## Structured Pseudospectra For Polynomial Eigenvalue Problems With Applications (2001)

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Venue: | SIAM J. MATRIX ANAL. APPL |

Citations: | 46 - 9 self |

### BibTeX

@ARTICLE{Tisseur01structuredpseudospectra,

author = {Françoise Tisseur and Nicholas J. Higham},

title = {Structured Pseudospectra For Polynomial Eigenvalue Problems With Applications},

journal = {SIAM J. MATRIX ANAL. APPL},

year = {2001},

volume = {23},

number = {1},

pages = {187--208}

}

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### Abstract

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrLx equation and thereby factorizes the quadratic h-matrLx. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.

### Citations

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Citation Context ...e set of eigenvalues of P is denoted by Λ(P ). When Am is nonsingular P has mn finite eigenvalues, while if Am is singular P has infinite eigenvalues. Good references for the theory of λ-matrices are =-=[8]-=-, [20], [21], [37]. Throughout this paper we assume that P has only finite eigenvalues (and pseudoeigenvalues); how to deal with infinite eigenvalues is described in [16]. For notational convenience, ... |

131 |
Hydrodynamic stability without eigenvalues
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Citation Context ...tions must be carried out for the numerical approximations to the modes to correctly determine the location of the modes. For more on the interpretation of pseudospectra for this problem, see [32]and =-=[44]-=-. Again, for comparison we computed the pseudospectra of the corresponding standard eigenvalue problem. The picture was qualitatively similar, but the contour levels were several orders of magnitude s... |

81 | Spectra and pseudospectra
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Citation Context ... for the standard eigenproblem, although attention has also been given to matrix pencils [4], [23], [33], [40], [46]. The literature on pseudospectra is large and growing. We refer to Trefethen [41], =-=[42]-=-, [43] for thorough surveys of pseudospectra and their computation for a single matrix; see also the Web site [3]. In this work we investigate pseudospectra for polynomial matrices (or λ-matrices) (1.... |

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Citation Context ...spond to parts of the spectrum of interest and can be computed using the Arnoldi process on the companion form pencil (F, G) or directly on P (λ) with the Jacobi–Davidson method or its variants [26], =-=[35]-=-. In the latter case, the matrix Vk is built during the Davidson process. 3.3.2. Direct approach. This approach consists of approximating ‖P (z) −1 ‖ at each grid point z. Techniques analogous to thos... |

63 |
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Citation Context ... of eigenvalues of P is denoted by Λ(P ). When Am is nonsingular P has mn finite eigenvalues, while if Am is singular P has infinite eigenvalues. Good references for the theory of λ-matrices are [8], =-=[20]-=-, [21], [37]. Throughout this paper we assume that P has only finite eigenvalues (and pseudoeigenvalues); how to deal with infinite eigenvalues is described in [16]. For notational convenience, we int... |

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Citation Context ...genvalues of P is denoted by Λ(P ). When Am is nonsingular P has mn finite eigenvalues, while if Am is singular P has infinite eigenvalues. Good references for the theory of λ-matrices are [8], [20], =-=[21]-=-, [37]. Throughout this paper we assume that P has only finite eigenvalues (and pseudoeigenvalues); how to deal with infinite eigenvalues is described in [16]. For notational convenience, we introduce... |

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Citation Context ...oundary conditions. We are interested in the eigenvalues λ that are the closest to the real axis, and we need Im(λ) > 0 for stability. The linear eigenvalue problem (Case 1) has been solved by Orszag =-=[29]-=-. The critical neutral point corresponding to λ and ω both real for minimum R was found at R = 5772 and λ =1.02056 with the frequency ω =0.26943 [2], [29]. For our calculations we set R and ω to these... |

49 | Backward error and condition of polynomial eigenvalue problems
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Citation Context ...n be expressed in terms of the backward error of λ as (2.7) Λɛ(P )={ λ ∈ C : η(λ) ≤ ɛ }. The following lemma gives an explicit expression for η(x, λ) and η(λ). This lemma generalizes results given in =-=[36]-=-for the 2-norm and earlier in [5], [10]for the generalized eigenvalue problem. Lemma 2.2. The normwise backward error η(x, λ) is given for x ̸= 0by ‖r‖ η(x, λ) = p(|λ|)‖x‖ , (2.8) where r = P (λ)x and... |

