Companion matrix pencils for hermite interpolants

Abstract

This abstract describes new methods for solving polynomial problems where the polynomials are expressed as (generalized) Hermite interpolants; that is, where the polynomials are given by values and by derivatives at certain nodes. We consider here the mixed case where at some nodes, only values are known, whereas at others, both values and derivatives are known. We neither consider the case where derivatives higher than the 1st are known, nor the Birkhoff case where some data is ‘missing’. In the full paper, we will give the general case for higher derivatives; we leave the Birkhoff case for a future paper. For example, suppose that we know that a polynomial has the values p(tm) = pm, p(tm+1/2) = pm+1/2, and p(tm+1) = pm+1, for distinct nodes τ1 = tm, τ2 = tm+1/2, τ3 = tm+1, and suppose further that we also know the derivatives p′(tm) and p′(tm+1), which we denote p ′ m and p ′ m+1. This can occur naturally in the context of the numerical solution of initial value problems for ordinary differential equations, for example. At the beginning of a numerical step, t = tm, the value is known and the derivative is calculated by a function evaluation, so we know pm and p ′ m. At the end of the step, t = tm+1, another function evaluation is carried out in order to continue the marching process and so we know p′m+1 as well as pm+1. However, it is not necessarily the case that derivatives are known in the interior of the interval (tm, tm+1), although for graphical and other purposes the values of the polynomial p(t) interpolating the numerical