The Ontario Research Centre for Computer Algebra
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Robert M. Corless, Azar Shakoori, D.A.~Aruliah, Laureano Gonzalez-Vega
Abstract: Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error control, or event location. Recent developments suggest that a uniform, stable and convenient interpolant may be provided directly by value and derivative data (Hermite data), because a new companion matrix for such data allows stable, robust and convenient root-finding by means of (usually built-in) generalized eigenvalue solvers such as \texttt{eig} in \textsc{Matlab} or \texttt{Eigenvalues} in \textsc{Maple}. Even though these solvers are not as efficient as a special-purpose Hermite interpolant root-finder might be, being $O(d^3)$ in cost instead of $O(d^2)$, for low or moderate degrees $d$ they are efficient enough. More, because all roots are found, the first root (and hence the event) is guaranteed to be found. Further, the excellent conditioning properties (compared to the monomial basis or to divided differences) suggest that the results will be as accurate as possible. The techniques of this paper apply to polynomial or rational interpolants such as the shape-preserving interpolants of Brankin and Gladwell. We give a sketch of barycentric Hermite interpolation and a sketch of the theory of conditioning of such interpolants. Moreover, we present the construction of the Hermite interpolation polynomial companion matrix pencil and a discussion of scaling and precomputation. We remark that the B\'{e}zout matrix can be used used to solve more complicated event location problems involving more than one polynomial, via polynomial eigenvalue problems.
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