38 |
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Citation Context ...ENPROBLEMS 205 are periodic in x and grow or decay in time depending on the sign of the imaginary part of ω. This case has been studied with the help of pseudospectra by Reddy, Schmid, and Henningson =-=[32]-=-. Case 2. Spatial stability. For most real flows, the perturbations are periodic in time, which means that ω is real. Then the sign of the imaginary part of λ determines whether the perturbations will... |

36 | Pseudozeros of polynomials and pseudospectra of companion matrices
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Citation Context ...ood of the polynomial P introduced by Mosier [28], that is, the set of all polynomials obtained by elementwise perturbations of P of size at most ɛ. This set is also investigated by Toh and Trefethen =-=[38]-=-, who call it the ɛ-pseudozero set. 2.2. Connection with backward error. A natural definition of the normwise backward error of an approximate eigenpair (x, λ) of (2.1) is (2.5) η(x, λ) :=min{ ɛ :(P (... |

35 | Calculation of pseudospectra by the Arnoldi iteration
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Citation Context ...problem and then compute the pseudospectra of the reduced problem, and those that approximate the norm of the resolvent directly. 3.3.1. Projection approach. For a single matrix, A, Toh and Trefethen =-=[39]-=- and Wright and Trefethen [48]approximate the resolvent norm by the Arnoldi method; that is, they approximate ‖(A − zI) −1‖2 by ‖(Hm − zI) −1‖2 or by σmin( ˜ Hm − zĨ), where Hm is the square Hessenber... |

34 |
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Citation Context ...ork is for the standard eigenproblem, although attention has also been given to matrix pencils [4], [23], [33], [40], [46]. The literature on pseudospectra is large and growing. We refer to Trefethen =-=[41]-=-, [42], [43] for thorough surveys of pseudospectra and their computation for a single matrix; see also the Web site [3]. In this work we investigate pseudospectra for polynomial matrices (or λ-matrice... |

33 |
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Citation Context ...e parameters in Θ, but that not all linear functions can be represented in this form. We choose this particular structure for the perturbations because it is one commonly used in control theory [17], =-=[18]-=-, [30] and it leads to more tractable formulae than a fully general approach. Note, for instance, that the system ˙x(t) =(A + DΘE)x(t), t > 0 (which leads to a polynomial eigenvalue problem with m = 1... |

32 | A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra
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(Show Context)
Citation Context ... in the analogous expression for the resolvent of a single matrix in the standard eigenproblem. For the 1- and ∞-norms we can efficiently estimate ‖P (z) −1‖ using the algorithm of Higham and Tisseur =-=[14]-=-, which requires only the ability to multiply matrices by P (z) −1 and P (z) −∗ . An alternative to the generalized Schur decomposition is the generalized Hessenberg-triangular form, which differs fro... |

29 | Structured backward error and condition of generalized eigenvalue problems
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Citation Context ... error of λ as (2.7) Λɛ(P )={ λ ∈ C : η(λ) ≤ ɛ }. The following lemma gives an explicit expression for η(x, λ) and η(λ). This lemma generalizes results given in [36]for the 2-norm and earlier in [5], =-=[10]-=-for the generalized eigenvalue problem. Lemma 2.2. The normwise backward error η(x, λ) is given for x ̸= 0by ‖r‖ η(x, λ) = p(|λ|)‖x‖ , (2.8) where r = P (λ)x and p(x) = ∑m k=0 αkxk .Ifλis not an eigen... |

23 | Locking and restarting quadratic eigenvalue solvers
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Citation Context ... correspond to parts of the spectrum of interest and can be computed using the Arnoldi process on the companion form pencil (F, G) or directly on P (λ) with the Jacobi–Davidson method or its variants =-=[26]-=-, [35]. In the latter case, the matrix Vk is built during the Davidson process. 3.3.2. Direct approach. This approach consists of approximating ‖P (z) −1 ‖ at each grid point z. Techniques analogous t... |

