q-Orthogonal polynomials and the oscillator quantum group

The oscillator quantum algebra is shown to provide a group-theoretic setting for the q-Laguerre and q-Hermite polynomials.On leave from Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Canada H3C 3J7.     

Global approximate Newton methods

Summary We derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

Monotonieeigenschaften des Steffensen-Verfahrens

Steffensen's method is slightly generalized by introducing a real parameter. In this way one can get different monotonicity properties, depending on the choice of this parameter. These monotonicity statements give the possibility of bracketing the solution of a given problem. In a special case they even ensure the convergence and the existence of a solution. Furthermore there are given sufficient conditions, which show that Steffensen's method converges at least as quickly as Newton's method. A numerical example shows the efficiency of the theorems and compares Steffensen's and Newton's method.

     

Newton interpolation is efficient for approximation of linear functionals

Summary The Newton interpolation approach is developed for approximation of linear functionals. It is shown that in numerical interpolation and numerical differentiation, the Newton interpolation approach is more efficient than solving the Vandermonde systems.This work was supported in part by the United States Air Force under a grant AFOSR 76-3020     

On patched variational principles with application to elliptic and mixed elliptic-hyperbolic problems

Summary Variational principles are important tools for the approximate solution of boundary-value problems. There are many types of variational principles, and each has its advantages and disadvantages. In this paper we show how to use a combination of variational principles, each for a given subregion of the underlying region of space, so as to best utilize the chief benefits of the individual principles. Such a patched principle is particularly useful in solving transonic flow problems, where we use different principles in the elliptic and hyperbolic regions. We present the results of some numerical experiments for the Tricomi problem. These seem to indicate that our patched principle, when used in conjunction with the finite element method, leads to accuracy which is second-order in the mesh spacing, as compared to the standard numerical methods of solving this problem, which are only first-order.     

General order Newton-Padé approximants for multivariate functions

Summary Padé approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x ₁, ...,x _p ). Therefore multivariate Newton-Padé approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Padé approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Padé approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Padé approximants for a univariate function are obtained.     

A shape preserving area true approximation of histogram by rational splines

For a given histogram, we consider an application of a simple rational spline to a shape preserving area true approximation of the histogram. An algorithm for determination of the spline is as easy as one with a quadratic polynomial spline, while the latter does not always preserve the shape of the histogram. Some numerical examples are given at the end of the paper.     

Solving systems of polynomial equations by bounded and real homotopy

Summary The homotopy method for solving systems of polynomial equations proposed by Chow, Mallet-Paret and Yorke is modified in two ways. The first modification allows to keep the homotopy solution curves bounded, the second one to work with real polynomials when solving a system of real equations. For the first method numerical results are presented.