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

✍ Scribed by François Dubeau

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 705 KB
• Volume: 224
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2008.04.014

✦ Synopsis

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.