Polynomial Data Fitting & Trigonometric Functions

Given a set of points sampled form a trig function in R^2, find the polynomial that contains them.


We choose random points on the graph of $$\sin{\pi x^2 / 2}$.

	x_given = -2+4*rand(1,15);
    y_given = sin ((pi*x_given.^2 )/2);
	%y_given = exp(x_given.^2);

	n = length(x_given);

The "Coefficient" Matrix

The polynomial is a_0 + a_1*x + ... + a_m*x^m where m = n-1; This general form determines one equation for every given pair

	M = ones(n,n+1);
	for point = 1:n
		for i = 1:n;
			M(point,i) = x_given(point)^(i-1);
		M(point,n+1) = y_given(point);

Solve the system

	A = rref(M);
	a = A(:,end);

The polynomial

For example, y = a(1) + a(2)*x + a(3)*x.^2;

	r = 8; % How wide is the range of the plotting window
	x = -r:.0001:r; % a sample of x values in the chosen range
	s = [num2str(a(1))]; % make a string out of the number held in a(1)

	% term by term, make a polynomial whose coeffs are taken from a
	for i = 2:length(a);
	 	% s is a character string that looks like the poly we want
		s = [s,' + ',num2str(a(i)),'*x.^',num2str(i-1)];
	% now we turn the character string into a set of y-values of the polynomial
	y = eval(s);

Plot the points and the curve in a psudo-animation

	close all; % close all other figures
	figure; % create a new figure
	ylim([-r r]); % the max/min of the y-axis
	xlim([-r r]); % the max/min of the x-axis
	hold on % put all the plots onto the same figure
	title(s); % puts the string s into the title

	xx = [-r:.001:r]; % sample x-values, the domain of the function
	yy = sin((pi*xx.^2)/2); % the exact function
	%yy = exp(xx.^2);
	plot(xx,yy,'sk','MarkerSize',1) % plot the exact function in black
	legend('Exact') % create a legend
	pause % pause the printing for greater appreciation

	% Plot the given points as large blue stars,
	legend('Exact', 'Interpolating Points')

	% Plot the graph of the polynomial along with the given points
	legend('Exact','Interpolating Points', 'Polynomial Approx')

One could evaluate the exact and approximate integrals by calling the script CompareIntegrals afterward.