%% Orthogonal Polynomials
%% Introduction
% In lectures, examples of orthogonal polynomials for a few weight
% functions and associated intervals will have been given. This GUI allows
% you to enter your own weight function and interval and see a few of the
% associated orthogonal polynomials.
%
% *Note:* _This weight function must be positive on any subset of the 
% interval having positive measure. For example, it could be positive 
% except at a finite number of points._

%% Theory
% Refer to "2.3: The three-term recurrence relation" in the lecture notes,
% in particular Theorem 2.4
%
% The inner products are calculated numerically using a quadrature routine
% (closed forms for the integrals will not be available, except in
% special cases).
%
%% The GUI
% To open the GUI (after downloading the relevant files below), type
orth_poly_gui
%%
% You can select the highest degree polynomial displayed, the weight
% function, and the interval of interest. You may also select from common
% families using the drop-down *"Presets..."*  list.
%
% If you change any of these parameters, press *"Show Plot"* to see the 
% polynomials plotted, or *"Show LaTeX"* to see them as LaTeX.
%
% The *"Rationals"* checkbox allows you to display the coefficients in LaTeX
% mode as rational numbers. If you change this, press *"Show LaTeX"* to 
% update the display.
%
% *WARNING:* This GUI is for educational purposes only. In practice there
% are better was of calculating orthogonal polynomials, depending on the
% exact application and weight function. See e.g. the Gaussian Quadrature 
% module. This approach suffers from accumulating roundoff error, so the
% results may not be completely accurate.

%% Code
%
% * <orth_poly_gui.m> (Run this)
% * <orth_poly_gui.fig> (Required - GUI figure)
%
% All files as .zip archive: <orthogonal_polynomials_all.zip>