ITERATIVE INTERVAL FORMULAS FOR SYSTEM OF EQUATION IN TOPOLOGICAL SPACE

Abstract

Given a map F : X  Y acting between two topologicalspaces X and Y , it is pertinent to ask if the path from a point X to a point Y is a closed path, and under what conditions can the topological space from X to a topological space Y be said to be contractible to a point?We give answers to this poised question using the concept of hemi-continuity for F-differentiable function and the Banach Fixed point theorem. Furthermore, solving the resulting linear system of equation for the map F : R n  R n using either Guassian or LU decomposition, we againask under what conditioncan wesay that Guassian elimination method or LU Factorization cannot compute exactly the inverse of the matrix.In this paper, wegive such errorbounds for the LU Factorization and the resulting residual error estimate for system of equation.We realized our solutions to systems of nonlinear equations using the interval LU Factorization, the interval Gauss-Siedel
iteration and the Krawczyk’s interval methodwith guaranteed error bounds. A ray tracing implicit surface for the obtained solution is described and a normalized distance between imaging and distortion of a ray tracing implicit surface in the obtained solutions from the nonlinear system is computed.