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

Abstract

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. *

A rational Even-IRA algorithm for the solution of T-even polynomial eigenvalue problems

“…T -even, linearization L P (λ) for P (λ) in the following Theorem 2.5. It is a block minimal bases pencil (as introduced in [9]) and was already used in [12]. In particular, Theorem 3.3 in [9] applies to the matrix pencil L P (λ) defined in (2.5) below and confirms that it is in fact a linearization for P (λ).…”

“…for storing matrix decompositions) can be decreased. These advantages of L P (λ) over other linearizations (see, e.g., [16]) have already been successfully applied in [12] to the Even-IRA algorithm. However, as the Even-IRA algorithm does not allow for changes of the shift ζ during the iteration, this feature is incorporated in our method.…”

“…We present an algorithm for real, T -even polynomial eigenvalue problems of large scale that takes into account all the aforementioned aspects. The method we propose is an implicitly-restarted rational Krylov-Schur approach based on the Even-IRA algorithm introduced in [24] (see also [12]). In contrast to the Even-IRA algorithm and motivated by [3], our approach explicitly allows for changes of the shift parameter during the iteration.…”

“…As outlined in Theorem 2.5, L P (λ) is T -even, so the Even-IRA algorithm can be applied to L P (λ) to determine a part of the finite spectrum of P (λ). In consideration of largescale-problems 2 , the sparsity and structure of L P (λ) can be exploited to significantly increase the computational speed in calculating K(ζ)v. This effective computational approach in explained in detail in this section (see also [12] and [28,Sec. 5.2]).…”