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A Review of Preconditioners for the Interval GaussSeidel Method
, 1991
"... . Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system ..."
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Cited by 52 (16 self)
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. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) +F 0 (X)( ~ X \Gamma M); if interval arithmetic is then used to bound the solutions of this system, the resulting box ~ X contains all roots of the nonlinear system. We may use the interval GaussSeidel method to find these solution bounds. In order to increase the overall efficiency of the interval Newton / generalized bisection algorithm, the linear interval system is multiplied by a preconditioner matrix Y before the interval GaussSeidel method is applied. Here, we review results we have obtained over the past few years concerning computation of such preconditioners. We emphasize importance and connecting relationships,...
A Review Of Techniques In The Verified Solution Of Constrained Global Optimization Problems
, 1996
"... Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previousl ..."
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Cited by 25 (6 self)
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Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed. 1 INTRODUCTION, BASIC IDEAS AND LITERATURE We consider the constrained global optimization problem minimize OE(X) subject to c i (X) = 0; i = 1; : : : ; m (1.1) a i j x i j b i j ; j = 1; : : : ; q; where X = (x 1 ; : : : ; xn ) T . A general constrained optimization problem, including inequality constraints g(X) 0 can be put into this form by introducing slack variables s, replacing by s + g(X) = 0, and appending the bound constraint 0 s ! 1; see x2.2. 2 Chapter 1 W...
An Interval Branch and Bound Algorithm for Bound Constrained Optimization Problems
 JOURNAL OF GLOBAL OPTIMIZATION
, 1992
"... In this paper, we propose modifications to a prototypical branch and bound algorithm for nonlinear optimization so that the algorithm efficiently handles constrained problems with constant bound constraints. The modifications involve treating subregions of the boundary identically to interior region ..."
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Cited by 13 (5 self)
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In this paper, we propose modifications to a prototypical branch and bound algorithm for nonlinear optimization so that the algorithm efficiently handles constrained problems with constant bound constraints. The modifications involve treating subregions of the boundary identically to interior regions during the branch and bound process, but using reduced gradients for the interval Newton method. The modifications also involve preconditioners for the interval GaussSeidel method which are optimal in the sense that their application selectively gives a coordinate bound of minimum width, a coordinate bound whose left endpoint is as large as possible, or a coordinate bound whose right endpoint is as small as possible. We give experimental results on a selection of problems with different properties.
On interval weighted threelayer neural networks
 In Proceedings of the 31 Annual Simulation Symposium
, 1998
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Parallel Reliable Computing with Interval Arithmetic
"... Reliability of computational results is crucial in computational science and engineering. In this paper, we report some current research results on parallel reliable computing with interval arithmetic. In section 1, a brief introduction to interval arithmetic is provided. In section 2, an interval a ..."
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Cited by 1 (0 self)
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Reliability of computational results is crucial in computational science and engineering. In this paper, we report some current research results on parallel reliable computing with interval arithmetic. In section 1, a brief introduction to interval arithmetic is provided. In section 2, an interval algorithm to reliably solving largescale sparse nonlinear systems of equations is presented. In section 3, polynomial interpolation with interval arithmetic is studied. We conclude this paper with section 4. I. Introduction Interval arithmetic, first introduced by Moore [24] in the 1960's, has become an active research area in scientific computing. Here is the definition of interval arithmetic. Definition 1.1: Let x and y be two real intervals 1 , and op be one of the arithmetic operations +; \Gamma; \Theta, \Xi. Then, x op y = fx op y : x 2 x; y 2 yg, provided that 0 62 y if op represents \Xi. For example, [1; 2] + [\Gamma1; 0] = [0; 2] and [2; 4] \Xi [1; 2] = [1; 4]. Some reasons for...
On Interval Weighted Threelayer Neural Networks
 Proc. of the 31 Annual Simulation Symposium
, 1998
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On the optimization of INTBIS on a Cray YMP
 Reliable Computing
, 1995
"... [NTBIS is a welltested software package which uses an interval Newton/generalized bisection method to find all numerical solutions to nonlinear systems of equations. Since INTBIS uses interval computations, its results are guaranteed to cnntain all solutions. To efficiently solve very, large nonlin ..."
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[NTBIS is a welltested software package which uses an interval Newton/generalized bisection method to find all numerical solutions to nonlinear systems of equations. Since INTBIS uses interval computations, its results are guaranteed to cnntain all solutions. To efficiently solve very, large nonlinear systems on a parallel vector computer, it is necessary to effectively utilize the architectural features of the machine [n this paper, we report our implementations of INTBIS for large nonlinear systems on the Cray YMP supercomputer. We first present the direct implementation of INTBIS nn a Cray. Then, we report our work on optimizing INTBIS on the Cray YMP
An Online Interval Calculator
 Society for Computer Simulation
, 1998
"... In this paper, we report the motivation, design, implementation, and usage of an online interval calculator which we developed very recently. ..."
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In this paper, we report the motivation, design, implementation, and usage of an online interval calculator which we developed very recently.
(1.1) æ A Review of Preconditioners for the Interval Gauss–Seidel Method
"... Abstract. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ⊂ R n of a system of nonlinear equations F(X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the sys ..."
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Abstract. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ⊂ R n of a system of nonlinear equations F(X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) + F ′ (X) ( ˜ X − M); if interval arithmetic is then used to bound the solutions of this system, the resulting box ˜ X contains all roots of the nonlinear system. We may use the interval Gauss–Seidel method to find these solution bounds. In order to increase the overall efficiency of the interval Newton / generalized bisection algorithm, the linear interval system is multiplied by a preconditioner matrix Y before the interval Gauss–Seidel method is applied. Here, we review results we have obtained over the past few years concerning computation of such preconditioners. We emphasize importance and connecting relationships, and we cite references for the underlying elementary theory and other details.