Numerical stability of a class (of systems) of nonlinear equations

Abstract

In this article we considerstability of nonlinear equations which have the following form:$$\begin{array}{}\text{(1)}& Ax+F(x)=b,\end{array}$$where $F$ is any function, $A$ is a linear operator, $b$ is givenand $x$ is an unknown vector. We give (under some assumptionsabout function $F$ and operator $A$) a generalization ofinequality:$$\begin{array}{}\text{(2)}& \frac{\Vert X_1-X_2\Vert }{\Vert X_1\Vert }\le \Vert A\Vert \Vert A^{-1}\Vert \frac{\Vert b_1-b_2\Vert }{\Vert b_1\Vert }\end{array}$$(equation (2) estimates the relative error of the solution whenthe linear equation $Ax=b_1$ becomes the equation $Ax=b_2$)and a generalization of inequality:$$\begin{array}{}\text{(3)}& \frac{\Vert X_1-X_2\Vert }{\Vert X_1\Vert }\le \Vert A_1^{-1}\Vert \Vert A_1\Vert (\frac{\Vert b_1-b_2\Vert }{\Vert b_1\Vert }+\Vert A_1\Vert \Vert A_2^{-1}\Vert \frac{\Vert b_2\Vert }{\Vert b_1\Vert }\cdot \frac{\Vert A_1-A_2\Vert }{\Vert A_1\Vert })\end{array}$$(equation (3) estimates the relative error of the solution whenthe linear equation $A_{1}x=b_1$ becomes the equation $A_{2}x=b_2$).