Strong Linearizations of Rational Matrices

Amparan, A.; Dopico, Froilán M.; Marcaida, Silvia; Zaballa, Ion

doi:10.1137/16m1099510

Cited by 24 publications

(77 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this subsection, we present some results and notations related to the REP. Interested readers can find more information in the summaries presented in sections 1 and 2 in the work of Alam and Behera 26 and section 2 in the work of Amparan et al 3 as well as in the classical works of Kailath 1 and Rosenbrock. 2 In this work, we assume that the rational matrix R( ) in (2) is regular, that is, det(R( )) ≢ 0.…”

Section: A Linearization For Repsmentioning

confidence: 99%

“…A formal definition of linearization of a rational matrix can be found in the work of Alam and Behera 26 and another one that includes the concept of strong linearization in the work of Amparan et al 3 In fact, it is proved in the aforementioned work 3 2 of the strictly proper part of R( ) (see section 8 in the work of Amparan et al 3 ). We emphasize that the requirement that −E(C − D) −1 F T is a minimal state-space realization is very mild (see section 8 in the work of Amparan et al 3 ) and that it is fully necessary to guarantee that for every eigenvalue of R( ), the matrix C − D is nonsingular (see example 3.2 in the work of Amparan et al 3 ). In the rest of this paper, we implicitly assume that E(C − D) −1 F T is a minimal realization, although the only result we will use explicitly is Theorem 3, which remains valid even when E(C − D) −1 F T is not minimal.…”

Section: A Linearization For Repsmentioning

confidence: 99%

“…The reason for this restriction is that the theory of linearizations of REPs is far less developed than the theory of linearizations of PEPs. Thus, although many linearizations of REPs have been introduced very recently in the works of Amparan et al 3 and Alam and Behera, 26 their properties are not yet fully understood.…”

Section: Theorem 3 (See Theorem 31 In the Work Of Su And Bai 4 )mentioning

confidence: 99%

“…Since many linearizations of rational matrices different from (9) have been developed very recently 3,26 and some of them include the option of considering that the matrix polynomial P( ) in (8) is expressed in nonmonomial bases, an interesting…”

Section: Conclusion and Lines Of Future Researchmentioning

confidence: 99%

“…where P( ) ∈ ℂ[ ] n×n is a matrix polynomial of degree d in the variable ; f i ( ) and g i ( ) are coprime scalar polynomials of degrees m i and n i , respectively; and m i < n i and E i ∈ ℂ n×n are constant matrices for i = 1, … , k. We emphasize that it is well known that every rational matrix can be written in the form given in (2) 1,2 (see also section 2 in the work of Amparan et al 3 ) and that such form appears naturally in many applications. 4 The REP has attracted considerable interest in recent years since it arises in different applications in some fields such as vibration of fluid-solid structures, 5 optimization of acoustic emissions of high-speed trains, 6 free vibration of plates with elastically attached masses, 7 free vibrations of a structure with a viscoelastic constitutive relation describing the behavior of a material, 8,9 and electronic structure calculations of quantum dots.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A compact rational Krylov method for large‐scale rational eigenvalue problems

Dopico

González-Pizarro

2018

Numerical Linear Algebra App

Self Cite

View full text Add to dashboard Cite

In this work, we propose a new method, termed as R-CORK, for the numerical solution of large-scale rational eigenvalue problems, which is based on a linearization and on a compact decomposition of the rational Krylov subspaces corresponding to this linearization. R-CORK is an extension of the compact rational Krylov method (CORK) introduced very recently in the literature to solve a family of nonlinear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R-CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size as the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R-CORK is more efficient from the point of view of memory and orthogonalization than the classical rational Krylov method applied directly to the linearization. Taking into account that the R-CORK method is based on a classical rational Krylov method, to implement implicit restarting is also possible, and we show how to do it in a memory-efficient way. Finally, some numerical examples are included in order to show that the R-CORK method performs satisfactorily in practice. KEYWORDSlarge-scale, linearization, rational eigenvalue problem, rational Krylov method where R( ) ∈ ℂ( ) n×n is a nonsingular rational matrix, that is, the entries of R( ) are scalar rational functions in the variable with complex coefficients and det(R( )) ≢ 0 is not identically zero, and x ∈ ℂ n is a nonzero vector. More Numer Linear Algebra Appl. 2018;26:e2214.wileyonlinelibrary.com/journal/nla † We remark that Su and Bai 4 considered the permutations of the matrices  and  in (10) for the linearized problem (9). More precisely, they ordered the matrices P i in reverse order and interchanged the identity blocks in  and . Since we are following, in this paper, the spirit of CORK, then  and  are written in (10) in the same way as in table 1 in the work of Van Beeumen et al. 20

show abstract

Section: A Linearization For Repsmentioning

confidence: 99%