Kantorovich Theorem and a Class of Successive Overrelaxation Methods on Error Reduction Factor Per Iteration in the Solution to Nonlinear System of Equations
Stephen Ehidiamhen Uwamusi *
Department of Mathematics, Faculty of Physical Sciences, University of Benin, Benin City, Edo State, Nigeria.
Andrew Esaborlupia Uwamusi
Department of Mathematics, College of Science, Federal University of Petroleum Resources, Effurun Delta State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This paper provides a basis for the analysis of Kantorovich theorem and a class of Successive Overrelaxation (SOR) methods in the nonlinear system of equations. We provide answers to the question whether a local homeomorphism of a differentiable function is a basis to mean that a Lipeomorphism of signifies a Lipschitz homeomorphism. Using Heinrich and Mankiewicz’s theorem that established every separable metric space is Lipschitz equivalent to a subset of a continuous function, we are led to investigating the matrix splitting as a preconditioner in the iterative process leading to solution of system of nonlinear equations. The emphasis is on the rigours concerning error reduction factor per iteration in the classical iterative methods of Jacobi, Gauss-Siedel, Successive Overrelaxation (SOR) in comparison with Kantorivich theorem. In this regard, we carefully avoided making comments regarding measurements of convergence speeds in terms of iteration matrices for Jacobi, Gauss−Siedel and Successive Overrelaxation methods since SOR has been traditionally known to be the fastest in this class. Further inquiry on performance among this class of linear solvers led to extension of ideas to the cyclic Chevbyshev semi- iterative method in the computation.
The first result we got in the findings is the proof that Newton operator on Jacobi, Gauss−Siedel, Successive overrelaxation (SOR) iterations and Chevbyshev−Semi iteration respectively, satisfied the Keller space.
Secondly, it is established that among these classes of methods , in terms of error reduction per step, it was discovered that Newton−Kantorovich method has the least number of iterations to attain desired accuracy followed closely by Newton −Gauss−Siedel method while Newton−Jacobi iteration maintained a third position with Newton−Chevbyshev −Semi−iterative method coming last in the analysis. Numerical examples given to support our claim proved so. Numerical toolbox used is programmed in Python C++ codes for Matlab routines.
Keywords: System of nonlinear equations, homeomorphism, Banach space, Kantorovich theorem, error reduction factor, Keller space