Guido Walz

Numerical Solution of Fractional Order Differential Equations by Extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples.

Identities for trigonometric B-splines with an application to curve design

In this paper we investigate some properties of trigonometric B-splines. We establish a complex integral representation for these functions, which is in certain analogy to the polynomial case, but the proof of which has to be done in a different and more complicated way. Using this integral representation, we can prove some identities concerning the evaluation of a trigonometric B-spline, its derivative and its partial derivative w.r.t. the knots. Finally we show that—in the case of equidistant knots—the trigonometric B-splines of odd order form a partition of a constant, and therefore the corresponding B-spline curve possesses the convex-hull property. This is illustrated by a numerical example.

Sequences of transformations and triangular recursion schemes, with applications in numerical analysis

Abstract In many fields of numerical analysis there appear transformations of the form T k v = Σ v + k i = v α k i,v T i . When v varies, a sequence of transformations is obtained. This approach covers, for example, the E− and Θ-algorithms, the recursion formulae for B-splines, Bernstein polynomials and orthogonal polynomials, Pade approximants, the divided difference scheme and projection methods. In this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme. The reciprocal of these results also holds. Furthermore, we will show that T v k can be represented in terms of a complex contour integral. Throughout the paper we will study several examples in some detail, and it will turn out that the application of our general theory leads to interesting new results in the special cases. Among others, we will derive a new determinantal representation formula for B-splines, a recurrence relation for generalized Bernstein polynomials, a generalization of the E-algorithm and we will prove that the Θ-algorithm can be represented as a quotient of determinants.

Multivariate approximation and splines

This volume presents refereed papers covering a variety of topics in the growing field of multivariate approximation and slines.

Trigonometric Be´zier and Stancu polynomials over intervals and triangles

We introduce a family of trigonometrie polynomials, denoted as Stancu polynomials, which covers as special cases the trigonometrie Lagrange and Bernstein polynomials. This family depends only on one real parameter, denoted as design parameter. Our approach works for curves as well as for surfaces over triangles. The resulting Stancu curves resp. surfaces therefore establish a link between trigonometrie interpolatory and Bernstein-Bezier curves resp. surfaces.

Computing the matrix exponential and other matrix functions

Abstract In this note we present an algorithm for the computation of matrix functions, especially the matrix exponential. It is a new application of the so-called ‘elimination method’, which was presented for the scalar case in previous papers (e.g. [5]).

Interpolation by Bivariate Splines on Crosscut Partitions

We give a survey of recent methods to construct Lagrange interpolation points for splines of arbitrary smoothness rand degree q on general crosscut partitions in IR2. For certain regular types of partitions, also results on Hermite interpolation sets and on the approximation order of the corresponding interpolating splines are given.

Asymptotic expansions for multivariate polynomial approximation

In this paper the approximation of multivariate functions by (multivariate) Bernstein polynomials is considered. Building on recent work of Lai, we can prove that the sequence of these Bernstein polynomials possesses an asymptotic expansion with respect to the index n. This generalizes a corresponding result due to Costabile, Gualtieri and Serra on univariate Bernstein polynomials, providing at the same time a new proof for it. After having shown the existence of an asymptotic expansion we can apply an extrapolation algorithm which accelerates the convergence of the Bernstein polynomials considerably; this leads to a new and very efficient method for polynomial approximation of multivariate functions. Numerical examples illustrate our approach.

Best Approximation by Free Knot Splines

We consider the problem of finding the best (uniform) approximation of a given continuous function by spline functions with free knots. Our approach can be sketched as follows. By using the Gauß transform with arbitrary positive real parameter t, we map the set of splines under consideration onto a function space, which is arbitrarily close to the spline set, but satisfies the local Haar condition and also possesses other nice structural properties. This enables us to give necessary and sufficient conditions for best approximations (in terms of alternants) and, under some assumptions, even full characterizations and a uniqueness result. By letting t → 0, we recover best approximation in the original spline space. Our results are illustrated by some numerical examples, which show in particular the nice alternation behavior of the error function.

A Unified Approach to B-Spline Recursions and Knot Insertion, with Application to New Recursion Formulas

We present a unified approach to and a generalization of almost all known recursion schemes concerning B-spline functions. This includes formulas for the computation of a B-splines values, its derivatives (ordinary and partial), and for a knot insertion method for B-spline curves. Furthermore, our generalization allows us to derive also some new relations for these purposes.