22 | Numerical analysis of a quadratic matrix equation
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Citation Context ...success of this method depends on two things: the existence of solvents and being able to compute one at a reasonable cost. Some sufficient conditions for the existence of a solvent are summarized in =-=[13]-=-. In particular, for an overdamped problem, one for which A2 and A1 are Hermitian positive definite, A0 is Hermitian positive semidefinite, and (x∗A1x) 2 > 4(x∗A2x)(x∗A0x) for all x ̸= 0, a solvent is... |

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Kelb B: Spectral value sets: a graphical tool for robustness analysis
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Citation Context ... of the parameters in Θ, but that not all linear functions can be represented in this form. We choose this particular structure for the perturbations because it is one commonly used in control theory =-=[17]-=-, [18], [30] and it leads to more tractable formulae than a fully general approach. Note, for instance, that the system ˙x(t) =(A + DΘE)x(t), t > 0 (which leads to a polynomial eigenvalue problem with... |

22 | Large-scale computation of pseudospectra using ARPACK and eigs
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(Show Context)
Citation Context ...seudospectra of the reduced problem, and those that approximate the norm of the resolvent directly. 3.3.1. Projection approach. For a single matrix, A, Toh and Trefethen [39] and Wright and Trefethen =-=[48]-=-approximate the resolvent norm by the Arnoldi method; that is, they approximate ‖(A − zI) −1‖2 by ‖(Hm − zI) −1‖2 or by σmin( ˜ Hm − zĨ), where Hm is the square Hessenberg matrix of dimension m ≪ n ob... |

21 | Computation of pseudospectra by continuation
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- 1997
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Citation Context ...σmin(P (z)) is the square root of λmin(P (z) ∗ P (z)), we can approximate ‖P (z) −1 ‖2 with the power iteration or Lanczos iteration applied to P (z) −1 P (z) −∗ . In the case of a single matrix, Lui =-=[25]-=-introduced the idea of using the Schur form of A in order to speed up the computation of λmin((A − zI) ∗ (A − zI)). Unfortunately, for matrix polynomials of degree m ≥ 2 no analogue of the Schur form ... |

19 |
Root neighborhoods of a polynomial
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- 1986
(Show Context)
Citation Context ...ion of the ɛ-pseudospectrum in Lemma 2.1 will be the basis of our algorithms for computing pseudospectra. We note that for n =1,Λɛ(P) is the root neighborhood of the polynomial P introduced by Mosier =-=[28]-=-, that is, the set of all polynomials obtained by elementwise perturbations of P of size at most ɛ. This set is also investigated by Toh and Trefethen [38], who call it the ɛ-pseudozero set. 2.2. Conn... |

16 | More on pseudospectra for polynomial eigenvalue problems and applications in control theory
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Citation Context ...r the theory of λ-matrices are [8], [20], [21], [37]. Throughout this paper we assume that P has only finite eigenvalues (and pseudoeigenvalues); how to deal with infinite eigenvalues is described in =-=[16]-=-. For notational convenience, we introduce (2.2) ∆P (λ) =λ m ∆Am + λ m−1 ∆Am−1 + ···+ ∆A0. We define the ɛ-pseudospectrum of P by (2.3) Λɛ(P )= { λ ∈ C :(P(λ)+∆P (λ))x = 0 for some x ̸= 0 and ∆P (λ) w... |

15 |
Portraits spectraux de matrices: un outil d’analyse de la stabilité
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Citation Context ...areas such as fluid mechanics, Markov chains, and control theory. Most of the existing work is for the standard eigenproblem, although attention has also been given to matrix pencils [4], [23], [33], =-=[40]-=-, [46]. The literature on pseudospectra is large and growing. We refer to Trefethen [41], [42], [43] for thorough surveys of pseudospectra and their computation for a single matrix; see also the Web s... |

