Spherical t-designs provide quadrature rules for the sphere which are exact for polynomials up to degree t. In this paper, we propose a computational algorithm based on interval arithmetic which, for given t, upon successful completion will have proved the existence of a t-design with (t+1)² nodes on the unit sphere S^{2 3} and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty. Since there is no theoretical result which proves the existence of a t-design with (t+1)² nodes for arbitrary t, our method contributes to the theory because it was tested successfully for t=1, 2, ..., 100. The t-design is usually not unique; our method aims at finding a well-conditioned one. The method relies on computing an interval enclosure for the zero of a highly nonlinear system of dimension (t+1)². We therefore develop several special approaches which allow us to use interval arithmetic efficiently in this particular situation. The computations were all done using the MATLAB toolbox INTLAB.

The verification of the existence of certain spherical t-designs involves the evaluation of a degree-t polynomial J_t at a very large number of (interval) arguments. To make the overall verification process feasible computationally, this evaluation must be fast, and the enclosures for the function values must be affected with only modest over-estimation. We discuss several approaches for multi-argument interval evaluation of the polynomial J_t and show how they can be adapted to other polynomials p. One particularly effective new method is based on expanding the polynomial p around several points ξ_j and truncating each resulting expansion p_{ξ_j} to a lower-degree polynomial.

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We discuss the use of interval arithmetic in the computation of the convex hull of n points in D dimensions. Convex hull algorithms rely on simple geometric tests, such as “does some point p lie in a certain half-space or affine subspace?” to determine the structure of the hull. These tests in turn can be carried out by solving appropriate (not necessarily square) linear systems. If interval-based methods are used for the solution of these systems then the correct hull can be obtained at lower cost than with exact arithmetic. In addition, interval-based methods can determine the topological structure of the convex hull even if the position of the points is not known exactly. In the present paper we compare various interval linear solvers with respect to their ability to handle close-to-pathological situations. This property determines how often interval arithmetic cannot provide the required information and therefore some computations must be redone with exact arithmetic.

In this paper we discuss the use of interval arithmetic in the computation of the convex hull of n points in D dimensions. If interval-based methods are used for the solution of certain appropriate (not necessarily square) linear systems then the correct topological structure of the convex hull can be obtained at lower cost than with exact rational arithmetic. In addition, interval-based methods can determine the topological structure of the convex hull even if the position of the points is not known exactly. In our experiments we compared various linear solvers with respect to their speed and their ability to handle close-to-pathological situations. The latter property determines how often interval arithmetic cannot provide the required information and therefore some computations must be redone with rational arithmetic.

Second-order enclosures for a function's range can be computed with centered forms, which involve a so-called slope vector. In this paper we discuss several techniques for determining such vectors and compare them with respect to tightness of the resulting enclosure. We advocate that a two-stage slope vector computation with symbolic preprocessing is optimal.

Understanding and controlling the behavior of chemical processes are important issues, for safety as well as economical reasons. Some processes can have multiple steady states and even switch between them in a complex way, the reasons for the multiplicity not always being well understood. A singularity theory approach for investigating such behavior leads to nonlinear systems whose solutions correspond to specific singular states of the process. In order to exclude certain types of singularities, rigorous methods must be used to check the solvability of the matching systems. As these systems are highly structured, our solution method combines a symbolic preprocessing phase (term manipulation for utilizing the structure) with a branch--and--bound type rigorous interval-based solver. We report on our experience with this approach for small--to--medium sized example problems.

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