## Backward error of polynomial eigenproblems solved by linearization (2006)

Venue: | Manchester Institute for Mathematical Sciences, The University of Manchester |

Citations: | 25 - 7 self |

### BibTeX

@TECHREPORT{Higham06backwarderror,

author = {Nicholas J. Higham and Ren-cang Li and Françoise Tisseur and Mims Eprint and Nicholas J. Higham and Ren-cang Li and Françoise Tisseur},

title = {Backward error of polynomial eigenproblems solved by linearization},

institution = {Manchester Institute for Mathematical Sciences, The University of Manchester},

year = {2006}

}

### OpenURL

### Abstract

Abstract. The most widely used approach for solving the polynomial eigenvalue problem P(λ)x = ��m i=0 λi � Ai x =0inn × n matrices Ai is to linearize to produce a larger order pencil L(λ) =λX + Y, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P, infinitely many linearizations L exist and approximate eigenpairs of P computed via linearization can have widely varying backward errors. We show that if a certain one-sided factorization relating L to P can be found then a simple formula permits recovery of right eigenvectors of P from those of L, and the backward error of an approximate eigenpair of P can be bounded in terms of the backward error for the corresponding approximate eigenpair of L. A similar factorization has the same implications for left eigenvectors. We use this technique to derive backward error bounds depending only on the norms of the Ai for the companion pencils and for the vector space DL(P) of pencils recently identified by Mackey, Mackey, Mehl, and Mehrmann. In all cases, sufficient conditions are identified for an optimal backward error for P. These results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of P. Other contributions of this work are a block scaling of the companion pencils

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Citation Context ...nvalue with the choice x = z2. This is confirmed by the boldface entries in the last two columns of Table 6.2. Our second problem is a simplified model of a nuclear power plant, as described in [12], =-=[23]-=-. The largest ratios η Q (x, α, β)/ηL(z,α,b) and corresponding upper bounds are displayed in Table 6.3. Similar conclusions to those for the wave problem can be drawn for this problem. Since ρ =7× 10 ... |

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Citation Context ...icle is prohibited.sBACKWARD ERROR OF POLYNOMIAL EIGENPROBLEMS 1219 with the same spectrum as P and solve the eigenproblem for L. This generalized eigenproblem is usually solved with the QZ algorithm =-=[20]-=- for small to medium size problems or a projection method for large sparse problems [1]. That L has the same spectrum as P is assured if � � P(λ) 0 (1.2) E(λ)L(λ)F(λ)= 0 I (m−1)n for some unimodular E... |

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Citation Context ... meanings; in particular, it can refer to the relative error of the eigenvalue or the backward error of the eigenpair. The relative error question has been investigated by Higham, Mackey, and Tisseur =-=[7]-=-, by analyzing the conditioning of both the polynomial P and the linearization L. The purpose of the present work is to investigate the backward error for a wide variety of linearizations. Two key asp... |

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Citation Context ...have the important property that they are always linearizations [19, sec. 4]. C1(λ) and C2(λ) belong to large sets of potential linearizations recently identified by Mackey et al. [19] and studied in =-=[6]-=- and [19]. With the notation Λ = [λ m−1 ,λ m−2 ,...,1] T , these sets are (3.3) (3.4) L1(P)= � L(λ):L(λ)(Λ ⊗ In)=v ⊗ P(λ), v∈ C m � , L2(P)= � L(λ):(Λ T ⊗ In)L(λ)=�v T ⊗ P(λ), �v ∈ C m � . There are m... |

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Citation Context ...s T-even since QT (−λ) =Q(λ). On the other hand, the quadratic Q(λ) =λ2A + λB + AT with B complex symmetric, arising in the study of vibration of rail tracks under the excitation of high speed trains =-=[10]-=-, [11], is T-palindromic since revQT (λ)=Q(λ). Linearizations in L1(P) that reflect the structure of these polynomials and therefore preserve symmetries in their spectra have recently been investigate... |

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Citation Context ...of quadratics arising from mechanical systems with damping, the condition (5.16) holds for systems that are not too heavily damped. A class of problems for which (5.16) is satisfied is the elliptic Q =-=[9]-=-, [13]: those for which A is Hermitian positive definite, B and C are Hermitian, and (x ∗ Bx) 2 < 4(x ∗ Ax)(x ∗ Cx) for all nonzero x ∈ C n . Our conclusions about the benefits to the backward error o... |