14 | Solving a quadratic matrix equation by Newton’s method with exact line search
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Citation Context ...mitian positive definite, A0 is Hermitian positive semidefinite, and (x∗A1x) 2 > 4(x∗A2x)(x∗A0x) for all x ̸= 0, a solvent is guaranteed to exist. Various methods are available for computing solvents =-=[12]-=-, [13]. One of the most generally useful is Newton’s method, optionally with exact line searches, which requires a generalized Sylvester equation in n×n matrices to be solved on each iteration, at a t... |

13 | On stability radii of generalized eigenvalue problems - Vermaut - 1997 |

12 | A note on the normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, Numerical Linear Algebra With Applications 5
- Frayssé, Toumazou
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(Show Context)
Citation Context ...kward error of λ as (2.7) Λɛ(P )={ λ ∈ C : η(λ) ≤ ɛ }. The following lemma gives an explicit expression for η(x, λ) and η(λ). This lemma generalizes results given in [36]for the 2-norm and earlier in =-=[5]-=-, [10]for the generalized eigenvalue problem. Lemma 2.2. The normwise backward error η(x, λ) is given for x ̸= 0by ‖r‖ η(x, λ) = p(|λ|)‖x‖ , (2.8) where r = P (λ)x and p(x) = ∑m k=0 αkxk .Ifλis not an... |

12 | Robust stability of linear systems described by higher order dynamic equations
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Citation Context ...meters in Θ, but that not all linear functions can be represented in this form. We choose this particular structure for the perturbations because it is one commonly used in control theory [17], [18], =-=[30]-=- and it leads to more tractable formulae than a fully general approach. Note, for instance, that the system ˙x(t) =(A + DΘE)x(t), t > 0 (which leads to a polynomial eigenvalue problem with m = 1), may... |

11 |
Differential eigenvalue problems in which the parameter appears nonlinearly
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(Show Context)
Citation Context ...r the perturbations will grow or decay in space. In this case, the parameter is λ, which appears to the fourth power in (4.3), so we obtain a quartic polynomial eigenvalue problem. Bridges and Morris =-=[2]-=-calculated the spectrum of (4.3) using a finite Chebyshev series expansion of φ combined with the Lanczos tau method and they computed the spectrum of the quartic polynomial by two methods: the QR alg... |

9 |
Efficient multivariable frequency response computations
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Citation Context ...ansfer function of the linear timeinvariant multivariate system described by ⎡ ⎤ 0 ... G ˙x(t) =Fx(t)+ ⎢ ⎥ ⎣ 0 ⎦ I u(t), y(t) =[I 0 ··· 0]x(t). Several algorithms have been proposed in the literature =-=[22]-=-, [27] to compute transfer functions at a large number of frequencies, most of them assuming that G = I. Our objective is to efficiently compute the norm of the transfer function, rather than to compu... |

9 | Transfer functions and resolvent norm approximation of large matrices
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Citation Context ...ently evaluate or approximate ‖P (z) −1 ‖ for many different z. 3.1. Transfer function approach. The idea of writing pseudospectra in terms of transfer functions is not new. Simoncini and Gallopoulos =-=[34]-=-used a transfer function framework to rewrite most of the techniques used to approximate ɛ-pseudospectra of large matrices, yielding interesting comparisons as well as better understanding of the tech... |

7 | V.(1996) Spectral portraits for matrix pencils
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Citation Context ... applications in areas such as fluid mechanics, Markov chains, and control theory. Most of the existing work is for the standard eigenproblem, although attention has also been given to matrix pencils =-=[4]-=-, [23], [33], [40], [46]. The literature on pseudospectra is large and growing. We refer to Trefethen [41], [42], [43] for thorough surveys of pseudospectra and their computation for a single matrix; ... |

7 |
Numerical range of matrix polynomials
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Citation Context ...n form (2.16) we deduce that m−1 ∑ max |λj(P )| ≤1+ j j=0 ‖A −1 m Aj‖p for any p-norm. Alternatively, we could bound maxj |λj(P )| by the largest absolute value of a point in the numerical range of P =-=[24]-=-, but computation of this number is itself a nontrivial problem. For much more on bounding the eigenvalues of matrix polynomials see [15]. For the 2-norm, ‖P (z) −1 ‖2 =(σmin(P (z))) −1 , where σmin d... |

