Fiedler-comrade and Fiedler--Chebyshev pencils

Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulas for eigenvectors, and minimal indices and bases

On the stability of computing polynomial roots via confederate linearizations

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Based on the analysis we suggest how to choose between QR and QZ. Finally, we focus on the special case of the Chebyshev basis and finding real roots of a general function on an interval, and discuss how to compute accurate roots. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary

A class of quasi-sparse companion pencils

In this paper, we introduce a general class of quasi-sparse potential companion pencils for arbitrary square matrix polynomials over an arbitrary field, which extends the class introduced in [B. Eastman, I.-J. Kim, B. L. Shader, K.N. Vander Meulen, Companion matrix patterns. Linear Algebra Appl. 436 (2014) 255-272] for monic scalar polynomials. We provide a canonical form, up to permutation, for companion pencils in this class. We also relate these companion pencils with other relevant families of companion linearizations known so far. Finally, we determine the number of different sparse companion pencils in the class, up to permutation.This work has been partially supported by theMinisterio de Economía y Competitividad of Spain through grants MTM2015-68805-REDT and MTM2015-65798-P

A note on generalized companion pencils in the monomial basis

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Terán et al. (Linear Algebra Appl 459:264&-333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360&-365, 2016), and Ma&-Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621&-625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Terán and Hernando (INdAM Series, Springer, Berlin, pp 157&-179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencilThis work has been partially supported by the Ministerio de Economía y Competitividad of Spain through Grants MTM2017-90682-REDT and MTM2015-65798-P

Fiedler matrices: numerical and structural properties

The first and second Frobenius companion matrices appear frequently in numerical application, but it is well known that they possess many properties that are undesirable numerically, which limit their use in applications. Fiedler companion matrices, or Fiedler matrices for brevity, introduced in 2003, is a family of matrices which includes the two Frobenius matrices. The main goal of this work is to study whether or not Fiedler companion matrices can be used with more reliability than the Frobenius ones in the numerical applications where Frobenius matrices are used. For this reason, in this work we present a thorough study of Fiedler matrices: their structure and numerical properties, where we mean by numerical properties those properties that are interesting for applying these matrices in numerical computations, and some of their applications in the field on numerical linear algebra. The introduction of Fiedler companion matrices is an example of a simple idea that has been very influential in the development of several lines of research in the numerical linear algebra field. This family of matrices has important connections with a number of topics of current interest, including: polynomial root finding algorithms, linearizations of matrix polynomials, unitary Hessenberg matrices, CMV matrices, Green’s matrices, orthogonal polynomials, rank structured matrices, quasiseparable and semiseparable matrices, etc.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Paul Van Dooren.- Secretario: Juan Bernardo Zaballa Tejada.- Vocal: Françoise Tisseu

Algebraic Companions and Linearizations

In this thesis, we look at a novel way of finding roots of a scalar polynomial using eigenvalue techniques. We extended this novel method to the polynomial eigenvalue problem (PEP). PEP have been used in many science and engineering applications such vibrations of structures, computer-aided geometric design, robotics, and machine learning. This thesis explains this idea in the order of which we discovered it. In Chapter 2, a new kind of companion matrix is introduced for scalar polynomials of the form $c(\lambda) = \lambda a(\lambda)b(\lambda)+c_0$, where upper Hessenberg companions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence\u27s Mandelbrot companion matrix. The construction was motivated by use of Narayana-Mandelbrot polynomials. In Chapter 3, we define Euclid polynomials $E_{k+1}(\lambda) = E_{k} (\lambda) (E_{k} (\lambda) - 1) + 1$ where $E_{1}(\lambda) = \lambda + 1$ in analogy to Euclid numbers $e_k = E_{k} (1)$. We show how to construct companion matrices $E_{k}$, so $E_{k} (\lambda) = \det(\lambda I - E_{k} )$ is of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties. In Chapter 4, we show how to construct linearizations of matrix polynomials z\mat{a}(z)\mat{d}_0 + \mat{c}_0, \mat{a}(z)\mat{b}(z), \mat{a}(z) + \mat{b}(z) (when \deg(\mat{b}(z)) \u3c \deg(\mat{a}(z))), and z\mat{a}(z)\mat{d}_0\mat{b}(z) + \mat{c}_0 from linearizations of the component parts, matrix polynomials \mat{a}(z) and \mat{b}(z). This extends the new companion matrix construction introduced in Chapter 2 to matrix polynomials. In Chapter 5, we define ``generalized standard triples\u27\u27 which can be used in constructing algebraic linearizations; for example, for \H(z) = z \mat{a}(z)\mat{b}(z) + \mat{c}_0 from linearizations for \mat{a}(z) and \mat{b}(z). 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. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. In Chapter 6, we investigate the numerical stability of algebraic linearization, which re-uses linearizations of matrix polynomials \mat{a}(\lambda) and \mat{b}(\lambda) to make a linearization for the matrix polynomial \mat{P}(\lambda) = \lambda \mat{a}(\lambda)\mat{b}(\lambda) + \mat{c}. Such a re-use \textsl{seems} more likely to produce a well-conditioned linearization, and thus the implied algorithm for finding the eigenvalues of \mat{P}(\lambda) seems likely to be more numerically stable than expanding out the product \mat{a}(\lambda)\mat{b}(\lambda) (in whatever polynomial basis one is using). We investigate this question experimentally by using pseudospectra