High-order nonlinear solver for multiple roots

A method of order four for finding multiple zeros of nonlinear functions is developed. The method is based on Jarratt's fifth-order method (for simple roots) and it requires one evaluation of the function and three evaluations of the derivative. The informational efficiency of the method is the same as previously developed schemes of lower order. For the special case of double root, we found a family of fourth-order methods requiring one less derivative. Thus this family is more efficient than all others. All these methods require the knowledge of the multiplicity.