Computation of Square and Cube Roots of $p$-Adic Numbers via Newton-Raphson Method

Abstract

The problem of finding square roots of p-adic integers in Z p , p 2, has been a classic application of Hensel's lemma. A recent development on this problem is the application and analysis of convergence of numerical methods in approximating p-adic numbers. For a p-adic number a, Zerzaihi, Kecies, and Knapp (2010) introduced a fixedpoint method to find the square root of a in Q p . Zerzaihi and Kecies (2011) later extended this problem to finding the cube root of a using the secant method. In this paper, we compute for the square roots and cube roots of p-adic numbers in Q p , using the Newton-Raphson method. We present findings that confirm recent results on the square roots of p-adic numbers, and highlight the advantages of this method over the fixed point and secant methods. We also establish sufficient conditions for the convergence of this method, and determine the speed of its convergence. Finally, we detemine how many iterations are needed to obtain a specified number of correct digits in the approximate.

The performance of the secant method in the field of $p$-adic numbers

“…These authors [14] then applied the Newton method to find the cubic roots of padic numbers in Q p . A similar problem also appeared in [8] wherein Ignacio et al computed the square roots of p-adic numbers via Newton-Raphson method.…”

Householder's method for solving the $p$-adic polynomial equations

“…A related study was also carried out in [14] where Kecies considered the problem of finding the square roots of p-adic numbers in Q p through the secant method. A similar problem also appeared in [11] wherein Ignacio et al computed the square and cube roots of p-adic numbers via Newton-Raphson method.…”

NEWTON-RAPHSON METHOD FOR COMPUTING p-ADIC ROOTS

“…There are many numerical methods to find roots of p-adic numbers, but we use the Newton-Raphson method because of the speed of convergence of the approximate solutions ( [1], [3]). …”

“…The researches continued on finding the cubic root of p-adic numbers. These were done using the same technique that was used in finding square roots of p-adic numbers ( [2], [3], [5]). The latest research was on computing fifth roots of p-adic numbers ( [7]).…”