The function chebyzeros.m computes the zeros of Chebyshev polynomials in the [-1,+1] interval. The function chebynodes.m maps those zeros into an arbitrary [a,b] interval. The function chebypol.m evaluates polynomials in the Chebyshev basis functions for a given vector of coefficients. Finally, cheby.m computes the Chebyshev-Vandermonde matrix, for the univariate and multivariate case.
The function chebyinterp.m performs Chebyshev interpolation of a generic function on a given interval. The script chebyinterp_test.m runs a quick test of goodness of fit for a simple example.
The scripts Chebytest.m and Chebytest2.m show how to use spectral methods (Chebyshev Collocation and Galerkin) to solve a functional equation, in the univariate and bivariate case. The solution is known, so it's easy to evaluate the goodness of fit.