This paper deals with the iterative solution of the linear systemx=Bx+c when its Jacobi matrixB is weakly 2-cyclic consistently ordered and has a complex eigenvalue spectrum which lies on a straight-line segment. The optimization problem of the following three methods is considered and solved: i) The extrapolation of the optimum Successive Overrelaxation (SOR) ii) The second order extrapolation of a “good” SOR and iii) The second order extrapolation of the Gauss-Seidel method. In addition a variant of the second order methods considered, suitable for the solution of the system even ifB isnot necessarily weakly 2-cyclic consistently ordered, is proposed. Finally a reference to a theoretical comparison of the various optimum methods in the paper is made and their asymptotic convergence factors for selected eigenvalue spectra are illustrated in a Table in support of the theory developed.

Optimum first and second order extrapolations of successive overrelaxation type methods for certain classes of matrices

I n this paper, we present the rate of convergence of Generalized Extrapolated Newton – Raphson (GEN-R) Method to find the multiple roots of non-linear equations. Numerical examples are given to illustrate the performance of the presented methods.

The cubic rate of convergence of generalized extrapolated newton - raphson method for solving nonlinear equations

In this paper, we present Modified Generalized Newton’s (MGN) Method for solving non-linear equation of the type f(x) =0 and is studied in a special case when the multiplicity of the root of the equation is not known in advance and Some numerical examples are given to illustrate the efficiency and the performance of the new method.

• modified generalized newton’s method without multiplicity

First and second degree Extrapolated Accelerated Gauss-Seidel (EAGS) methods for solving a system of linear algebraic equations are studied and it is shown that the second degree EAGS method is superior to the Accelerated Overtaxation (AOR) under the assumption that the matrix coefficient of the system is consistently ordered and positive definite and zero is not an eigenvalue of the corresponding Jacobi matrix.

S.R. Koneru

Papers

Extrapolated accelerated gauss-seidel methods