CONVERGENCE OF A RELAXED NEWTON METHOD FOR CUBIC EQUATIONS

This paper presents new results on the behavior of Newton's method whe n it is used to search for the roots of a cubic equation. When the equ ation 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 chaot ic and periodic behavior is also discussed. The applicability of the r esults to more general nonlinear functions and to the solution of equa tions of state is discussed.