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 $\mathbb{Z}_p$, $p\neq 2$, 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 fixed-point method to find the square root of $a$ in $\mathbb{Q}_p$. 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 $\mathbb{Q}_p$, 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.