Interval Arithmetic Methods Research Articles

Iterative methods for systems of equations with interval coefficients and linear form

We introduce iterative methods for systems of equations with interval coefficients and linear form by suitable matrix splittings. When compared to the iterative methods for systems amenable to iteration introduced in [1], improved convergence and inclusion properties can be proved under suitable conditions. The method can also be used in the solution of specific nonlinear systems of equations by interval arithmetic methods.

An interval arithmetic method for the solution of nonlinear systems of equations on a vector computer

We describe a vectorized algorithm for an interval arithmetic Newton-like method for a class of systems of nonlinear equations arising from discretizations of nonlinear elliptic PDE. This method converges to a solution under relatively weak conditions. It is founded on a combination of a Newton-like interval method and interval arithmetic ‘fast’ direct solver. In the present paper we focus our attention on aspects of the vectorization, in particular that of a simulation of an interval arithmetic and that of the direct solver.

Interval arithmetic methods for systems of nonlinear equations arising from discretizations of quasilinear elliptic and parabolic partial differential equations

We introduce Newton-like interval arithmetic methods for large systems of equations arising from discretizations of quasilinear elliptic and parabolic problems on rectangles. The methods are combined with an interval arithmetic fast solver for which we discuss a version for a class of nonsymmetric block tridiagonal matrices. The nonlinear methods converge (almost) globally to the solution. For suitable parabolic problems we introduce an implicit method yielding in each time step arbitrarily close inclusions of the solution with automatically computed starting vectors and guaranteed convergence of the Newton-like methods.

Interval arithmetic multistep methods for nonlinear systems of equations

In the present paper we introduce interval arithmetic multistep methods for nonlinear systems of equations. These methods are derived from several types of singlestep methods discussed in previous papers. An interval arithmetic particularity consists in the fact that both the function defining the system and its Jacobi matrix can be kept constant in interval multistep methods for a well defined number of steps while it is still possible to prove superlinear convergence. The solution of nonlinear systems of equations by interval arithmetic methods has been discussed by many authors. See [1, 7] or some papers cited in [11], for example. For interval methods convergence results can be obtained which go beyond those for “point”, i.e. noninterval methods. There are, for example, methods converging to the solution without requiring any convexity condition. The inclusion of the solution by all iterates can be proved and the convergence to the solution requires the existence of a computable initial inclusion as main condition, i.e. the convergence can be said to be almost global. We report numerical results for applications of some of our methods on a vector and a sequential computer.

A truncated cyclic reduction algorithm for interval arithmetic tridiagonal systems of equations

In many numerical problems the solution of tridiagonal systems of equations consumes an important part of the computation time. For their efficient solution on vector or parallel computers the recursive Gauss algorithm has often to be replaced by a method with a higher degree of parallelism. Among other methods cyclic reduction has been widely discussed. In the present paper we discuss some aspects of the numerical treatment of tridiagonal systems with interval coefficients which arise, for example, as part of interval arithmetic Newton-like methods combined with a “fast Poisson solver” [8, 9]. We have discussed interval arithmetic cyclic reduction in [10]. Here we introduce a truncated version dedicated to reduce the computation time. In contrast to the non-interval case we have to preserve inclusion properties. Instead of really truncating steps, we replace them by easily computable intervals. In contrast to the non-interval case we can “truncate” both the reduction and the solution phase.

Krawczyk-Like Algorithms for the Solution of Systems of Nonlinear Equations

In this paper we present interval arithmetic methods for the solution of systems of nonlinear equations. These methods are based on the algorithm of R. Krawczyk [5] and a modification introduced in [3]. Starting with an interval vector containing a solution of the given system, these methods converge to that solution under relatively weak conditions, i.e. it can be said that the convergence is almost global.

On interval arithmetic range approximation methods of polynomials and rational functions

Interval arithmetics seems to offer a variety of simple numerical approximation methods well applicable in computational algorithms of curves and sculptured surfaces. In this work we analyse and compare several interval arithmetic methods to approximate the range of polynomials and rational functions.

Intervallanalytische Behandlung konvexer Optimierungsaufgaben

A nonlinear convex programming problem is solved by methods of interval arithmetic which take into account the input errors and the round-off errors. The problem is reduced to the solution of a nonlinear parameter dependent system of equations. Moreover error estimations are developed for special problems with uniformly convex cost functions.

An interval arithmetic method for global optimization

An interval arithmetic method is described for finding the global maxima or minima of multivariable functions. The original domain of variables is divided successively, and the lower and the upper bounds of the interval expression of the function are estimated on each subregion. By discarding subregions where the global solution can not exist, one can always find the solution with rigorous error bounds. The convergence can be made fast by Newton's method after subregions are grouped. Further, constrained optimization can be treated using a special transformation or the Lagrange-multiplier technique.

Fehlerabsehätzung beim verfahren von wolfe zur lösung quadratischer optimierungsproblem mit hilfe der intervallarithmetik

This paper is dealt with error estimations for solutions of quadratic programming problems. These estimations are given by solution of a linear system of equations by means of the methods of interval arithmetic. Therefore it is quite certain that the round off errors are also included in these estimations.

One-dimensional problems in unconstrained signomial programming

One-dimensional problems arise in both primal and dual signomial programs. Primal problems are of interest since many multidimensional search procedures involve one-dimensional searches. Dual problems occur naturally since many engineering design problems can be expressed as signomial programs with a single degree of difficulty. This paper applies an interval arithmetic version of Newton's method to one-dimensional unconstrained problems in signomial programming. The interval arithmetic method is guaranteed to generate all the stationary points.