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A block Newton method for
nonlinear eigenvalue problems
Journal Article
Author(s):
Kressner, Daniel
Publication date:
2009-12
Permanent link:
https://doi.org/10.3929/ethz-b-000019530
Rights / license:
In Copyright - Non-Commercial Use Permitted
Originally published in:
Numerische Mathematik 114(2), https://doi.org/10.1007/s00211-009-0259-x
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Numer. Math. (2009) 114:355–372
DOI 10.1007/s00211-009-0259-x
Numerische
Mathematik
A block Newton method for nonlinear eigenvalue
problems
Daniel Kressner
Received: 2 February 2009 / Revised: 20 July 2009 / Published online: 15 September 2009
© Springer-Verlag 2009
Abstract We consider matrix eigenvalue problems that are nonlinear in the
eigenvalue parameter. One of the most fundamental differences from the linear case
is that distinct eigenvalues may have linearly dependent eigenvectors or even share
the same eigenvector. This has been a severe hindrance in the development of gen-
eral numerical schemes for computing several eigenvalues of a nonlinear eigenvalue
problem, either simultaneously or subsequently. The purpose of this work is to show
that the concept of invariant pairs offers a way of representing eigenvalues and eigen-
vectors that is insensitive to this phenomenon. To demonstrate the use of this concept
in the development of numerical methods, we have developed a novel block Newton
method for computing such invariant pairs. Algorithmic aspects of this method are
considered and a few academic examples demonstrate its viability.
Mathematics Subject Classification (2000) Primary 65F15; Secondary 15A18 ·
47A56
1 Introduction
Given a function T : → C
n×n
holomorphic on an open set ⊆ C, we consider the
nonlinear eigenvalue problem of finding pairs (x,λ) ∈ C
n
× with x = 0 such that
T (λ)x = 0. (1)
For any such pair (x,λ), we call x an eigenvector and λ an eigenvalue. This formula-
tion includes linear eigenvalue problems, for which T (λ) = A − λI with A ∈ C
n×n
,
as well as polynomial eigenvalue problems, for which T is a matrix polynomial in λ.
D. Kressner (
B
)
Seminar für Angewandte Mathematik, HG G 57.1, Rämistrasse 101, 8092 Zurich, Switzerland
e-mail: kressner@math.ethz.ch
123
356 D. Kressner
To avoid degenerate situations, we assume that T is regular, i.e., det
(
T (·)
)
≡ 0onany
of the components of , throughout this paper. For a recent overview on the numerics
and numerous applications of such nonlinear eigenvalue problems, we refer to [20].
In contrast to the linear case, there may be eigenvector/eigenvalue pairs (λ
1
, x
1
),
..., (λ
k
, x
k
) of (1), for which the eigenvalues λ
1
,...,λ
k
are pairwise distinct but
{x
1
,...,x
k
} is linearly dependent. This possibility is already evident from the fact
that k can be larger than n. Another example [12]isgivenby
T (λ) =
012
−214
+ λ
−1 −6
2 −9
+ λ
2
10
01
, (2)
for which the eigenvalues 3 and 4 share the same eigenvector
1
1
. The occurrence
of such linear dependencies is an annoyance when attempting to develop numeri-
cal methods for computing more than one eigenvalue of (1). For example, standard
Newton methods [10,11] for the simultaneous computation of several eigenvalues
crucially depend on the existence of a basis for the invariant subspace belonging
to the eigenvalues of interest. In methods that determine several eigenvalues sub-
sequently, such as Krylov subspace or Jacobi-Davidson methods [2], repeated con-
vergence towards an eigenvalue is usually avoided by reorthogonalization against
converged eigenvectors. If such an idea was directly applied t o nonlinear eigen-
value problems, eigenvalues could be missed due to linear dependencies among eigen-
vectors.
In the case that the nonlinear eigenvalue problem admits a minimum–maximum
characterization [26,31], its eigenvalues can be ordered and numbered. Voss and his
co-authors [4,5,7,27–30] have developed Arnoldi-type and Jacobi-Davidson-type
methods that employ this numbering as a safety scheme for avoiding repeated con-
vergence towards the same eigenvalue. Unfortunately, for many applications such
minimum–maximum characterizations do not exist or are difficult to verify.
In this work, we will propose a different approach for dealing with several eigen-
values, very much inspired by the work of Beyn and Thümmler [9] on continuation
methods for quadratic eigenvalue problems. For this purpose, it will be more conve-
nient to assume that the nonlinear eigenvalue problem (1) takes the form
(
f
1
(λ)A
1
+ f
2
(λ)A
2
+···+ f
m
(λ)A
m
)
x = 0. (3)
for holomorphic functions f
1
,..., f
m
: → C and constant matrices A
1
,...,A
m
∈
C
n×n
. This is no restriction as we could turn (1)into(3) by choosing m = n
2
,
f
(i−1)n+ j
(λ) = t
ij
(λ) and A
(i−1)n+ j
= e
i
e
T
j
, with e
i
and e
j
denoting the ith and jth
unit vectors of length n, respectively. However, many applications of nonlinear eigen-
value problems already come in the form (3) and such a reformulation is not needed.
For example, in eigenvalue problems related to the stability of time-delay systems [21],
the functions f
j
are exponentials or polynomials. In applications related to vibrating
mechanical structures [30], the functions f
j
are rational and model different material
properties.
