Zeros of unilateral quaternionic polynomials

Abstract

Abstract. The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the strehgth of this method, we compare it with the Niven algorithm and show where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For the convenience of the readers, we explicitly solve some examples of second and third order unilateral quaternionic polynomials. The leading idea of the practical solution method proposed in this work can be summarized in following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.

Weierstrass method for quaternionic polynomial root‐finding

“…Nowadays, other quaternionic root‐finding algorithms are available that essentially replace the problem of computing the roots of a quaternionic polynomial of degree n , by the problem of determining the roots of a real or complex polynomial of degree 2 n (usually with multiple roots), relying in this way on algorithms for complex polynomial root finding (see De Leo et al, Janovská and Opfer, Serôdio and Siu, and the references therein). Several experiments performed by Falcão and Miranda and Falcão have shown the substantial gain in computational effort that can be achieved when using a direct quaternionic approach to this problem.…”