L Wuytack

Nonlinear methods in numerical analysis

I. Continued Fractions. Since these play an important role, the first chapter introduces their basic properties, evaluation algorithms and convergence theorems. From the section dealing with convergence it can be seen that in certain situations nonlinear approximations are more powerful than linear approximations. The recent notion of branched continued fraction is introduced in the multivariate section and is later used for the construction of multivariate rational interpolants. II. Pade Approximants. A survey of the theory of these local rational approximants for a given function is presented, including the problems of existence, unicity and computation. Also considered are the convergence of sequences of Pade approximants and the continuity of the Pade operator which associates with a function its Pade approximant of a certain order. A special section is devoted to the multivariate case. III. Rational Interpolants. These rational functions fit a given function at some given points. Many results of the previous chapter remain valid for this more general case where the interpolation conditions are spread over several points. In between the rational interpolation case and Pade approximation case lies the theory of rational Hermite interpolation where each interpolation point can be assigned more than one interpolation condition. Some results on the convergence of sequences of rational Hermite interpolants are mentioned and multivariate rational interpolants are introduced in two different ways. IV. Applications. The previous types of rational approximants are used here to develop several numerical methods for the solution of classical problems such as convergence acceleration, nonlinear equations, ordinary differential equations, numerical quadrature, partial differential equations and integral equations. Many numerical examples illustrate the different techniques, and it is seen that nonlinear methods are very useful in situations involving singularities. Subject Index.

On the osculatory rational interpolation problem

The problem of the existence and construction of a table of osculating rational functi, ns rj m for 1, m > 0 is considered. First, a survey is given of some results from the th -ory of osculatory rational interpolation of order Si 1 at points xi for i > 0. Using these results, we prove the existence of continued fractions of the form Co + c1 .(x yO) + * * * + Ck (X yO) . . . (x Yk-1) Ck + 1(X -YO) . . (X Yk) Ck+ 2 (XYk+1 ) + F 1 1 Ck+3 (x -Yk+2) + F with the Yk suitably selected from among the xi, whose convergents form the elements rk,O, rk+1,0, rk+l,1 rk+2,1.... of the table. The properties of these continued fractions make it possible to derive an algorithm for constructing their coefficients Ci for i > 0. This algorithm is a generalization of the qd-algorithm.

On the continuity of the multivariate Padé operator

Abstract Continuity of the univariate Pade operator was proved in [5,6]. We discuss the limitations of a multivariate generalization and prove a multivariate analogon of the continuity property.

On the computation of non-normal Padé approximants

Abstract Several basic techniques are described for computing Pade approximants which are not normal. These techniques are modifications of some known methods for the computation of normal Pade approximants.