In this contribution the isolation of real roots and the computation of the topological degree...

. Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration. In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example. Intervalliterationen Konnen in Verbindung mit anderen Verfahren verwendet werden, um alle Losungen eines nichlinearen Gleichungsystems in einem geg...

. Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root is contained in a box in one of the sets, the second set may have several boxes in clusters near a single root. Thus, the output is of higher quality if there are relatively more boxes in the first set. In contrast to previously implemented similar techniques, a box expansion technique in this paper, based on using an approximate root finder, ffl-inflation and exact set complementation, decreases the size of the second set, increases the size of the first set, and never loses roots. In addition to the expansion technique, use of second-order extensions to eliminate small boxes that do not contain roots, and interval slopes versus interval d...

Abstract not found

Linear systems whose coefficients have large uncertainties arise routinely in finite element calculations for structures with uncertain geometry, material properties, or loads. However, a true worst case analysis of the influence of such uncertainties was previously possible only for very small systems and uncertainties, or in special cases where the coefficients do not exhibit dependence. This paper presents a method for computing rigorous bounds on the solution of such systems, with a computable overestimation factor that is frequently quite small. The merits of the new approach are demonstrated by computing realistic bounds for some large, uncertain truss structures, some leading to linear systems with over 5000 variables and over 10000 interval parameters, with excellent bounds for up to about 10 % input uncertainty. Also discussed are some counterexamples for the performance of traditional approximate methods for worst case uncertainty analysis.

Methods are presented for performing a rigorous sensitivity analysis of numerical prob-lems with independent, noncorrelated data for general systems of linear and nonlinear equations. The methods may serve for the following two purposes. First, to bound the dependency of the solution on changes in the input data. In contrast to condi-tion numbers a componentwise sensitivity analysis of the solution vector is performed. Second, to estimate the true solution set for problems the input data of which are afflicted with tolerances. The methods presented are very effective with the additional property that, due to an automatic error control mechanism, every computed result is guaranteed to be correct. Examples are given for linear systems demonstrating that the computed bounds are in general very sharp. Interesting comparisons to traditional condition numbers are given.

Abstract. Recently, an alternative interval approximation F ( X) for enclosing a factorable function f(x) in a given box X has been suggested. The enclosure is in the form of an affine interval function n i=1 F ( X) = a X + B where only the additive term B is an interval, the coefficients ai being real i i numbers. The approximation is applicable to continuously differentiable, continuous and even discontinuous functions. In this paper, a new algorithm for determining the coefficients ai and the interval B of F(X) is proposed. It is based on the introduction of a specific generalized representation of intervals which permits the computation of the enclosure considered to be fully automated. 1.

Abstract. Many verification algorithms use an expansion f(x) ∈ f(˜x)+S· (x−˜x), f: Rn → Rn for x ∈ X, where the set of matrices S is usually computed as a gradient or by means of slopes. In the following, an expansion scheme is described which frequently yields sharper inclusions for S. This allows also to compute sharper inclusions for the range of f over a domain. Roughly speaking, f has to be given by means of a computer program. The process of expanding f can then be fully automatized. The function f need not be differentiable. For locally convex or concave functions special improvements are described. Moreover, in contrast to other methods, ˜x ∩ X may be empty without implying large overestimations for S. This may be advantageous in practical applications. 0. Notation We denote by IR the set of real intervals X ∈ IR ⇒ X = [inf(X), sup(X)] = { x ∈ R | inf(X) ≤ x ≤ sup(X)}. By PT we denote the power set over a given set T, and we use the canonical embedding IR ⊆ PR. Thesetofn-dimensional interval vectors is denoted by IRn, i.e., X ∈ IR n ⇒ X = { (xi) ∈ R n | xi ∈ Xi} with Xi ∈ IR, 1 ≤ i ≤ n. Interval vectors are compact. Interval operations and power set operations are defined in the usual way. Details can be found in standard books on interval analysis, among others [10, 2, 11]. If not explicitly noted otherwise, all operations are power set operations. 1. Expansion of nonlinear functions A differentiable function f: D ⊆ Rn → R can be locally expanded by its gradient. For ˜x ∈ D, X ⊆ D, and[g]∈IRn with ∇f(˜x ∪ X) ⊆ [g] there holds ∀ x ∈ X ∃ g ∈ [g] : f(x)−f(˜x)=g T (1.0) ·(x−˜x). Here, ∪ denotes the convex hull, and ∇f(˜x ∪ X) denotes the range of ∇f over ˜x ∪ X. The gradient, for real and for interval arguments, can be computed using automatic differentiation [4, 12]. This process is fully automatized. This approach has three disadvantages: (1) f needs to be differentiable,

Branch and bound methods for nding all zeros of a nonlinear system of equations in a box frequently have the diculty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the eect that near each zero, many small boxes are created by repeated splitting, whose processing may dominate the total work spent on the global search. This paper discusses the reasons for the occurrence of this so-called cluster eect, and how to reduce the cluster eect by de ning exclusion regions around each zero found, that are guaranteed to contain no other zero and hence can safely be discarded. Such exclusion regions are traditionally constructed using uniqueness tests based on the Krawczyk operator or the Kantorovich theorem. These results are reviewed; moreover, re nements are proved that signi cantly enlarge the size of the exclusion region. Existence and uniqueness tests are also given. Keywords: zeros, system of equations, validated enclosure, existence test, uniqueness test, inclusion region, exclusion region, branch and bound, cluster eect, Krawczyk operator, Kantorovich theorem, backboxing, ane invariant 2000 MSC Classi cation: primary 65H20, secondary 65G30 1

. In this paper we introduce a pruning technique based on slopes in the context of interval branch-and-bound methods for nonsmooth global optimization. We develop the theory for a slope pruning step which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. This pruning step offers the possibility to cut away a large part of the box currently investigated by the optimization algorithm. We underline the new technique's efficiency by comparing two variants of a global optimization model algorithm: one equipped with the monotonicity test and one equipped with the pruning step. For this reason, we compared the required CPU time, the number of function and derivative or slope evaluations, and the necessary storage space when solving several smooth global optimization problems with the two variants. The paper concludes on the test results for several nonsmooth examples. Keywords: Global optimization, interval ar...