Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems

Dopico, Froilán M.; Marcaida, Silvia; Quintana, María C.; Dooren, Paul Van

doi:10.1016/j.laa.2020.07.004

Cited by 18 publications

(27 citation statements)

References 30 publications

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“…By exhibiting a rational unimodular equivalence, the paper [1] showed that the general form of this was at least a local linearization for matrix polynomials ( ), and also demonstrated that this was true for the reversal as well, showing that it was a strong (local) linearization. They also gave indirect arguments, equivalent to the notion of patched local linearizations introduced in [6], showing that the construction gave a genuine strong linearization. Here, we wish to see if we can explicitly construct unimodular matrix polynomials ( ) and ( ) which show equivalence, directly demonstrating that this is a linearization.…”

Section: Lagrange Interpolational Basesmentioning

confidence: 97%

“…The paper [6] introduces a new notion in their Definition 2, that of a local linearization of rational matrix functions: this definition allows ( ) and ( ) to be rational unimodular transformations, and defines a local linearization only on a subset Σ of F. See also [14] which extended the matrix polynomial theory to use local linearization. Specifically, by this definition, two rational matrices 1 ( ) and 2 ( ) are locally equivalent if there exist rational matrices ( ) and ( ), invertible for all ∈ Σ, such that 1 ( ) = ( ) 2 ( ) ( ) for all ∈ Σ.…”

Section: Contributed Papermentioning

confidence: 99%

“…As in [1] we may use factoring to show that is also equivalent to at the nodes, essentially using Proposition 2.1 of [6]. We also have a purely algebraic proof based on local equivalence of Smith forms, not shown here.…”

Section: Hermite Form Of the Linearizationmentioning

confidence: 99%

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Equivalences for Linearizations of Matrix Polynomials

Corless

Sevyeri

Saunders

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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One useful standard method to compute eigenvalues of matrix polynomials ( ) ∈ C × (︀ ⌋︀ of degree at most ℓ in (denoted of grade ℓ, for short) is to first transform ( ) to an equivalent linear matrix polynomial ( ) = − , called a companion pencil, where and are usually of larger dimension than ( ) but ( ) is now only of grade 1 in . The eigenvalues and eigenvectors of ( ) can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of ( ), including those for infinite eigenvalues, can also be recovered from eigenvectors of ( ) if ( ) is what is called a "strong linearization" of ( ). In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors ( ) and ( ) which, via the equation ( ) ( ) ( ) = diag( ( ), , . . . , ), explicitly show the equivalence of ( ) and ( ). By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.

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Section: Lagrange Interpolational Basesmentioning

confidence: 97%

Section: Contributed Papermentioning

confidence: 99%

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Equivalences for Linearizations of Matrix Polynomials

Corless

Sevyeri

Saunders

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…In Section 1.3, we show how to use a generalized standard triple in the construction of algebraic linearizations. We also give a proof, by construction of the necessary equivalence matrices E(z) and F (z), that algebraic linearizations are local linearizations (a very useful notion from [13]) in the same sense that its components are. In particular, if the components A and B are local linearizations in the sets Σ A and Σ B , respectively, then the 'algebraic linearization' is a local linearization in the intersection…”

Section: Introductionmentioning

confidence: 99%

“…In the paper [13], we find the powerful notion of a local linearization. In the notation of their Definition 2.1, two rational matrices G 1 (z) and G 2 (z) are said to be equivalent in a nonempty set Σ ⊂ F if there exist rational matrices R 1 (z) and R 2 (z) each nonsingular in…”

Section: Introductionmentioning

confidence: 99%

Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

Chan

Corless

Sevyeri

2021

ELA

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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.

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