ITERATIVE INTERVAL FORMULAS FOR SYSTEM OF EQUATION INTOPOLOGICAL SPACE
Keywords:
multivariate functions, interval arithmetic, nearness in topology, topological vector space, suslin space, K- analytic, ray tracing implicit surfaceAbstract
Given a map F : X Y acting between two topological spaces 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 Fdifferentiable function and the Banach Fixed point theorem. Furthermore,
solving the resulting linear system of equation for the map n n F : R R
using either Guassian or LU decomposition, we again ask under what condition can we say that Guassian elimination method or LU Factorization cannot compute exactly the inverse of the matrix. .In this paper, we give such error bounds 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 method with 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.
