Similarly, if |λ|<1,we have
ηS2(λ,Λm−1⊗x,L;v)≤
√2kΛmk2
kΛm−1k2
s
τ2δ−2v + 1
k(1, λ)k42η(λ, x,P).
Hence the result follows by (5.2). The proof is similar when L is an H-anti-palindromic linearization. Hence the proof.¥
The morale of the Theorem 5.4.2 is as follows. For|λ|= 1,it does not really make any difference in the magnified backward error due to eitherH-palindromic orH-anti-palindromic linearization and hence any of H-palindromic or H-anti-palindromic linearization can be chosen. For|λ| 6= 1,two cases arise. Let us defineτ :=τpforH-palindromic linearization and τ :=τa forH-anti-palindromic linearization. Then for a given (λ, x) ifτa ≤τa then choose H-palindromic linearization else chooseH-anti-palindromic linearization.
µu= (λmP(λ))Hu=µu,we have P(λ)u+4P(λ)u=λmµ u−λmµu= 0.Further, we have
|||4P|||= σmin(P(λ)) kΛmk2
=η(λ,P).
Hence the results follow. ¥
structuredeigenvaluechoiceofβS :=ηS (λ,x,L)/η(λ,Λm−1⊗x,P)βS :=ηS (λ,x,L)/η(λ,Λm−1⊗x,P) polynomialofinterestlinearization|||·|||≡|||·|||F|||·|||≡|||·|||2 T-palindromic/re(λ)≥0T-palindromicq 1−2re(λ)|1+λ|−2δ−2 v≤αS ≤√ 21√ 2≤βS ≤√ 2q 1+|1+λ|−2δ−2 v T-anti-palindromicre(λ)≤0T-anti-palindromicq 1+2re(λ)|1−λ|−2δ−2 v≤αS≤√ 21√ 2≤βS≤√ 2q 1+|1−λ|−2δ−2 v H-palindromic/|λ|=1H-palindromic/√ 2−δ−2 v√ 2≤βS ≤√ 21√ 2≤βS ≤1 H-anti-palindromicH-anti-palindromic |λ|6=1H-palindromic1√ 2≤βS≤2q 1+τ2 pδ−2 vk(1,λ)k2 21√ 2≤βS≤2q k(1,λ)k−2 2+τ2 pδ−2 vk(1,λ)k2 2if|λ|<1 1√ 2≤βS ≤2q k(1,λ−1)k−2 2+τ2 pδ−2 vk(1,λ)k2 2if|λ|>1 H-anti-palindromic1√ 2≤βS≤2q 1+τ2 aδ−2 vk(1,λ)k2 21√ 2≤βS≤2q k(1,λ)k−2 2+τ2 aδ−2 vk(1,λ)k2 2if|λ|<1 1√ 2≤βS ≤2q k(1,λ−1)k−2 2+τ2 aδ−2 vk(1,λ)k2 2if|λ|>1 Table5.4:Thechoiceofstructuredlinearizationsforpalindromicpolynomials,whereδv=kΛm−1k2/|Λ∗ m−1v|,∗∈{T,H}andvistheansatz vector. Hereτp:=
° ° ° ° °·¸† 1+re(λ)im(λ) im(λ)1−re(λ)
° ° ° ° ° 2 andτa:=
° ° ° ° °·¸† 1−re(λ)−im(λ) −im(λ)1+re(λ)
° ° ° ° ° 2
.
Chapter 6
Structured eigenvalue condition numbers and linearizations for matrix polynomials
This chapter is concerned with the sensitivity analysis of eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. As mentioned before, numerical methods for solving such eigenvalue problem proceed by linearizing the matrix polynomial into a matrix pencil of larger size. A question of practical importance is whether this process of linearization increases the sensitivity of the eigenvalue with respect to structured perturbations. For all structures under consideration, we show that this is not the case: there is always a linearization for which the structured condition number of an eigenvalue does not differ significantly. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.
6.1 Introduction
Consider an n×nmatrix polynomial
P(λ) =A0+λA1+λ2A2+· · ·+λmAm, (6.1) with A0, . . . , Am∈Cn×n. An eigenvalue λ∈Cof P, defined by the relation det(P(λ)) = 0, is called simple ifλis a simple root of the polynomial det(P(λ)).
This chapter is concerned with the sensitivity of a simple eigenvalueλunder perturbations of the coefficients Aj. The condition number of λis a first-order measure for the worst-case effect of perturbations onλ. Tisseur [93] has provided an explicit expressions for this condition number. Subsequently, this expression was extended to polynomials in homogeneous form by Dedieu and Tisseur [26], see also [1, 14, 21], and to semi-simple eigenvalues in [54]. In the more general context of nonlinear eigenvalue problems, the sensitivity of eigenvalues and eigenvectors has been investigated in, e.g., [4, 58, 60, 61]. We consider the same classes of structured polynomials as discussed in Table 1.2.
In certain situations, it is reasonable to expect that perturbations of the polynomial re- spect the underlying structure. For example, if a strongly backward stable eigenvalue solver is applied to a palindromic matrix polynomial then the computed eigenvalues would be the exact eigenvalues of a slightly perturbed palindromic eigenvalue problems. Also, structure- preserving perturbations are physically more meaningful in the sense that the spectral sym- metries induced by the structure are not destroyed. Restricting the admissible perturbations might have a positive effect on the sensitivity of an eigenvalue. This question has been studied for linear eigenvalue problems in quite some detail recently [19, 38, 49, 50, 51, 54, 73, 78, 81].
It often turns out that the desirable positive effect is not very remarkable: in many cases the worst-case eigenvalue sensitivity changes little or not at all when imposing structure. No- table exceptions can be found among symplectic, skew-symmetric, and palindromic eigenvalue problems [51, 54]. In the first part of this chapter, we will extend these results to structured matrix polynomials.
Due to the lack of a robust genuine polynomial eigenvalue solver, the eigenvalues of P are usually computed by first reformulating (6.1) as an mn×mn linear generalized eigen- value problem and then applying a standard method such as the QZ algorithm [34] to the linear problem. This process of linearization introduces unwanted effects. Besides the obvi- ous increase of dimension, it may also happen that the eigenvalue sensitivities significantly deteriorate. Fortunately, one can use the freedom in the choice of linearization to minimize this deterioration for the eigenvalue region of interest, as proposed for quadratic eigenvalue problems in [29, 42, 93]. For the general polynomial eigenvalue problem (6.1), Higham et al. [39, 41] have identified linearizations with minimal eigenvalue condition number/backward error among the set of linearizations described in [67]. For structured polynomial eigenvalue problem, rather than usingany linearization it is of course advisable to use one which has a similar structure. For example, it was shown in [68] that a palindromic matrix polynomial can usually be linearized into a palindromic or anti-palindromic matrix pencil, offering the possibility to apply structure-preserving algorithms to the linearization. It is natural to ask whether there is also a structured linearization that has no adverse effect on the structured condition number. For a small subset of structures from Table 1.2, this question has already been discussed in [42]. In the second part of this chapter, we extend the discussion to all structures from Table 1.2.
The rest of this chapter is organized as follows. In Section 6.2, we first review the derivation of the unstructured eigenvalue condition number for a matrix polynomial and then provide explicit expressions for structured eigenvalue conditions numbers. Most but not all of these expressions are generalizations of known results for linear eigenvalue problems. In Section 6.4, we apply these results to find good choices from the set of structured linearizations described in [68].