Nonlinear resistive networks, which can be characterized by the equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x) =y</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\cdot)</tex> is a continuous piecewise linear mapping of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> into itself, are discussed. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is a point in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> and represents a set of chosen network variables and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> is an arbitrary point in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> and represents the input to the network. New theorems on the existence of solutions together with a convergent method for obtaining at least one of the solutions are given. Also dealt with is an efficient computational algorithm which is especially suited for analysis of large piecewise-linear networks. The effectiveness of the method in terms of the amount of computation and data handling and storage is demonstrated.