Block Kronecker linearizations of matrix polynomials and their backward errors

Dopico, Froilán M.; Lawrence, Piers W.; Pérez, Javier; Dooren, Paul Van

doi:10.1007/s00211-018-0969-z

Cited by 71 publications

(298 citation statements)

References 71 publications

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“…This artificially turns P (λ) into a T -even matrix polynomial P (λ) = d+1 j=0 P j λ j of grade d + 1. Now d + 1 is odd and the T -even matrix pencil L P (λ) will still be a linearization for P (λ) and has the same finite spectrum as P (λ) [3]. In this situation, all simplifications presented in Section 2 equally apply to L P (λ).…”

Section: Numerical Results and Extension Of The Algorithmmentioning

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: Numerical Results and Extension Of The Algorithmmentioning

confidence: 99%

“…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]).…”

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confidence: 99%

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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“…As we have demonstrated numerically in this paper, this is connected to a graded structure in the backward error generated by the QZ algorithm. Currently a lot of research is done proving the backward stability of applying the QZ algorithm to certain types of linearizations, e.g., [8]. However, because only the well-known backward error properties of the QZ algorithm are used, the backward error on the matrix polynomial is one where all the coefficients are taken together normwise, without taking into account the possible difference in norm of each of the coefficients separately.…”

Section: Discussionmentioning

confidence: 99%

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Barel

Tisseur

2018

Linear Algebra and its Applications

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We propose a novel approach to solve polynomial eigenvalue problems via linearization. The novelty lies in (a) our choice of linearization, which is constructed using input from tropical algebra and the notion of well-separated tropical roots, (b) an appropriate scaling applied to the linearization and (c) a modified stopping criterion for the QZ iterations that takes advantage of the properties of our scaled linearization. Numerical experiments show that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even when they are very different in magnitude.

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“…Quoting Nick Higham [17, p. 56], "it is the relative condition number that is of interest, but it is more convenient to state results for the absolute condition number". The relative with respect to the norm of P eigenvalue condition number corresponds to perturbations in the coefficients of P coming from the backward errors of solving PEPs by applying a backward stable generalized eigenvalue algorithm to any reasonable linearization of P [13,35]. Observe that κ p θ ((α 0 , β 0 ), P ) = P ∞ κ a θ ((α 0 , β 0 ), P ) and, therefore, one of these condition numbers can be easily computed from the other.…”

Section: Properties Of Möbius Transformationsmentioning

confidence: 99%

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

2019

Self Cite

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Möbius transformations have been used in numerical algorithms for computing eigenvalues and invariant subspaces of structured generalized and polynomial eigenvalue problems (PEPs). These transformations convert problems with certain structures arising in applications into problems with other structures and whose eigenvalues and invariant subspaces are easily related to the ones of the original problem. Thus, an algorithm that is efficient and stable for some particular structure can be used for solving efficiently another type of structured problem via an adequate Möbius transformation. A key question in this context is whether these transformations may change significantly the conditioning of the problem and the backward errors of the computed solutions, since, in that case, their use might lead to unreliable results. We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of PEPs. By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that, if the matrix inducing the Möbius transformation is well conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the whole polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the Möbius transformations induced by well-conditioned matrices only if a penalty factor, depending on the norms of those matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP is used.

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Block Kronecker linearizations of matrix polynomials and their backward errors

Cited by 71 publications

References 71 publications

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

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

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

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