Weierstrass method for quaternionic polynomial root‐finding

Abstract

MOS Classification: 65H04; 30G35; 12D10Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce.In this paper, we propose a Weierstrass-like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.

A Two-Step Quaternionic Root-Finding Method

“…We introduce now some definitions and results concerning H[x], which will play an important role in the sequel (see [11,15] for other details). We mainly follow the notions and notations of [5,9].…”

“…This is a free-derivative method relying on the factorization of the polynomial which makes its extension to the quaternion setting possible. Such generalization was derived in [5], where it was also proved that, as in the classical case, the method has quadratic order of convergence for the simple roots of a polynomial.…”

A blockchain-based traceable and secure data-sharing scheme

“…(2) is satisfied by the elliptic curve E , Eq. (1) is referred to as the Weierstrass equation ( Falcão et al, 2018 ). By further simplifying Eq.…”

Mathematica Tools for Quaternionic Polynomials

“…The explicit relation between factor-terms and zeros of a quaternionic polynomial is addressed in the following results. The first result is useful if one knows a factorization of the polynomial; for example in [10] a numerical method for computing a Weierstrass factorization of a quaternionic polynomial is proposed and the non-spherical zeros are obtained via (11). The second result plays an important role in the construction of polynomials with prescribed zeros.…”