Convergence Analysis of Acceleration Schemes: Spectral Radius versus Condition Number

Abstract

The theory of matrix splitting plays an important role in convergence analysis for the acceleration schemes. Spectral radius allows us to make a complete description of eigenvalues of a matrix and is independent of any particular matrix norm. The aim of this research article is to find the relationship between spectral radius and condition number in terms of stability of algorithms using various classes of linear systems and determine whether the condition number can be used to predict the convergence of acceleration schemes as well as iterative methods in the solutions of systems of linear equation. From the results obtained using matlab programming language, there is strong relationship between spectral radius and condition number for dense symmetric positive definite (SPD) and tridiagonal system. The relationship is shown as k(A)=||A|| ||A-1||=ρ(A)(A-1) and the value of condition number helps to predict the rate of convergence and iterations for a given linear system.