Convergence properties of continuous iterative methods

One of the solution methods for algebraic equations is the continuous iterative method. This is an iterative method in which a differential equation is constructed with the solution of a given algebraic equation as the equilibrium solution, and the solution of the difference equation obtained by discretizing that equation is sought until the solution converges. The method is an extension of various basic iterative methods, and coincides with the original iterative method when the discretization step width is 1. This paper shows first that the continuous iterative method converges globally if the function of the given algebraic equation is a product of a positive-definite matrix, and the gradient of a scalar function and the step width is somewhat small. Especially, for the case where the algebraic equation is linear, it is shown that the continuous interative method may converge even if the original iterative method diverges, or that it converges faster than the original iterative method if the parameters are appropriately set. Then a case is considered where the function of the given algebraic equation is the gradient of a scalar function. A method is proposed which adjusts the coefficients of the continuous iterative method adaptively so that the convergence is as fast as possible. The effectiveness of the method is verified by simple examples.