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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 ..."
Abstract

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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,...
(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.