Iteration functions for complex solutions of cubic polynomial equations

Abstract

Iterative methods provide an alternative to finding the solutions of equations where analytical methods are inconvenient or even impossible to use. This study which focuses on cubic polynomial equation f(t)= at3 + bt2 + ct + d = 0, a > 0 with real coefficients and having an imaginary root, found that the fixed-point iteration xn+1 = h(xn) where h(t)=13[f(t)f′(t)−ba] will always converge to the real part x of the imaginary root of f(t) = 0 whenever b2 - 3ac < 0. The only real root of g(t)=12f′(t)f″(t)−af(t)=0 was found to be the real part x of the imaginary root of f(t) = 0 and is always outside the interval formed by the critical numbers of the function f.