Constructing Strong Linearizations of Matrix Polynomials Expressed in Chebyshev Bases

Lawrence, Piers W.; Pérez, Javier

doi:10.1137/16m105839x

Cited by 18 publications

(39 citation statements)

References 29 publications

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“…Block Kronecker ansatz equations may be defined for other polynomial bases as well (see, e.g., [10] for a clever generalization of block Kronecker pencils for the Chebyshevbasis). Moreover, as we pointed out in Remark 1, the conceptual ideas presented here may even be formulated in the abstract framework of dual bases (i.e.…”

Section: Discussionmentioning

confidence: 99%

Block Kronecker ansatz spaces for matrix polynomials

Faßbender¹,

Saltenberger²

2018

Linear Algebra and its Applications

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Section: Discussionmentioning

confidence: 99%

Block Kronecker ansatz spaces for matrix polynomials

Faßbender¹,

Saltenberger²

2018

Linear Algebra and its Applications

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“…Since any pencil L(λ) of the form (6) or (8) can be uniquely identified with the tuple (v, B) we obtain the isomorphism…”

Section: Generalized Ansatz Spacesmentioning

confidence: 99%

On vector spaces of linearizations for matrix polynomials in orthogonal bases

Faßbender

Saltenberger

2017

Linear Algebra and its Applications

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Regular and singular matrix polynomials P (λ) = k i=0 P i φ i (λ), P i ∈ R n×n given in an orthogonal basis φ 0 (λ), φ 1 (λ), . . . , φ k (λ) are considered. Following the ideas in [9], the vector spaces, called M 1 (P ), M 2 (P ) and DM(P ), of potential linearizations for P (λ) are analyzed. All pencils in M 1 (P ) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M 1 (P ) is a (strong) linearization of P (λ) are given. The equivalence of some of them to the Z-rank-condition [9] is pointed out. Results on the vector space dimensions, the genericity of linearizations in M 1 (P ) and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Moreover, an extension of these results to degree-graded bases is presented. Throughout the paper, structural resemblances between the matrix pencils in L 1 , i.e. the results obtained in [9], and their generalized versions are pointed out.

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“…The Frobenius companion forms are degenerate strong block minimal bases pencils. Moreover, from Theorem 4.2 and Lemma 3.6, they are strong linearizations of [22], the Chebyshev pencils [37], the extended block Kronecker pencils [9], the linearization for product bases in [46], and the pencils in block-Kronecker ansatz spaces [26]. (iii) Fiedler and Fiedler-like pencils.…”

Section: Proof Of Part (B)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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Constructing Strong Linearizations of Matrix Polynomials Expressed in Chebyshev Bases

Cited by 18 publications

References 29 publications

Block Kronecker ansatz spaces for matrix polynomials

Block Kronecker ansatz spaces for matrix polynomials

On vector spaces of linearizations for matrix polynomials in orthogonal bases

Block minimal bases ℓ-ifications of matrix polynomials

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