A Note on the Perturbation of arithmetic expressions

In this paper we present the theoretical foundation of forward error analysis of numerical algorithms under; Approximations in built-in functions. Rounding errors in arithmetic floating-point operations. Perturbations of data.The error analysis is based on linearization method. The fundamental tools of the forward error analysis are system of linear absolute and relative a prior and a posteriori error equations and associated condition numbers constituting optimal of possible cumulative round – off errors. The condition numbers enable simple general, quantitative bounds definitions of numerical stability. The theoretical results have been applied a Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.