We describe algorithms for computing the greatest common divisor GCD of two univariate polynomials with inexactlyknown coe cients. Assuming that an estimate for the GCD degree is available e.g., using an SVD-based algorithm, we formulate and solve a nonlinear optimization problem in order to determine the coe cients of the best" GCD. We discuss various issues related to the implementation of the algorithms and present some preliminary test results.
Abstract. A robust technique for solving primitive-variable formulations of the incompressibleNavier-Stokes equations is to use Newton iteration for the fully-implicit nonlinear equations. A direct sparse matrix method can be used to solve the Jacobian but is costly for large problems; an alternative is to use an iterative matrix method. This paper investigates effective ways of using a conjugate gradient type method with an incomplete LU factorization preconditioner for two-dimensional incompressible viscous flow problems. Special attention is paid to the ordering of unknowns, with emphasis on a minimum updating matrix (MUM) ordering. Numerical results are given for several test problems.
This article presents a concise review of the scientific-technical computing system Maple 1 and its application potentials in Operations Research, systems modeling and optimization. The primary emphasis is placed on nonlinear optimization models that may involve complicated functions, and/or may have multiple -global and local -optima. We introduce the Global Optimization Toolbox to solve such models, and illustrate its use by numerical examples.Keywords: advanced systems modeling and optimization; Maple; Global Optimization Toolbox; application examples.
Model Development and Optimization: An O.R. FrameworkIn today's competitive global economy, government organizations and private businesses all aim for resource-efficient operations that deliver high quality products and services. This demands effective and timely decisions in an increasingly complex and dynamic environment. Operations Research (O.R.) provides a scientifically established methodology to assist analysts and decision-makers in finding "good" (feasible) or "best" (optimal) solutions. O.R. has undergone remarkable progress since its beginnings: for example, the 50 th Anniversary Issue of the journal Operations Research (2002) reviews an impressive range of real-world applications.To motivate the present discussion, we shall review the key steps of the O.R. modeling framework. A formal procedure aimed at making optimized decisions consists of the following main steps.1. Conceptual description of the decision problem at a suitable level of abstraction, retaining only the essential attributes, while omitting secondary details and circumstances.2. Development of a quantitative model (or system of models) that captures the key elements of the decision problem, in terms of decision variables and the relevant functional relationships among these, expressed by constraints and objectives.3. Development and/or adaptation of an algorithmic solution procedure, in order to explore the set of feasible solutions and to select the best decision (or a list of good alternative decisions).1 Maple is a trademark of Maplesoft, a division of Waterloo Maple Inc.
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