The rest of this paper is organized as follows. In “Invariant pairs”, the concept of
invariant pairs for the nonlinear eigenvalue Problem (3) is introduced. We believe this
123
A block Newton method for nonlinear eigenvalue problems 357
to be the most suitable extension of an eigenvalue/eigenvector pair to several eigen-
values. Several useful properties are shown to substantiate this belief. In “A Newton
method for simple invariant pairs”, a Newton method for computing such invariant
pairs is developed, along with some algorithmic details and numerical experiments.
2 Invariant pairs
Definition 1 Let the eigenvalues of S ∈ C
k×k
be contained in and let X ∈ C
n×k
.
Then (X, S) ∈ C
n×k
× C
k×k
is called an invariant pair of the nonlinear eigenvalue
problem (3)if
A
1
Xf
1
(S) + A
2
Xf
2
(S) +···+ A
m
Xf
m
(S) = 0. (4)
Note that the matrix functions f
1
(S),..., f
m
(S) are well defined under the given
assumptions [16]. As an example, let (x
1
,λ
1
) and (x
2
,λ
2
) be eigenvector/eigenvalue
pairs of (3). Then (X, S) with X =[x
1
, x
2
] and S = diag(λ
1
,λ
2
) is an invariant pair.
To avoid trivial invariant pairs, such as X = 0, an additional property needs to be
imposed. However, we have already seen that requiring X to have full column rank is
not reasonable in the context of nonlinear eigenvalue problems. Instead, we use the
concept of minimal invariant pairs from [6,9].
Definition 2 A pair (X, S) ∈ C
n×k
× C
k×k
is called minimal if there is l ∈ N such
that the matrix
V
l
(X, S) =
⎡
⎢
⎢
⎢
⎣
X
XS
.
.
.
XS
l−1
⎤
⎥
⎥
⎥
⎦
(5)
has rank k. The smallest such l is called the minimality index of (X, S).
Example 3 For the example (2), the pair (X, S) with X =
11
11
and S = diag(3, 4)
is invariant and minimal with minimality index 2.
It has been shown in [6, Theorem 3] that any non-minimal pair can be turned into
a minimal one in the following sense. If V
l
(X, S) has rank
˜
k < k then there is a
minimal pair (
X,
S) ∈ C
n×
˜
k
× C
˜
k
×
˜
k
such that span
X = span X and span V
l
(
X,
S) =
span V
l
(X, S). The following Lemma reveals the connection of minimal invariant pairs
to the nonlinear eigenvalue problem (3).
Lemma 4 Let (X, S) ∈ C
n×k
× C
k×k
be a minimal invariant pair of (3). Then the
following statements hold.
1. For any invertible matrix Z ∈ C
k×k
, (XZ, Z
−1
SZ) is also a minimal invariant
pair of (3).
2. The eigenvalues of S are eigenvalues of (3).
123
358 D. Kressner
Proof 1. Using f
j
(Z
−1
SZ) = Z
−1
f
j
(S)Z, the relation (4) can be written as
A
1
XZf
1
(Z
−1
SZ)Z
−1
+ A
2
XZf
2
(Z
−1
SZ)Z
−1
+···
+ A
m
XZf
m
(Z
−1
SZ)Z
−1
= 0,
which is equivalent to
A
1
XZf
1
(Z
−1
SZ) + A
2
XZf
2
(Z
−1
SZ) +···+ A
m
XZf
m
(Z
−1
SZ) = 0, (6)
and shows that (XZ, Z
−1
SZ) is an invariant pair. Its minimality follows from
V
l
(XZ, Z
−1
SZ) = V
l
(X, S)Z.
2. By the Schur decomposition, we can choose Z orthogonal such that
S = Z
−1
SZ
is upper triangular with any eigenvalue λ of S appearing in the (1, 1) position
of
S. Setting x = XZe
1
, the first column of V
l
(Z
−1
SZ, XZ) has the entries
x, xλ, . . . , xλ
l−1
. Hence, x = 0 since otherwise V
l
(Z
−1
SZ, XZ) would be rank
deficient for any l. Moreover,
XZf
j
(Z
−1
SZ)e
1
= f
j
(λ)x
and thus the first column of (6)impliesthat(x,λ) is an eigenvector/eigenvalue
pair.
Let us briefly discuss the practical consequences of Lemma 4. Once a minimal invariant
pair is computed we can extract the corresponding eigenvalues of T (·) by computing
the eigenvalues of S. Moreover, if S admits a diagonalization Z
−1
SZ then the columns
of XZ contain the corresponding eigenvectors.
The following lemma shows that for checking minimality, it is sufficient to check
the rank of V
k
(X, S).
Lemma 5 If a pair (X, S) ∈ C
n×k
×C
k×k
is minimal then its minimality index cannot
exceed k.
Proof Since (X, S) is minimal, there is l ∈ N such that rank
(
V
l
(X, S)
)
= k.For
l ≤ k there is nothing to prove. For l > k, the Cayley-Hamilton theorem yields the
existence of coefficients α
ij
∈ C such that
XS
k+i
= α
i0
X + α
i1
XS+···+α
i,k−1
XS
k−1
, i ≥ 0.
Hence, there is a square invertible matrix W such that
WV
l
(X, S) =
V
k
(X, S)
0
,
implying rank
(
V
k
(X, S)
)
= k.
123
