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In this paper, a new trust-region subproblem combining with the BFGS update is proposed for solving nonlinear equations, where the trust region radius is defined by a new way. The global convergence without the nondegeneracy assumption and the quadratic convergence are obtained under suitable conditions. Numerical results show that this method is more effective than the norm method. [ABSTRACT FROM AUTHOR]

The idea of containment sets (csets) is due to Walster and Hansen, and the theory is mainly due to the first author. Now that floating point computation with infinities is widely accepted, it is necessary to achieve the same for interval computation. The cset of a function over a set in its domain space is the set of all limits of normal function values over that set. Csets form a sound basis for defining a number of practical models for interval arithmetic that handle division by zero and related operations in an intuitive and consistent manner. Cset-based systems offer new opportunities for compiler optimization by rearranging interval expressions, achieving tighter bounds by reducing dependencies within the expression. This paper presents basic theory. It discusses division by zero, the case for a global flag to support ``loose'' evaluation, performance, and semantics. It presents numerical examples using a trial Matlab implementation. [ABSTRACT FROM AUTHOR]