A Modified Quaternionic Weierstrass Method

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].…”

“…Our next example concerns a polynomial P with a spherical zero, i.e., P does not fulfill the assumptions of Theorem 1. [i] Deflation Spherical 0 -Recently, we have proposed a deflation algorithm [9] to be used together with the quaternionic Weierstrass method which allows to obtain quadratic order of convergence for isolated and spherical roots without requiring higher order precision. The same technique can be used now for the two-step method.…”

“…More recently [9], an approach combining a deflation procedure with the Weierstrass method allowed to obtain approximations also to the non-isolated zeros of P . In this work, we suggest an improvement of the quaternionic Weierstrass method, by using a two-step strategy and prove that this new procedure has cubic order of convergence.…”

“…4 we illustrate the performance of the method by considering some examples and computing the corresponding computational order of convergence. We also apply, in Examples 3 and 4, the technique described in [9], to obtain successfully both the isolated and non-isolated zeros of the polynomial under consideration. The paper ends with some remarks and conclusions.…”