Abstract

In this article we consider stability 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 given and $x$ is an unknown vector. We give (under some assumptions about function $F$ and operator $A$) a generalization of inequality: $$ \frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq \|A\|\|A^{-1}\|\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|} \tag2 $$ (equation (2) estimates the relative error of the solution when the 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 when the linear equation $A_1x=b_{1}$ becomes the equation $A_2x=b_{2}$).

Keywords: Numerical stability, nonlinear equations.

MSC: 65J15

Pages:  27$-$33     

Volume  57 ,  Issue  1$-$2 ,  2005