Malaysian Journal of Mathematical Sciences, January 2017, Vol. 11, No. 1
Solving Higher-Order \(p\)-Adic Polynomial Equations via Newton-Raphson Method
Julius Fergy T. Rabago
Corresponding Email: jtrabago@upd.edu.ph
Received date: 11 November 2015
Accepted date: 16 December 2016
Abstract:
We consider the root-finding problem \(f(x) = 0\), \(f \in \mathbb{Z}_{p}[x]\), and seek a root in \(\mathbb{Z}_{p}\) of this equation through a \(p\)-adic analogue of Newton-Raphson method. We show in particular that, under appropriate assumptions, the sequence of approximants generated by the iterative formula of the Newton-Raphson method converges to a unique root of \(f\) in \(\mathbb{Z}_{p}\). Also, we give the rate of convergence of this method in the \(p\)-adic setting. Our work generalizes previous results concerning \(q\)-th roots of \(p\)-adic numbers due to Kecies and Zerzaihi (2013) and Ignacio et al. (2016).
Keywords: Newton-Rapshon method, \(p\)-adic polynomials, \(p\)-adic numbers, roots of polynomials
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