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Summary Newton's method for the evaluation of the zeros of analytic functions is generalized by the recurrence relation zi+i = Zi- (ki-li)fi\zi)/fli+1)(zi), where ki and k are determined so that ideally U + 1 = ki = k in the vicinity of a zero of order k. The process so defined is a second-order one and yields accurate approximations to zeros for all values of k, provided I,, + 1 = k{ = k.

A unified process for the evaluation of the zeros of polynomials over the complex number field