2 citations · 3 across the 6 of their papers we have counts for
10 papers
Root finding techniques that work
Aaron Melman
In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or…
An octagon containing the numerical range of a bounded linear operator
Aaron Melman
A polygon is derived that contains the numerical range of a bounded linear operator on a complex Hilbert space, using only norms. In its most general form, the polygon is an octago…
A unifying framework for generalizations of the Enestrom-Kakeya theorem
Aaron Melman
The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polynomial with positive coefficients that are explicit functions of those coefficients.…
Improved Cauchy radius for scalar and matrix polynomials
A. Melman
We improve the Cauchy radius of both scalar and matrix polynomials, which is an upper bound on the moduli of the zeros and eigenvalues, respectively, by using appropriate polynomia…
Polynomial eigenvalue bounds from companion forms
Aaron Melman
We show how -ifications, which are companion forms of matrix polynomials, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix po…
Eigenvalue bounds for matrix polynomials in generalized bases
Aaron Melman
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bound…
Cauchy-like and Pellet-like results for polynomials
Aaron Melman
We obtain several Cauchy-like and Pellet-like results for the zeros of a general complex polynomial by considering similarity transformations of the squared companion matrix and th…
Geometric aspects of Pellet's and related theorems
Aaron Melman
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, according to their moduli. We refine one of those regions and replace it with the clos…
Implementation of Pellet's theorem
Aaron Melman
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but doe…
Generalization and variations of Pellet's theorem for matrix polynomials
Aaron Melman
We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouché theorem for matrix-valued functions, to generate upper, lower, and internal bounds…