A Review of Preconditioners for the Interval Gauss-Seidel Method (original) (raw)
Related papers
Preconditioners for the Interval Gauss-Seidel Method
1990
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.
On Preconditioners and Splitting in the Interval Gauss – Seidel Method
2005
Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as finding mathematically rigorous but rapid location of all solutions to nonlinear systems or finding global optima, involves bounding the solution sets to systems of equations with wide interval coefficients. The interval Gauss–Seidel algorithm has various properties that make it suited to this task. However, the system must in general be preconditioned for the interval Gauss–Seidel method to be effective. The most common preconditioner has been the “inverse midpoint” preconditioner; however, we have proposed other classes of preconditioners that obey certain optimality conditions, and have shown empirically advantages of their use. In this paper, we revisit similar preconditioners, previously applied only in the context of interval Newton methods, that are appropriate when the solution set may have more than one semi-infinite component. We first review our previous...
International Journal of Physical Sciences, 2011
We discuss Hansen-Sengupta operator in the context of circular interval arithmetic for the algebraic inclusion of zeros of interval nonlinear systems of equations. It was demonstrated by showing the effects of applying repeatedly preconditioners of inverses of the midpoint interval matrices on the well known Trapezoidal Newton method at each iteration cycle wherein, the work of Shokri (2008) was our major tool of investigation. It was shown that the Trapezoidal interval Newton method with inverse midpoint interval matrix as preconditioner is not a H-continuous map and that Baire category failed to hold in the sense of Aguelov et al. (2007). This was more so since it produced from our numerical example, not only overestimated results but, also results that are not finitely bounded which we compare with results computed previously given in Uwamusi.
Interval linear constraint solving using the preconditioned interval gauss-seidel method
1995
Abstract We propose the use of the preconditioned interval Gauss-Seidel method as the backbone of an efficient linear equality solver in a CLP (Interval) language. The method, as originally designed, works only on linear systems with square coefficient matrices. Even imposing such a restriction, a naive incorporation of the traditional preconditioning algorithm in a CLP language incurs a high worst-case time complexity of O (n4), where n is the number of variables in the linear system.
Applied Mathematics and Computation, 1983
We introduce an interval Newton method for bounding solutions of systems of nonlinear equations. It entails. three subalgorithms.
A general iterative sparse linear solver and its parallelization for interval Newton methods
Reliable Computing, 1995
Interval Newton/Generalized Bisection methods reliably find all numerical solutions within a given domain. Both computational complexity analysis and numerical experiments have shown that solving the corresponding interval linear system generated by interval Newton's methods can be computationally expensive (especially when the nonlinear system is large).
On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic
Lecture Notes in Computer Science, 2006
This paper investigates the impact of the selection of a transversal on the speed of convergence of interval methods based on the nonlinear Gauss-Seidel scheme to solve nonlinear systems of equations. It is shown that, in a marked contrast with the linear case, such a selection does not speed up the computation in the general case; directions for researches on more flexible methods to select projections are then discussed.
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.
Global Journal of Mathematical Sciences, 2014
The paper is a modificationofNguyen and Revol"s method for the solution set to the linear interval system. The presented methoddoes not require solving Kahan"s arithmetic which may be a hindrance to that of Nguyen and Revol"s method as Nguyen and Revol"s method relies mainly on interval data inputs.Our method under consideration first advances solutionusing real floating point LU Factorization to the real point linear system and then solves a preconditioned residual linear interval system for the error term by incorporating Rohn"s method which does not make use of interval data inputs wherein, the use of united solution set in the sense of Shary comes in handy as a tool for bounding solution for the linear interval system. Special attention is paid to the regularity of the preconditioned interval matrix. Numerical exampleis used to illustrate the algorithm and remarks are made based on the strength of our findings.