On Lommel Matrix Polynomials

The main aim of this paper is to introduce a new class of Lommel matrix polynomials with the help of hypergeometric matrix function within complex analysis. We derive several properties such as an entire function, order, type, matrix recurrence relations, differential equation and integral representations for Lommel matrix polynomials and discuss its various special cases. Finally, we establish an entire function, order, type, explicit representation and several properties of modified Lommel matrix polynomials. There are also several unique examples of our comprehensive results constructed.

Bounds for the zeros of unilateral octonionic polynomials

In the present work it is proved that the zeros of a unilateral octonionic polynomial belong to the conjugacy classes of the latent roots of an appropriate lambda-matrix. This allows the use of matricial norms, and matrix norms in particular, to obtain upper and lower bounds for the zeros of unilateral octonionic polynomials. Some results valid for complex and/or matrix polynomials are extended to octonionic polynomials.

Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases.

The limit empirical spectral distribution of complex matrix polynomials

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas for Fourier cosine and sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms several results are obtained, which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials

An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.

Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.

Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus

In this paper, we introduce a matrix version of the generalized heat polynomials. Some analytic properties of the generalized heat matrix polynomials are obtained including generating matrix functions, finite sums, and Laplace integral transforms. In addition, further properties are investigated using fractional calculus operators.