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The Quadratic Eigenvalue Problem

Françoise Tisseur, +1 more
- 01 Feb 2001 - 
- Vol. 43, Iss: 2, pp 235-286
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This work surveys the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques.
Abstract
We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

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The quadratic eigenvalue problem
Tisseur, Françoise and Meerbergen, Karl
2001
MIMS EPrint: 2006.256
Manchester Institute for Mathematical Sciences
School of Mathematics
The University of Manchester
Reports available from: http://eprints.maths.manchester.ac.uk/
And by contacting: The MIMS Secretary
School of Mathematics
The University of Manchester
Manchester, M13 9PL, UK
ISSN 1749-9097

SIAM REVIEW
c
2001 Society for Industrial and Applied Mathematics
Vol. 43, No. 2, pp. 235–286
The Quadratic Eigenvalue
Problem
Fran¸coise Tisseur
Karl Meerbergen
Abstract. We survey the quadratic eigenvalue problem, treating its many applications, its mathe-
matical properties, and a variety of numerical solution techniques. Emphasis is given to
exploiting both the structure of the matrices in the problem (dense, sparse, real, com-
plex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify
numerical methods and catalogue available software.
Key words. quadratic eigenvalue problem, eigenvalue, eigenvector, λ-matrix, matrix polynomial,
second-order differential equation, vibration, Millennium footbridge, overdamped sys-
tem, gyroscopic system, linearization, backward error, pseudospectrum, condition num-
ber, Krylov methods, Arnoldi method, Lanczos method, Jacobi–Davidson method
AMS subject classification. 65F30
PII. S0036144500381988
1. Introduction. On its opening day in June 2000, the 320-meter-long Millen-
nium footbridge over the river Thames in London (see Figure 1.1) started to wobble
alarmingly under the weight of thousands of people; two days later the bridge was
closed. To explain the connection between this incident and the quadratic eigenvalue
problem (QEP), the subject of this survey, we need to introduce some ideas from
vibrating systems. A natural frequency of a structure is a frequency at which the
structure prefers to vibrate. When a structure is excited by external forces whose
frequencies are close to the natural frequencies, the vibrations are amplified and
the system becomes unstable. This is the phenomenon of resonance. The exter-
nal forces on the Millennium Bridge were pedestrian-induced movements. On the
opening day, a large crowd of people initially walking randomly started to adjust
their balance to the bridge movement, probably due to high winds on that day. As
they walked, they became more synchronized with each other and the bridge started
to wobble even more. The lateral vibrations experienced on the bridge occurred
because some of its natural modes of vibration are similar in frequency to the side-
ways component of pedestrian footsteps on the bridge.
1
The connection with this
survey is that the natural modes and frequencies of a structure are the solution of
Received by the editors November 3, 2000; accepted for publication (in revised form) December
4, 2000; published electronically May 2, 2001.
http://www.siam.org/journals/sirev/43-2/38198.html
Department of Mathematics, University of Manchester, Manchester M13 9PL, England (ftisseur@
ma.man.ac.uk, http://www.ma.man.ac.uk/˜ftisseur/). This author’s work was supported by Engi-
neering and Physical Sciences Research Council grant GR/L76532.
Free Field Technologies, 16, place de l’Universit´e, 1348 Louvain-la-Neuve, Belgium (Karl.
Meerbergen@fft.be).
1
More details on the Millennium Bridge, its construction, and the wobbling problem can be found
at http://www.arup.com/MillenniumBridge.
235

