Newton's method and high-order algorithms for the nth root computation

Authors:
François Dubeau
Affiliations:
Département de mathématiques, Faculté des sciences, Université de Sherbrooke, 2500, boul. de l'Université, Sherbrooke (Qc), Canada, J1K 2R1
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

Citing 5
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Abstract

Two modifications of Newton's method to accelerate the convergence of the nth root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p@?N,p=2. We consider affine combinations of the two modified pth-order methods which lead to a family of methods of order p with arbitrarily small asymptotic constants. Moreover the methods are of order p+1 for some specific values of a parameter. Then we consider affine combinations of the three methods of order p+1 to get methods of order p+1 again with arbitrarily small asymptotic constants. The methods can be of order p+2 with arbitrarily small asymptotic constants, and also of order p+3 for some specific values of the parameters of the affine combination. It is shown that infinitely many pth-order methods exist for the nth root computation of a strictly positive real number for any p=3.