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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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. 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,...
Mathematik 9 by SpringerVerlag 1980 Global Optimization Using Interval Analysis
"... Summary. We show how interval analysis can be used to compute the global minimum of a twicecontinuously differentiable function of n variables over an ndimensional parallelopiped with sides parallel to the coordinate axes. Our method provides infallible bounds on both the globally minimum value of ..."
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Summary. We show how interval analysis can be used to compute the global minimum of a twicecontinuously differentiable function of n variables over an ndimensional parallelopiped with sides parallel to the coordinate axes. Our method provides infallible bounds on both the globally minimum value of the function and the point(s) at which the minimum occurs. Subject Classification: AMS(MOS): 65K05, 90C30. 1.
INTERVAL MATHEMATICS TECHNIQUES FOR CONTROL THEORY COMPUTATIONS
"... Various types of nonlinear equations or systems of equations arise in elementary and advanced control theory. For example, the transfer function corresponding to a single linear controlled ordinary differential equation is a rational function (cf. eg., [2], Sect. 1.2.). Stability of such systems dep ..."
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Various types of nonlinear equations or systems of equations arise in elementary and advanced control theory. For example, the transfer function corresponding to a single linear controlled ordinary differential equation is a rational function (cf. eg., [2], Sect. 1.2.). Stability of such systems depends
(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.
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, 1983
"... J Interval analysis is applied to the fixedpoint problem x *(x) for continuous +: S + S, where the space S is constructed from Cartesian products of the set R of real numbers, with componentwise definitions of arithmetic operations, ordering, and the product topology. Spaces of this kind include m ..."
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J Interval analysis is applied to the fixedpoint problem x *(x) for continuous +: S + S, where the space S is constructed from Cartesian products of the set R of real numbers, with componentwise definitions of arithmetic operations, ordering, and the product topology. Spaces of this kind include many of the usual Hilbert and Banach spaces important in analysis. With the aid of an interval inclusion 0: IS + IS in the interval space IS corresponding to S, the interval iteration process XN+ 1 X n 4(XN) is shown to converge if the initial interval X0 contains a fixed point x * of *i on the other hand, divergence of the iteration (XN4+ = 0 for some N) proves that X0 contains no fixed points of *, while O(N) c X for some N establishes the existence of a fixed point x * E X0 and guarantees the convergence of the interval iteration. Each step of interval iteration provides lower and upper bounds for fixed points of * in the initial interval, from which approximate values and guaranteed error
LIBRARIES
, 1992
"... Design automation requires reliable methods for solving the equations describing the performance of the engineering system. While progress has been made to provide good algorithms for polynomial systems, we propose new techniques for the solution of general nonlinear algebraic systems. Moreover, t ..."
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Design automation requires reliable methods for solving the equations describing the performance of the engineering system. While progress has been made to provide good algorithms for polynomial systems, we propose new techniques for the solution of general nonlinear algebraic systems. Moreover, the interval arithmetic approach we have chosen also guarantees numerical reliability. In this thesis we present a number of new algorithms that improve both the quality and the efficiency of existing interval arithmetic techniques for enclosing the solution of nonlinear algebraic equations. More specifically, we propose an exact algorithm for the solution of midpoint preconditioned linear interval equations. We extend existing techniques for automatic compilation of fast partial derivatives to include interval slopes and have conceived a number of graph algorithms to improve their efficiency for general computational graphs. Furthermore, we have devised variable precision techniques to automatically control the required precision based on interval width. Finally, we have unified a number of enclosure languages using denotational semantics.