A Hummel & Seebeck family of iterative methods with improved convergence and efficiency

Several improvements to a family of iterative methods deduced from a Hummel & Seebeck theorem are presented. A symbolic computation that allows to find the best coefficients respect to the local order of convergence is also given. The theoretical and computational order of convergence for all methods is increased. Furthermore, the efficiency of these methods applied to the functions tested is improved. Adapting the strategy presented here a new iteration function with a new evaluation of the function is obtained and using adaptive multi-precision arithmetic a smaller cost is got. The numerical results computed carrying out this procedure, with a floating point system representing 1000 decimal digits, support this theory.