLocal.m and dopDiffLocal.m have been improved and weighted polynomials have been added dopWeight.m

23 May 2013

Corrected version number

03 Jun 2013

1) The functions dopFit.m and dopVal.m have been added.
2) The derivation of the equations for weighted regression using weighted basis functions has been added.
3) Many small typing errors have been corrected.

29 Jul 2013

A new method of synthesizing constrained discrete orthogonal polynomials ihas been added, together with two additional demonstrations.

19 Aug 2013

New examples with constrained polynomials have been added.

The function dopDiffLocal has been improved.

04 Sep 2013

A code error in dopDiffLocal was corrected. The function now works correctly with sparse matrices.

04 Mar 2014

An error in the function dopVal has been corrected

04 Mar 2014

An error in the function dopVal.m has been corrected

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