Götz Alefeld

Interval analysis: theory and applications

We give an overview on applications of interval arithmetic. Among others we discuss verification methods for linear systems of equations, nonlinear systems, the algebraic eigenvalue problem, initial value problems for ODEs and boundary value problems for elliptic PDEs of second order. We also consider the item software in this field and give some historical remarks.

On the Convergence Speed of Some Algorithms for the Simultaneous Approximation of Polynomial Roots

This note gives an analysis of the order of convergence of some modified Newton methods. The modifications we are concerned with are well-known methods—a total-step method and a single-step methods—for refining all roots of an nth-degree polynomial simultaneously. It is shown that for the single-step method the R-order of convergence, used by Ortega and Rheinboldt in [6], is at least

On square roots of M-matrices

2 + \sigma _n > 3

The Cholesky method for interval data

, where

The basic properties of interval arithmetic, its software realizations and some applications

\sigma _n > 1

A QUADRATICALLY CONVERGENT KRAWCZYK-LIKE ALGORITHM*

is the unique positive root of the polynomial

On the Convergence of Some Interval-Arithmetic Modifications of Newton’s Method

p_n (\sigma ) = \sigma ^n - \sigma - 2

Some efficient methods for enclosing simple zeros of nonlinear equations

.

On the solution sets of particular classes of linear interval systems

Abstract The question of the existence and uniqueness of an M -matrix which is a square root of an M -matrix is discussed. The results are then used to derive some new necessary and sufficient conditions for a real matrix with nonpositive off diagonal elements to be an M -matrix.

On the Symmetric and Unsymmetric Solution Set of Interval Systems

Abstract We apply the well-known Cholesky method to bound the solutions of linear systems with symmetric matrices and right-hand sides both of which are varying within given intervals. We derive criteria to guarantee the feasibility and the optimality of the method. Furthermore, we discuss some general properties.