New simultaneous root-finding methods with accelerated convergence for analytic functions

A new iterative method of the fourth order for the simultaneous determination of zeros of a class of analytic functions, is proposed. Further improvements of the basic method are attained by using Newton’s and Halley’s corrections giving the orders of convergence five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. Numerical examples demonstrate a good convergence properties, fitting very well theoretical results.