Helmut Werner

Remarks on Newton Type Multivariate Interpolation for Subsets of Grids

In this paper the construction of the Newton interpolation is carried over to several variables in case the interpolation points are contained in a grid. This is frequently the case for finite elements. Coalescence of points, i.e. multiple knots may be included.ZusammenfassungIn dieser Note wird die Konstruktion der Newton-Interpolation auf mehrere Variable übertragen für den Fall, daß die Interpolationspunkte auf einem Gitter liegen, eine Situation, die man häufig bei Finiten Elementen antrifft. Man kann die Technik auf das Zusammenfallen von Interpolationspunkten, d. h. Hermite-Interpolation, ausdehnen.

Interpolation and Integration of Initial Value Problems of Ordinary Differential Equations by Regular Splines

We give the definition of regular splines, slightly improving the axioms stated earlier by R. Schaback. Considering splines that are twice continuously differentiable, we can prove that the problem of interpolation is solvable if the interpolation points are close enough together. The solution converges to the interpolated function with the fourth order of the maximal distance between adjacent points.The regular splines are then used to define an implicit scheme for the integration of initial value problems of ordinary differential equations, following Loscalzo–Talbot and Runge. Again fourth order convergence is established. The method may be particularly useful in treating solutions with movable singularities.

On the Continuity of the Padé Operator

Let

Algorithm 51. A reliable and numerically stable program for rational interpolation of lagrange data

r_{m,n}

Multivariate Pade´ approximation

be the Pade approximant of order

Ein Algorithmus zur Rationalen Interpolation

(m,n)

Calculations of Singularities for Solutions of Algebraic Differential Equations

for a given power series f. Let

Beitrag zur augenseitigen Optik

T_{m,n}

Eine Faktorisierung der bei der rationalen Interpolation auftretenden Matrizen

, be the operator that maps f on

Zur Diagnose Verletzter Knie — Ein Identifikationsproblem —

r_{m,n}