1er Symposium canadien en analyse numérique et calcul scientifique
A New Family of Matrices for Computation
Stenger, Frank and Fernando Guevara Vasquez
University of Utah, Salt Lake City

We start with an interpolation method for interpolating a function $f$ on an interval (a,b), i.e., \[ p(x) = \sum_1^n b_k(x) f(x_k),\] which could be the usual formula for polynomial interpolation, or for trigonometric polynomial interpolation, or for spline function interpolation, or for rational function interpolation, or for interpolation by Sinc methods, etc., and where $a < x_1 <  ... < x_n < b$. We assume that this interpolation results from using basis functions that are orthogonal over $(a,b)$ with respect to a weight function $w$ that is positive a.e. on $(a,b)$. For example for the case of Chebyshev polynomials, take the interval $(-1,1)$ and weight function $(1-x^2)^{-1/2}$, so that the $x_j$ are the zeros of the Chebyshev polynomial $T_n(x)$. We investigate properties and use of the matrices $B$ with entries $b_{jk} = \int_a^{x_j} w(x) b_k(x) dx$, and $C$ with entries $c_{jk} = \int_{x_j}^b w(x) b_k(x) dx$. These matrices enable new and simple methods of arbitrary accuracy for approximate integration, differentiation, solving differential equations, inverting Laplace transforms, etc.

Lundi, 17 juin, 16h30
Salle Des Plaines C