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

Abstract

In this article we considerstability of nonlinear equations which have the following form:$$Ax+F(x)=b, \tag1$$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:$$\frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq\|A\|\|A^{-1}\|\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|} \tag2$$(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:$$\frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq\|A_{1}^{-1}\|\|A_{1}\|\left(\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|}+\|A_{1}\|\|A_{2}^{-1}\|\frac{\|b_{2}\|}{\|b_{1}\|}\cdot\frac{\|A_{1}-A_{2}\|}{\|A_{1}\|}\right) \tag3$$(equation (3) estimates the relative error of the solution whenthe linear equation $A_1x=b_{1}$ becomes the equation $A_2x=b_{2}$).