Interpolation Polynomials and Linear Algebra

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators ...

We give a survey of the analytic theory of matrix orthogonal polynomials.

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reord...

For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=\delta_1 X_1^{k_1} + \delta_2 X_2^{k_2} + \cdots +\delta_m X_m^{k_m}$ in terms of the solution of certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$, the diagonal map is surjective when (a) $\mathbb K= \mathbb C$, (b) $\mathbb K= \mathbb F_q$...

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $n\times n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor resolution of linear systems of differential equations, nor resolution of linear systems with constant coefficients, nor matrix inversion, nor complex integration, nor functional analysis. The basic tools are power series and the method of pa...

One strategy to solve a nonlinear eigenvalue problem $T(\lambda)x=0$ is to solve a polynomial eigenvalue problem (PEP) $P(\lambda)x=0$ that approximates the original problem through interpolation. Then, this PEP is usually solved by linearization. Most of the literature about linearizations assumes that $P(\lambda)$ is expressed in the monomial basis but, because of the polynomial approximation techniques, in this context, $P(\lambda)$ is expressed in a non-monomial basis. Th...

An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to be comparable in operation count and convergence with the state--of--the--art method which is based on a Pade approximation of the exponential matrix function. The present polynomial form, however, is more reliable because the evaluation req...

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in non-monomial bases and allows to represent polynomials expressed in product families, that is as a linear combination of elements of the form $\phi_i(\lambda)...

We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a B\'{e}zout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our expositi...

In this paper some algorithms will be presented which can be used for the calculation of zeros of polynomials and eigenvalues of polynomial matrices with a multiplicity larger than one. The numerical values calculated with MATLAB are used as starting values. The reliability of the algorithms is demonstrated by means of 8 examples.