Iterative methods, or methods of successive approximation, are often preferred in solving the algebraic equations which arise in approximating,~ ordinary or I partial differential equations. Several iterative schemes are in standard use and many others could be devised. But to determine under what circumstances a given one will converge and, if it converges, at what rate, are problems that are by no means trivial. The purpose of the present paper is to develop a fairly general technique that will often provide good convergence criteria. Most of the known criteria for the convergence of particular iterations fall out as special t cases. As by-products, applications will be made to questions concerning the nonsingularity of matrices, and the location of latent roots. Let the system of linear algebraic equations be written in the form
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