Effects of repeatedly used preconditioner on computational accuracy for nonlinear interval system of equations (original) (raw)
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Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that nd all real roots within a speci ed box X R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in nite-precision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) + F 0 (X)(X ? M); if interval arithmetic is then used to bound the solutions of this system, the resulting boxX contains all roots of the nonlinear system. We may use the interval Gauss{Seidel method to nd these solution bounds. In order to increase the overall e ciency 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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Global Journal of Mathematical Sciences, 2014
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A Review of Preconditioners for the Interval Gauss-Seidel Method
1991
Interval Newton methods in conjunction with generalized bi- section can form the basis of algorithms that Þnd all real roots within a speciÞed box X öRn of a system of nonlinear equations F(X) = 0 with mathematical certainty, even in Þnite-precision arithmetic. In such methods, the system F(X) = 0 is transformed into a linear interval system 0 = F(M)
A new class of interval methods with higher order of convergence
Computing, 1989
A New Class of Interval Methods with Higher Order of Convergence. In this paper we introduce a new class of interval methods for enclosing a simple root of a nonlinear equation. For each nonnegative integer p we describe an iterative procedure belonging to this class which requires p + 1 function values and an interval evaluation of the second derivative per step. The order of convergence of the iterative procedure grows exponentially with p. For p_>4 this order is strictly greater than Key words." Nonlinear equations, order of convergence.
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Numerical Solution of Interval Nonlinear System of Equations
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I hereby certify that the work which is being presented in the thesis entitled "Numerical Solution of Interval Nonlinear System of Equations" in partial fulfilment of the requirement for the award of the degree of Master of Science, submitted in the Department of Mathematics, National Institute of Technology Rourkela is an authentic record of my own work carried out under the supervision of Prof. S. Chakraverty. The matter embodied in this thesis has not been submitted in other institution or university for the award of any other degree or diploma.