Computable Chain-reachable Mapping of Linear and Quadratic Backward
SOR Iteration for Newton Operator.

Stephen Ehidiamhen UWAMUSI
Page No. 18-27

The contraction mapping of Linear and Quadratic backward SOR methods for Newton operator in
the nonlinear system of equation in real floating point arithmetic is presented. It is showed that if
the computable reachable set of linear backward SOR method is lower chain-reachable to the outer
computable chain-reachable Quadratic SOR method, the quadratic backward SOR method is not
only finer in topology but also faster than backward linear SOR method if the arithmetic
computational complexity involved in the execution of backward quadratic SOR is overlooked. This
was demonstrated by a numerical example with the two methods where quadratic backward SOR
method with Newton operator is showed to have superiority over the linear backward SOR with
Newton operator. The computed results for the two methods were compared with results earlier
obtained from Uwamusi where interval Gauss-Siedel method was used.
Keyword : nonlinear system, newton method, Brouwer fixed point theorem, Hahn –Banach extension
theorem, SOR iteration matrix
MSC 2010 Category: 15A09, 15A 29, 97 I 20, 34G20