Abstract
This paper presents new results on the behavior of Newton's method when it is used to search for the roots of a cubic equation. When the equation has one real root and two real singular points, Newton's method can exhibit chaotic or periodic behavior. Asymptotic methods are used to derive results for the period p solutions of the iterative map for large p. The effect of introducing a relaxation parameter on the chaotic and periodic behavior is also discussed. The applicability of the results to more general nonlinear functions and to the solution of equations of state is discussed.
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