Jacobi-Perronのアルゴリズムによるn乗根の計算法

概要

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Jacobi-Perron Algorithm of n tuple (1, α, ..., α^<n-1>) with α=L^<1/n> is periodic for special classes of positive integer L. A fast algorithm to approximate n tuple (1, α, ..., α^<n-1>) with any positive number L is derived by using the exact formulae of the sequences {A^<(υ)>_κ} of the periodical JPA. Then a fourth-order converging algorithm of an irrational L^<1/n> is derived by simplifying a set of quadratic converging recurrence formulae for L^<1/n>. This algorithm is composed of those formulae and the average calculated by the formulae. The former is shown to be closely related to Newton's formula x_1=x_0-fm(x_0)/f′_m(x_0) with f_m(x)=x^<n-m>-L/X^m=0 (m=0, ..., n-1). The latter is a quadratic converging operation that is easily proved by using those Newton's formulae.
日本応用数理学会の論文
1997-12-15