Newton’s method and high-order algorithms for the nth root computation |
| |
Authors: | François Dubeau |
| |
Institution: | Département de mathématiques, Faculté des sciences, Université de Sherbrooke, 2500, boul. de l’Université, Sherbrooke (Qc), Canada, J1K 2R1 |
| |
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. |
| |
Keywords: | 65-01 11B37 65B99 65D99 65H05 |
本文献已被 ScienceDirect 等数据库收录! |
|