236 FRANC¸ OISE TISSEUR AND KARL MEERBERGEN
Fig. 1.1 The Millennium footbridge over the river Thames.
an eigenvalue problem that is quadratic when damping effects are included in the
model.
The QEP is currently receiving much attention because of its extensive applica-
tions in areas such as the dynamic analysis of mechanical systems in acoustics and
linear stability of flows in fluid mechanics. The QEP is to find scalars λ and nonzero
vectors x, y satisfying
(λ
2
M + λC + K)x =0,y
(λ
2
M + λC + K)=0,
where M, C, and K are n ×n complex matrices and x, y are the right and left eigen-
vectors, respectively, corresponding to the eigenvalue λ. A major algebraic difference
between the QEP and the standard eigenvalue problem (SEP),
Ax = λx,
and the generalized eigenvalue problem (GEP),
Ax = λBx,
is that the QEP has 2n eigenvalues (finite or infinite) with up to 2n right and 2n left
eigenvectors, and if there are more than n eigenvectors they do not, of course, form a
linearly independent set.
QEPs are an important class of nonlinear eigenvalue problems that are less fa-
miliar and less routinely solved than the SEP and the GEP, and they need special
attention. A major complication is that there is no simple canonical form analogous
to the Schur form for the SEP or the generalized Schur form for the GEP.
A large literature exists on QEPs, spanning the range from theory to applications,
but the results are widely scattered across disciplines. Our goal in this work is to
gather together a collection of applications where QEPs arise, to identify and classify
their characteristics, and to summarize current knowledge of both theoretical and
algorithmic aspects. The type of problems we will consider in this survey and their
spectral properties are summarized in Table 1.1.
The structure of the survey is as follows. In section 2 we discuss a number
of applications, covering the motivation, characteristics, theoretical results, and algo-
rithmic issues. Section 3 reviews the spectral theory of QEPs and discusses important
classes of QEPs coming from overdamped systems and gyroscopic systems. Several

THE QUADRATIC EIGENVALUE PROBLEM 237
Tab le 1 . 1 Matrix properties of QEPs considered in this survey with corresponding spectral proper-
ties. The first column refers to the section where the problem is treated. Properties can
be added: QEPs for which M, C, and K are real symmetric have properties P 3 and P 4
so that their eigenvalues are real or come in pairs (λ,
¯
λ) and the sets of left and right
eigenvectors coincide. Overdamped systems yield QEPs having the property P 6.InP 6,
γ(M, C,K) = min{(x
Cx)
2
4(x
Mx)(x
Kx):||x||
2
=1}. Gyroscopic systems yield
QEPs having at least the property P 7.
Matrix properties Eigenvalue properties Eigenvector properties
P1
§3.1
M nonsingular 2n finite eigenvalues
P2
§3.1
M singular
Finite and infinite eigen-
values
P3
§3.1
M, C, K real Eigenvalues are real or
come in pairs (λ,
¯
λ)
If x is a right eigenvector of λ
then ¯x is a right eigenvector
of
¯
λ
P4
§3.8
M, C, K Hermitian Eigenvalues are real or
come in pairs (λ,
¯
λ)
If x is a right eigenvector of
λ then x is a left eigenvector
of
¯
λ
P5
§3.8
M Hermitian positive
definite, C, K Hermitian
positive semidefinite
Re(λ) 0
P6
§3.9
M, C symmetric positive
definite, K symmetric
positive semidefinite,
γ(M, C,K) > 0
λs are real and negative,
gap between n largest and
n smallest eigenvalues
n linearly independent
eigenvectors associated with
the n largest (n smallest)
eigenvalues
P7
§3.10
M, K Hermitian,
M positive definite,
C = C
Eigenvalues are purely
imaginary or come in
pairs (λ,
¯
λ)
If x is a right eigenvector of
λ then x is a left eigenvector
of
¯
λ
P8
§3.10
M, K real symmetric
and positive definite,
C = C
T
Eigenvalues are purely
imaginary
linearizations are introduced and their properties discussed. In this paper, the term
linearization means that the nonlinear QEP is transformed into a linear eigenvalue
problem with the same eigenvalues. Our treatment of the large existing body of the-
ory is necessarily selective, but we give references where further details can be found.
Section 4 deals with tools that offer an understanding of the sensitivity of the problem
and the behavior of numerical methods, covering condition numbers, backward errors,
and pseudospectra.
A major division in numerical methods for solving the QEP is between those
that treat the problem in its original form and those that linearize it into a GEP of
twice the dimension and then apply GEP techniques. A second division is between
methods for dense, small- to medium-size problems and iterative methods for large-
scale problems. The former methods compute all the eigenvalues and are discussed
in section 5, while the latter compute only a selection of eigenvalues and eigenvectors
and are reviewed in section 6. The proper choice of method for particular problems
is discussed. Finally, section 7 catalogues available software, and section 8 contains
further discussion and related problems.
We shall generally adopt the Householder notation: capital letters A, B, C, ...
denote matrices, lower case roman letters denote column vectors, Greek letters denote
scalars, ¯α denotes the conjugate of the complex number α, A
T
denotes the transpose
of the complex matrix A, A
denotes the conjugate transpose of A, and ·is any
vector norm and the corresponding subordinate matrix norm. The values taken by

