Efficient Ehrlich–Aberth iteration for finding intersections of interpolating polynomials and rational functions

Robol, Leonardo; Vandebril, Raf

doi:10.1016/j.laa.2017.05.010

Cited by 4 publications

(2 citation statements)

References 19 publications

(51 reference statements)

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“…The idea of transforming the rational problem (35) into an eigenvalue problem is not new [21]. An algorithm based on the Ehrlich-Aberth iteration that uses this approach can be found in [20].…”

Section: Application To Scalar Rational Equationsmentioning

confidence: 99%

Linearizations of rational matrices from general representations

Pérez¹,

Quintana²

2020

Preprint

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We construct a new family of linearizations of rational matrices R(λ) written in the general form R(λ) = D(λ) + C(λ)A(λ) −1 B(λ), where D(λ), C(λ), B(λ) and A(λ) are polynomial matrices. Such representation always exists and are not unique. The new linearizations are constructed from linearizations of the polynomial matrices D(λ) and A(λ), where each of them can be represented in terms of any polynomial basis. In addition, we show how to recover eigenvectors, when R(λ) is regular, and minimal bases and minimal indices, when R(λ) is singular, from those of their linearizations in this family.

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Section: Application To Scalar Rational Equationsmentioning

confidence: 99%

Linearizations of rational matrices from general representations

Pérez¹,

Quintana²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…There are many (in fact, infinitely many) choices available in the literature for constructing strong linearizations and strong ℓ-ifications of matrix polynomials. From a numerical analyst point of view, this situation is very desirable, since one can choose the most favorable construction in terms of various criteria, such as conditioning and backward errors [31,32], the basis in which the polynomial is represented [1], preservation of algebraic structures [33,41], exploitation of matrix structures in numerical algoritghms [38,49,47], etc However, there has not been a framework providing a way to construct and analyze all these strong linearizations and strong ℓ-ifications in a consistent manner. Providing such a framework is one of the main goals of this work.…”

mentioning

confidence: 99%

Block minimal bases ℓ-ifications of matrix polynomials

Dopico

Pérez

Dooren

2019

Linear Algebra and its Applications

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The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong ℓ-ification of a matrix polynomial, which is a matrix polynomial of degree ℓ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong ℓ-ifications of matrix polynomials of size m × n and grade d when ℓ < d, and ℓ divides nd or md. This method is based on a family called "strong block minimal bases matrix polynomials", and relies heavily on properties of dual minimal bases. We show how strong block minimal bases ℓ-ifications can be constructed from the coefficients of a given matrix polynomial P (λ). We also show that these ℓ-ifications satisfy many desirable properties for numerical applications: they are strong ℓ-ifications regardless of whether P (λ) is regular or singular, the minimal indices of the ℓ-ifications are related to those of P (λ) via constant uniform shifts, and eigenvectors and minimal bases of P (λ) can be recovered from those of any of the strong block minimal bases ℓ-ifications. In the special case where ℓ divides d, we introduce a subfamily of strong block minimal bases matrix polynomials named "block Kronecker matrix polynomials", which is shown to be a fruitful source of companion ℓ-ifications.

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A simplified approach to Fiedler-like pencils via block minimal bases pencils

Bueno

Dopico

Pérez

et al. 2018

Linear Algebra and its Applications

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The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix polynomial into a matrix pencil, transforming the problem into an equivalent generalized eigenvalue problem. Such pencils are known as linearizations. Many of the families of linearizations for matrix polynomials available in the literature are extensions of the so-called family of Fiedler pencils. These families are known as generalized Fiedler pencils, Fiedler pencils with repetition and generalized Fiedler pencils with repetition, or Fiedler-like pencils for simplicity. The goal of this work is to unify the Fiedler-like pencils approach with the more recent one based on strong block minimal bases pencils introduced in [15]. To this end, we introduce a family of pencils that we have named extended block Kronecker pencils, whose members are, under some generic nonsingularity conditions, strong block minimal bases pencils, and show that, with the exception of the non proper generalized Fiedler pencils, all Fiedler-like pencils belong to this family modulo permutations. As a consequence of this result, we obtain a much simpler theory for Fiedler-like pencils than the one available so far. Moreover, we expect this unification to allow for further developments in the theory of Fiedler-like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via Fiedler-like linearizations.Key words. Fiedler pencils, generalized Fiedler pencils, Fiedler pencils with repetition, generalized Fiedler pencils with repetition, matrix polynomials, strong linearizations, block minimal bases pencils, block Kronecker pencils, extended block Kronecker pencils, minimal basis, dual minimal bases AMS subject classifications. 65F15, 15A18, 15A22, 15A54 1. Introduction. Matrix polynomials and their associated polynomial eigenvalue problems appear in many areas of applied mathematics, and they have received in the last years considerable attention. For example, they are ubiquitous in a wide range of problems in engineering, mechanic, control theory, computer-aided graphic design, etc. For detailed discussions of different applications of matrix polynomials, we refer the reader to the classical references [23,28,42], the modern surveys [2, Chapter 12] and [37,43] (and their references therein), and the references [32,33,34]. For those readers not familiar with the theory of matrix polynomials and polynomial eigenvalue problems, those topics are briefly reviewed in Section 2.The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to linearize the polynomial into a matrix pencil (i.e., matrix polynomials of grade 1), known as linearization [13,22,23]. The linearization process transforms the polynomial eigenvalue problem into an equivalent generalized eigenvalue problem, which, then, can be solved using mature and well-understood eigensolvers such as the QZ algorithm or the

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Efficient Ehrlich–Aberth iteration for finding intersections of interpolating polynomials and rational functions

Cited by 4 publications

References 19 publications

Linearizations of rational matrices from general representations

Linearizations of rational matrices from general representations

Block minimal bases ℓ-ifications of matrix polynomials

A simplified approach to Fiedler-like pencils via block minimal bases pencils

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