7 |
Generalized epsilon-pseudospectra
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6 | J.L.M.(1997) Pseudospectra for matrix pencils and stability of equilibria
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4 |
private communication
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- 2006
(Show Context)
Citation Context ...s the mass matrix, C is the damping matrix, and K the stiffness matrix. We give in Figure 4.5 the sparsity pattern of the three matrices M, C, and K of order 107 arising from a model of a speaker box =-=[1]-=-. These matrices are symmetric and the sparsity patterns of M and K are identical. There is a large variation in the norms: ‖M‖2 =1,‖C‖2 =0.06, ‖K‖2 =9.9× 106 . We plot in Figure 4.6 pseudospectra wit... |

4 |
The quadratic eigenvalue problem, Numerical Analysis Report 370
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- 2000
(Show Context)
Citation Context ...ues of P is denoted by Λ(P ). When Am is nonsingular P has mn finite eigenvalues, while if Am is singular P has infinite eigenvalues. Good references for the theory of λ-matrices are [8], [20], [21], =-=[37]-=-. Throughout this paper we assume that P has only finite eigenvalues (and pseudoeigenvalues); how to deal with infinite eigenvalues is described in [16]. For notational convenience, we introduce (2.2)... |

3 |
Numerical solution of matrix polynomial equations by Newton’s method
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- 1987
(Show Context)
Citation Context ...degrees m greater than 2 classes of problem for which a factorization into linear factors exists are less easily identified and the cost of Newton’s method (for example) is much higher than for m = 2 =-=[19]-=-. 3.3. Large-scale computation. All the methods described above are intended for small- to medium-scale problems for which Schur and other reductions are possible. For large, possibly sparse, problems... |

3 |
Nouvelles Approches de Calcul du ɛ-Spectre de Matrices et de Faisceaux de
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Citation Context ...ications in areas such as fluid mechanics, Markov chains, and control theory. Most of the existing work is for the standard eigenproblem, although attention has also been given to matrix pencils [4], =-=[23]-=-, [33], [40], [46]. The literature on pseudospectra is large and growing. We refer to Trefethen [41], [42], [43] for thorough surveys of pseudospectra and their computation for a single matrix; see al... |

3 |
A determinant identity and its application in evaluating frequency response matrices
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Citation Context ... function of the linear timeinvariant multivariate system described by ⎡ ⎤ 0 ... G ˙x(t) =Fx(t)+ ⎢ ⎥ ⎣ 0 ⎦ I u(t), y(t) =[I 0 ··· 0]x(t). Several algorithms have been proposed in the literature [22], =-=[27]-=- to compute transfer functions at a large number of frequencies, most of them assuming that G = I. Our objective is to efficiently compute the norm of the transfer function, rather than to compute the... |

3 |
Spectra and pseudospectra, in The Graduate Student’s Guide to Numerical Analysis ’98
- Trefethen
- 1999
(Show Context)
Citation Context ...he standard eigenproblem, although attention has also been given to matrix pencils [4], [23], [33], [40], [46]. The literature on pseudospectra is large and growing. We refer to Trefethen [41], [42], =-=[43]-=- for thorough surveys of pseudospectra and their computation for a single matrix; see also the Web site [3]. In this work we investigate pseudospectra for polynomial matrices (or λ-matrices) (1.1) P (... |

1 |
Bounds for Eigenvalues of Matrix Polynomials, Numerical Analysis Report 371
- Higham, Tisseur
- 2001
(Show Context)
Citation Context ... largest absolute value of a point in the numerical range of P [24], but computation of this number is itself a nontrivial problem. For much more on bounding the eigenvalues of matrix polynomials see =-=[15]-=-. For the 2-norm, ‖P (z) −1 ‖2 =(σmin(P (z))) −1 , where σmin denotes the smallest singular value. If the grid is ν × ν and σmin is computed using the Golub–Reinsch SVD algorithm then the whole comput... |