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This paper discusses the problem of errors that arise when using approximate element characteristics to represent the nonlinear elements in a network. In particular, we study the use of piecewise linear approximations, and consider both resistive and dynamic networks. The logarithmic derivative of a mapping as well as the measure of a square matrix are used to compute bounds on the solution errors in terms of properties of either the original nonlinear system or of the approximating system. Applications of these results to nonlinear networks and approximating piecewise linear counterparts are illustrated using numerical examples.