Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations

Several optimal eighth order methods to obtain simple roots are analyzed. The methods are based on two step, fourth order optimal methods and a third step of modified Newton. The modification is performed by taking an interpolating polynomial to replace either f(zn) or f′(zn). In six of the eight methods we have used a Hermite interpolating polynomial. The other two schemes use inverse interpolation. We discovered that the eighth order methods based on Jarratt's optimal fourth order methods perform well and those based on King's or Kung-Traub's methods do not. In all cases tested, the replacement of f(z) by Hermite interpolation is better than the replacement of the derivative, f′(z).