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Interval arithmetic: From principles to implementation
 J. ACM
"... We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithme ..."
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Cited by 77 (12 self)
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We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard’s specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed. 1
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 51 (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,...
Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems
, 1991
"... 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 accumul ..."
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Cited by 20 (9 self)
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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.
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.
A Comparison of Some Methods for Bounding Connected and Disconnected Solution Sets of Interval Linear Systems
, 2007
"... Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval c ..."
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Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval coefficients. In many cases, singular systems are admitted within the intervals of uncertainty of the coefficients, leading to unbounded solution sets with more than one disconnected component. This, combined with the fact that computing exact bounds on the solution set is NPhard, limits the range of techniques available for bounding the solution sets for such systems. However, the componentwise nature and other properties make the interval Gauss–Seidel method suited to computing meaningful bounds in a predictable amount of computing time. For this reason, we focus on the interval Gauss–Seidel method. In particular, we study and compare various preconditioning techniques we have developed over the years but not fully investigated, comparing the results. Based on a study of the preconditioners in detail on some simple, speciallydesigned small systems, we propose two heuristic algorithms, then study the behavior of the preconditioners on some larger, randomly generated systems, as well as a small selection of systems from the Matrix Market collection.
æ Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems
"... Abstract. 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 ..."
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Abstract. 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 Können in Verbindung mit anderen Verfahren verwendet werden, um alle Lösungen eines nichlinearen Gleichungsystems
(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 ..."
Abstract
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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.