NUMERICAL STABILITY OF A CLASS (OF SYSTEMS) OF NONLINEAR EQUATIONS

In this article we consider stability of nonlinear equations which have the following form: Ax + F (x) = b, (1) where F is any function, A is a linear operator, b is given and x is an unknown vector. We give (under some assumptions about function F and operator A) a generalization of inequality: parallel to X-1 - X-2 parallel to/parallel to X-1 parallel to <= parallel to A parallel to parallel to A(-1)parallel to parallel to b(1) - b(2)parallel to/parallel to b(1)parallel to (2) (equation (2) estimates the relative error of the solution when the linear equation Ax = b(1) becomes the equation Ax = b(2)) and a generalization of inequality: parallel to X-1 - X-2 parallel to/parallel to X-1 parallel to <= parallel to A(1)(-1)parallel to parallel to A(1)parallel to (b(1) - b(2)parallel to/parallel to b(1)parallel to + parallel to A(1)parallel to parallel to A(2)(-1)parallel to parallel to b(2)parallel to / parallel to b(1)parallel to . parallel to A(1) - A(2) parallel to/parallel to A(1)parallel to (3) (equation (3) estimates the relative error of the solution when the linear equation A(1)x = b(1) becomes the equation A(2)x = b(2)).