Block Kronecker ansatz spaces for matrix polynomials

Faßbender, Heike; Saltenberger, Philip

doi:10.1016/j.laa.2017.03.019

Cited by 12 publications

(37 citation statements)

References 13 publications

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“…Moreover, L P (λ) is a (strong) linearization for P (λ) (see [3,Thm. 3.3], [2]). In particular, the spectra σ(P ) and σ(L P ) of P (λ) and L P (λ) coincide and the spectral symmetries are preserved.…”

Section: Tu Braunschweig Universitätsplatz 2 38106 Braunschweigmentioning

confidence: 99%

“…2]). In particular, there existsṽ ∈ R n such that x = Λ(µ)ṽ + w. Backward substitution of this expression into (2) gives…”

Section: Tu Braunschweig Universitätsplatz 2 38106 Braunschweigmentioning

confidence: 99%

“…Equation (3) can now be used to uniquely determineṽ and in turn to compute x = Λ(µ)ṽ +w. As soon as x is known, y can be computed by (2) as the solution of L(−µ) T y = z 1 −M P (µ)x . Certainly, the solution of L(µ) T [ x T y T ] = z can be carried out in a similar manner leading to an analogous equation involving P (µ) T instead of P (µ).…”

Section: Tu Braunschweig Universitätsplatz 2 38106 Braunschweigmentioning

confidence: 99%

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On a modification of the EVEN‐IRA algorithm for the solution of T‐even polynomial eigenvalue problems

Faßbender

Saltenberger

2018

Proc Appl Math and Mech

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We discuss the numerical solution of T -even n × n polynomial eigenvalue problems and show how a small portion of the spectrum can be obtained using just O(n 3 ) arithmetic operations. For that purpose, we apply the EVEN-IRA algorithm proposed in [1] to a special structure-preserving linearization. In this particular situation, the Arnolid iteration as a main part of the EVEN-IRA algorithm can be realized very efficiently.We discuss the numerical solution of polynomial eigenvalue problems given by a T -even matrix polynomial of the formObserve that P (λ) T = P (−λ) holds. The spectrum of P (λ) is symmetric with respect to the real and imaginary axis. Therefore, numerical methods respecting this spectral symmetry are particularly suitable for the eigenvalue computation of T -even eigenvalue problems. To locate some finite eigenvalues of P (λ) as in (1), we first construct a T -even matrix pencil L P (λ) = λX + Y which is a linearization for P (λ) (e.g. see [2, Sec. 2.1]) as follows: assume P (λ) has odd degree d > 2 (for even degree see Section 3) and let = (d + 1)/2. We defineThen the matrix pencil L P (λ) :Moreover, L P (λ) is a (strong) linearization for P (λ) (see [3, Thm. 3.3], [2]). In particular, the spectra σ(P ) and σ(L P ) of P (λ) and L P (λ) coincide and the spectral symmetries are preserved. Now, the EVEN-IRA algorithm from [1] can be efficiently applied to L P (λ) for the computation of some eigenvalues of P (λ) keeping the computational costs within reasonable limits. In fact, L P (λ) never has to be formed explicitly.1 The EVEN-IRA algorithm applied to L P (λ)Assume det(P (λ)) = 0 (i.e. P (λ) is regular, [2, Sec. 2]) and let L P (λ) = λX +Y be the linearization for P (λ) as constructed before. Notice that Y = Y T and X = −X T holds. In [1] the authors introduce the EVEN-IRA algorithm for computing a small portion of the spectrum of a T -even linear matrix polynomial L(λ) = λX + Y . The main ideas of this algorithm applied to L(λ) = L P (λ) are summarized in the following: given some µ ∈ R \ σ(P ), the eigenvalues λ 1 , λ 2 , . . . of L P (λ) (i.e. of P (λ), respectively) are related to the eigenvalues θ 1 , θ 2 , . . . ofIn EVEN-IRA, the Arnoldi iteration applied to K along with implicit Krylov-Schur restarting is proposed to compute some eigenvalues θ 1 , . . . , θ , dn, of K. This way, ±λ j = ± 1/θ j + µ 2 , j = 1, . . . , , will approximate σ(P ) near µ. Matrix-vector products Kv, v ∈ R dn , required by the Arnoldi iteration involve the solution of linear systems of equations with L P (µ) and L P (µ) T . Not taking the special structure of L P (λ) into account, this can be expensive. But, due to the particular form of L P (λ), these can be computed very efficiently by the use of a modified null space method [4, Sec. 6]. The following section presents the main simplifications in that regard. *

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Section: Tu Braunschweig Universitätsplatz 2 38106 Braunschweigmentioning

confidence: 99%

“…2]). In particular, there existsṽ ∈ R n such that x = Λ(µ)ṽ + w. Backward substitution of this expression into (2) gives…”

Section: Tu Braunschweig Universitätsplatz 2 38106 Braunschweigmentioning

confidence: 99%

Section: Tu Braunschweig Universitätsplatz 2 38106 Braunschweigmentioning

confidence: 99%

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On a modification of the EVEN‐IRA algorithm for the solution of T‐even polynomial eigenvalue problems

Faßbender

Saltenberger

2018

Proc Appl Math and Mech

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“…These ideas were further extended in [3]. Inspired by [21], the recent work [14] considers linearizations of rectangular matrix polynomials in a vector space setting. Referred to as Block Kronecker ansatz spaces, these vector spaces contain Block Kronecker linearizations as well as Fiedler linearizations and their extensions modulo permutations and share some of the important properties that L 1 (P) and L 2 (P) have when P(λ ) is square.…”

Section: Introductionmentioning

confidence: 99%

“…Although the pencils in our proposed vector spaces are not linearizations of the non-square polynomial P(λ ) in the conventional sense, we show that almost every such pencil in these spaces can give rise to many linearizations of P(λ ) from which the finite and infinite eigenvalues and corresponding elementary divisors as well as left and right minimal indices and bases of P(λ ) can be easily extracted. We also give the relationship between these linearizations and those in some of the Block Kronecker ansatz spaces in [14], thus showing how g-linearizations and linearizations arising from them, interact with some of the important linearizations for rectangular matrix polynomials in the literature.…”

Section: Introductionmentioning

confidence: 99%

Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

Das

Bora

2019

ELA

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The seminal work [21] introduced vector spaces of matrix pencils, with the property that almost all the pencils in the spaces are strong linearizations of a given square regular matrix polynomial. This work was subsequently extended to include the case of square singular matrix polynomials in [5]. We extend this work to non-square matrix polynomials by proposing similar vector spaces of rectangular matrix pencils that are equal to the ones in [21] when the polynomial is square. Moreover, the properties of these vector spaces are similar to those in [5] for the singular case. In particular, the complete eigenvalue problem associated with the matrix polynomial can be solved by using almost every matrix pencil from these spaces. Further, almost every pencil in these spaces can be 'trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily solve the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in [7]. Further, the global backward error analysis in [10] applied to these linearizations, shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

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