238 FRANC¸ OISE TISSEUR AND KARL MEERBERGEN
any integer variable are described using the colon notation: i =1:n means the
same as i =1, 2,...,n.” We write A>0(A 0) if A is Hermitian positive definite
(positive semidefinite) and A<0(A 0) if A is Hermitian negative definite (negative
semidefinite). A definite pair (A, B) is defined by the property that A, B C
n×n
are
Hermitian and
min
zC
n
z
2
=1
(z
Az)
2
+(z
Bz)
2
> 0,
which is certainly true if A>0orB>0.
2. Applications of QEPs. A wide variety of applications require the solution
of a QEP, most of them arising in the dynamic analysis of structural mechanical,
and acoustic systems, in electrical circuit simulation, in fluid mechanics, and, more
recently, in modeling microelectronic mechanical systems (MEMS) [28], [155]. QEPs
also have interesting applications in linear algebra problems and signal processing.
The list of applications discussed in this section is by no means exhaustive, and, in
fact, the number of such applications is constantly growing as the methodologies for
solving QEPs improve.
2.1. Second-Order Differential Equations. To start, we consider the solution
of a linear second-order differential equation
M ¨q(t)+C ˙q(t)+Kq(t)=f(t),(2.1)
where M, C, and K are n × n matrices and q(t)isannth-order vector. This is the
underlying equation in many engineering applications. We show that the solution can
be expressed in terms of the eigensolution of the corresponding QEP and explain why
eigenvalues and eigenvectors give useful information.
Two important areas where second-order differential equations arise are the fields
of mechanical and electrical oscillation. The left-hand picture in Figure 2.1 illustrates
a single mass-spring system in which a rigid block of mass M is on rollers and can
move only in simple translation. The (static) resistance to displacement is provided
by a spring of stiffness K, while the (dynamic) energy loss mechanism is represented
by a damper C; f(t) represents an external force. The equation of motion governing
this system is of the form (2.1) with n =1.
As a second example, we consider the flow of electric current in a simple RLC
circuit composed of an inductor with inductance L, a resistor with resistance R, and
a capacitor with capacitance C, as illustrated on the right of Figure 2.1; E(t)is
the input voltage. The Kirchhoff loop rule requires that the sum of the changes in
potential around the circuit must be zero, so
L
di(t)
dt
+ Ri(t)+
q(t)
C
E(t)=0,(2.2)
where i(t) is the current through the resistor, q(t) is the charge on the capacitor, and
t is the elapsed time. The charge q(t) is related to the current i(t)byi(t)=dq(t)/dt.
Differentiation of (2.2) gives the second-order differential equation
L
d
2
i(t)
dt
2
+ R
di(t)
dt
+
1
C
i(t)=
dE(t)
dt
,(2.3)
which is of the same form as (2.1), again with n =1.

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References
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Book ChapterDOI

A Constrained Eigenvalue Problem

TL;DR: In this article, the authors considered the problem of computing an x such that an x = t such that ==================¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−1−1, −1.
Journal ArticleDOI

Orthogonality and partial pole assignment for the symmetric definite quadratic pencil

TL;DR: In this article, the eigenvectors of a symmetric matrix can be chosen to form an orthogonal set with respect to the identity and to the matrix itself, and the same can be said of the symmetric definite quadratic pencil.
Journal ArticleDOI

Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils

TL;DR: This work develops a structure-preserving skew-Hamiltonian, isotropic, implicitly restarted shift-and-invert Arnoldi algorithm (SHIRA), and several numerical examples demonstrate the superiority of SHIRA over a competing unstructured method.
Journal ArticleDOI

Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications

TL;DR: The usual definitions of pseudospectra are extended in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory.
Journal ArticleDOI

Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

TL;DR: The rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil and is a generalization of the shifted and inverted Arnoldi algorithm.
Frequently Asked Questions (17)
Q1. What can be done to solve the projected problem?