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Citation Context ...first part it suffices to note that for w as defined by (3.12) we have w ∗ � ∗ T y P(α, β)(e C1(α, β)= 1 ⊗ In), α �=0, y∗P(α, β)(eT m ⊗ In), α=0. For the next part, since C1 is a strong linearization =-=[4]-=- and P is regular, any eigenvalue (α, β) ofC1 of geometric multiplicity k is also an eigenvalue of P of geometric multiplicity k. Anyklinearly independent eigenvectors y of P for (α, β) clearly yield ... |

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Citation Context ...kernel {0}. Since L ∈ L1(P) is a linearization and P is regular, L is a strong linearization 1 [19, Thm. 4.3]. Hence the geometric multiplicity of any eigenvalue (including ∞) is the same for L and P =-=[14]-=-; that is, K1 and K2 have the same dimension. It follows that the map is a bijection, and the result is proved. 1 L is a strong linearization of P if it is a linearization for P and revL is a lineariz... |

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(Show Context)
Citation Context ...or of order u. Finally, we mention that further numerical illustration of the bounds developed here, on a symmetric QEP arising from a finite element model of a simply supported beam, can be found in =-=[8]-=-. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.s10 12 10 8 10 4 10 0 10 12 10 8 10 4 10 0 BACKWARD ERROR OF POLYNOMIAL EIGENPROBLEMS 1239 Ratios for right eigenpairs Co... |

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Citation Context ...at least to within some constant factor, but this is not necessarily the case. Consider, for example, the pencil � � � � A A B − A C L(λ)=λ + ∈ DL(λ A B − C C C 2 A + λB + C), which corresponds to v ==-=[11]-=- T in (3.3). Suppose A = B = I and C = ɛI with 0 < ɛ ≪ 1, and let ΔA = δI and ΔB = ΔC = 0. These perturbations have relative size max(�ΔA�2/�A�2, �ΔB�2/�B�2, �ΔC�2/�C�2) =δ, but for the pencil max(�ΔX... |

13 |
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Citation Context ...adratics arising from mechanical systems with damping, the condition (5.16) holds for systems that are not too heavily damped. A class of problems for which (5.16) is satisfied is the elliptic Q [9], =-=[13]-=-: those for which A is Hermitian positive definite, B and C are Hermitian, and (x ∗ Bx) 2 < 4(x ∗ Ax)(x ∗ Cx) for all nonzero x ∈ C n . Our conclusions about the benefits to the backward error of scal... |

13 |
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Citation Context ...g the identity blocks of C1 can significantly improve the backward error of the recovered eigenvectors of P. We can, of course, employ more sophisticated two-sided scalings, including balancing [16], =-=[25]-=-. However, these scalings produce a new pencil not belonging to L1, so our backward error bounds are not applicable to them. 3.4. DL(P ) linearizations. From section 2.2 and the definition of L1 in (3... |

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Citation Context ...anion linearization, C1. Our first problem comes from applying the Galerkin method to a PDE describing the wave motion of a vibrating string with clamped ends in a spatially inhomogeneous environment =-=[3]-=-, [9]. The quadratic Q is elliptic. Table 6.2 displays the smallest and largest ratios η Q (x, α, β)/ηL(z,α,b) over all computed eigenvalues for several linearizations and for the two ways of recoveri... |

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Citation Context ...y eigenvalue with the choice x = z2. This is confirmed by the boldface entries in the last two columns of Table 6.2. Our second problem is a simplified model of a nuclear power plant, as described in =-=[12]-=-, [23]. The largest ratios η Q (x, α, β)/ηL(z,α,b) and corresponding upper bounds are displayed in Table 6.3. Similar conclusions to those for the wave problem can be drawn for this problem. Since ρ =... |

4 |
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(Show Context)
Citation Context ...scaling the identity blocks of C1 can significantly improve the backward error of the recovered eigenvectors of P. We can, of course, employ more sophisticated two-sided scalings, including balancing =-=[16]-=-, [25]. However, these scalings produce a new pencil not belonging to L1, so our backward error bounds are not applicable to them. 3.4. DL(P ) linearizations. From section 2.2 and the definition of L1... |