A structure-preserving matrix method for the deconvolution of two Bernstein basis polynomials

Abstract

This paper describes the application of a structure-preserving matrix method to the deconvolution of two Bernstein basis polynomials. Specifically, the deconvolution hˆ/fˆ yields a polynomial gˆ provided the exact polynomial fˆ is a divisor of the exact polynomial hˆ and all computations are performed symbolically. In practical situations, however, inexact forms, h and f of, respectively, hˆ and fˆ are specified, in which case g=h/f is a rational function and not a polynomial. The simplest method to calculate the coefficients of g is the least squares minimisation of an over-determined system of linear equations in which the coefficient matrix is Tœplitz, but the solution is a polynomial approximation of a rational function. It is shown in this paper that an improved result for g is obtained when the Tœplitz structure of the coefficient matrix is preserved, that is, a structure-preserving matrix method is used. In particular, this method guarantees that a polynomial solution to the deconvolution h/f is obtained, and thus an essential property of the theoretically exact solution is retained in the computed solution. Computational examples that show the improvement in the solution obtained from the structure-preserving matrix method with respect to the least squares solution are presented.