As an orthogonal projection preserves symmetry and skew-symmetry, the numerical methods of sections 5.2 and 5.3 can be used to solve the projected problem. 

The projection method approximates an eigenvector x of L(λ) by a vector x̃ = V ξ ∈ Kk with corresponding approximate eigenvalue λ̃. AsW ∗L(λ̃)x̃ = W ∗L(λ̃)V ξ = Lk(λ̃)ξ = 0, the projection method forces the residual r = L(λ̃)x̃ to be orthogonal to Lk. 

The reduction to a Hamiltonian eigenproblem uses the fact that when the skew-Hamiltonian matrix B is nonsingular, it can be written in factored form asB = B1B2 = [ The author12C 0 M ] [ M 12C 0 The author] with BT2 J = JB1.(5.2)Then H = B−11 AB −1 2 is Hamiltonian. 

It produces a non-Hermitian tridiagonal matrix Tk and a pair of matrices Vk and Wk such that W ∗kVk = The authorand whose columns form bases for the Krylov subspaces Kk(S, v) and Kk(S∗, w), where v and w are starting vectors such that w∗v = 1. 

To measure the robustness of the system, one can take as a global measureν2 = 2n∑ k=1 ω2kκ(λk) 2,where the ωk are positive weights. 

One important advantage of working with a linearization of Q(λ) and a Krylov subspace method is that one can get at the same time the partial Schur decomposition of the single matrix S that is used to define the Krylov subspaces. 

The shift-and-invert spectral transformation f(λ) = 1/(λ − σ) and the Cayley spectral transformation f(λ) = (λ− β)/(λ− σ) (for β = σ), used to approximate eigenvalues λ closest to the shift σ, are other possible spectral transformations that are discussed in [7], for example. 

If (λ̃, x̃) is a converged Ritz eigenpair that belongs to the set of desired eigenvalues, one may want to lock it and then continue to compute the remaining eigenvalues without altering (λ̃, x̃). 

Parlett and Chen [114] introduced a pseudo-Lanczos algorithm for symmetric pencils that uses an indefinite inner product and respects the symmetry of the problem. 

G(λ)∗ = G(−λ̄),(3.19)the distribution of the eigenvalues of G(λ) in the complex plane is symmetric with respect to the imaginary axis. 

Most algorithms for large QEPs proceed by generating a sequence of subspaces {Kk}k≥0 that contain increasingly accurate approximations to the desired eigenvectors. 

One approach, adopted by MATLAB 6’s polyeig function for solving the polynomial eigenvalue problem and illustrated in Algorithm 5.1 below, is to use whichever part of ξ̃ yields the smallest backward error (4.7). 

If the pencil is indefinite, the HR [21], [23], LR [124], and Falk–Langemeyer [45] algorithms can be employed in order to take advantage of the symmetry of the GEP, but all these methods can be numerically unstable and can even break down completely. 

(Note that the matrix Hk is the Galerkin projection of S and not of A− λB.)A major disadvantage of the shift-and-invert Arnoldi and Lanczos methods is that a change of shift σ requires building a new Krylov subspace: all information built with the old σ is lost. 

If the converged (λ̃, x̃) does not belong to the set of wanted eigenvalues, one may want to remove it from the current subspace Kk. 

In this case Lk is the orthogonal projection of L(λ) onto Kk. When W = V , Lk is the oblique projection of L(λ) onto Kk along Lk. 

This process does not guarantee orthogonality of the columns of Vk in floating point arithmetic, so reorthogonalization is recommended to improve the numerical stability of the method [32], [135].