- Source: Nonlinear eigenproblem
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form
M
(
λ
)
x
=
0
,
{\displaystyle M(\lambda )x=0,}
where
x
≠
0
{\displaystyle x\neq 0}
is a vector, and
M
{\displaystyle M}
is a matrix-valued function of the number
λ
{\displaystyle \lambda }
. The number
λ
{\displaystyle \lambda }
is known as the (nonlinear) eigenvalue, the vector
x
{\displaystyle x}
as the (nonlinear) eigenvector, and
(
λ
,
x
)
{\displaystyle (\lambda ,x)}
as the eigenpair. The matrix
M
(
λ
)
{\displaystyle M(\lambda )}
is singular at an eigenvalue
λ
{\displaystyle \lambda }
.
Definition
In the discipline of numerical linear algebra the following definition is typically used.
Let
Ω
⊆
C
{\displaystyle \Omega \subseteq \mathbb {C} }
, and let
M
:
Ω
→
C
n
×
n
{\displaystyle M:\Omega \rightarrow \mathbb {C} ^{n\times n}}
be a function that maps scalars to matrices. A scalar
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
is called an eigenvalue, and a nonzero vector
x
∈
C
n
{\displaystyle x\in \mathbb {C} ^{n}}
is called a right eigevector if
M
(
λ
)
x
=
0
{\displaystyle M(\lambda )x=0}
. Moreover, a nonzero vector
y
∈
C
n
{\displaystyle y\in \mathbb {C} ^{n}}
is called a left eigevector if
y
H
M
(
λ
)
=
0
H
{\displaystyle y^{H}M(\lambda )=0^{H}}
, where the superscript
H
{\displaystyle ^{H}}
denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to
det
(
M
(
λ
)
)
=
0
{\displaystyle \det(M(\lambda ))=0}
, where
det
(
)
{\displaystyle \det()}
denotes the determinant.
The function
M
{\displaystyle M}
is usually required to be a holomorphic function of
λ
{\displaystyle \lambda }
(in some domain
Ω
{\displaystyle \Omega }
).
In general,
M
(
λ
)
{\displaystyle M(\lambda )}
could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix.
Definition: The problem is said to be regular if there exists a
z
∈
Ω
{\displaystyle z\in \Omega }
such that
det
(
M
(
z
)
)
≠
0
{\displaystyle \det(M(z))\neq 0}
. Otherwise it is said to be singular.
Definition: An eigenvalue
λ
{\displaystyle \lambda }
is said to have algebraic multiplicity
k
{\displaystyle k}
if
k
{\displaystyle k}
is the smallest integer such that the
k
{\displaystyle k}
th derivative of
det
(
M
(
z
)
)
{\displaystyle \det(M(z))}
with respect to
z
{\displaystyle z}
, in
λ
{\displaystyle \lambda }
is nonzero. In formulas that
d
k
det
(
M
(
z
)
)
d
z
k
|
z
=
λ
≠
0
{\displaystyle \left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0}
but
d
ℓ
det
(
M
(
z
)
)
d
z
ℓ
|
z
=
λ
=
0
{\displaystyle \left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0}
for
ℓ
=
0
,
1
,
2
,
…
,
k
−
1
{\displaystyle \ell =0,1,2,\dots ,k-1}
.
Definition: The geometric multiplicity of an eigenvalue
λ
{\displaystyle \lambda }
is the dimension of the nullspace of
M
(
λ
)
{\displaystyle M(\lambda )}
.
Special cases
The following examples are special cases of the nonlinear eigenproblem.
The (ordinary) eigenvalue problem:
M
(
λ
)
=
A
−
λ
I
.
{\displaystyle M(\lambda )=A-\lambda I.}
The generalized eigenvalue problem:
M
(
λ
)
=
A
−
λ
B
.
{\displaystyle M(\lambda )=A-\lambda B.}
The quadratic eigenvalue problem:
M
(
λ
)
=
A
0
+
λ
A
1
+
λ
2
A
2
.
{\displaystyle M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.}
The polynomial eigenvalue problem:
M
(
λ
)
=
∑
i
=
0
m
λ
i
A
i
.
{\displaystyle M(\lambda )=\sum _{i=0}^{m}\lambda ^{i}A_{i}.}
The rational eigenvalue problem:
M
(
λ
)
=
∑
i
=
0
m
1
A
i
λ
i
+
∑
i
=
1
m
2
B
i
r
i
(
λ
)
,
{\displaystyle M(\lambda )=\sum _{i=0}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),}
where
r
i
(
λ
)
{\displaystyle r_{i}(\lambda )}
are rational functions.
The delay eigenvalue problem:
M
(
λ
)
=
−
I
λ
+
A
0
+
∑
i
=
1
m
A
i
e
−
τ
i
λ
,
{\displaystyle M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },}
where
τ
1
,
τ
2
,
…
,
τ
m
{\displaystyle \tau _{1},\tau _{2},\dots ,\tau _{m}}
are given scalars, known as delays.
Jordan chains
Definition: Let
(
λ
0
,
x
0
)
{\displaystyle (\lambda _{0},x_{0})}
be an eigenpair. A tuple of vectors
(
x
0
,
x
1
,
…
,
x
r
−
1
)
∈
C
n
×
C
n
×
⋯
×
C
n
{\displaystyle (x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}}
is called a Jordan chain if
∑
k
=
0
ℓ
M
(
k
)
(
λ
0
)
x
ℓ
−
k
=
0
,
{\displaystyle \sum _{k=0}^{\ell }M^{(k)}(\lambda _{0})x_{\ell -k}=0,}
for
ℓ
=
0
,
1
,
…
,
r
−
1
{\displaystyle \ell =0,1,\dots ,r-1}
, where
M
(
k
)
(
λ
0
)
{\displaystyle M^{(k)}(\lambda _{0})}
denotes the
k
{\displaystyle k}
th derivative of
M
{\displaystyle M}
with respect to
λ
{\displaystyle \lambda }
and evaluated in
λ
=
λ
0
{\displaystyle \lambda =\lambda _{0}}
. The vectors
x
0
,
x
1
,
…
,
x
r
−
1
{\displaystyle x_{0},x_{1},\dots ,x_{r-1}}
are called generalized eigenvectors,
r
{\displaystyle r}
is called the length of the Jordan chain, and the maximal length a Jordan chain starting with
x
0
{\displaystyle x_{0}}
is called the rank of
x
0
{\displaystyle x_{0}}
.
Theorem: A tuple of vectors
(
x
0
,
x
1
,
…
,
x
r
−
1
)
∈
C
n
×
C
n
×
⋯
×
C
n
{\displaystyle (x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}}
is a Jordan chain if and only if the function
M
(
λ
)
χ
ℓ
(
λ
)
{\displaystyle M(\lambda )\chi _{\ell }(\lambda )}
has a root in
λ
=
λ
0
{\displaystyle \lambda =\lambda _{0}}
and the root is of multiplicity at least
ℓ
{\displaystyle \ell }
for
ℓ
=
0
,
1
,
…
,
r
−
1
{\displaystyle \ell =0,1,\dots ,r-1}
, where the vector valued function
χ
ℓ
(
λ
)
{\displaystyle \chi _{\ell }(\lambda )}
is defined as
χ
ℓ
(
λ
)
=
∑
k
=
0
ℓ
x
k
(
λ
−
λ
0
)
k
.
{\displaystyle \chi _{\ell }(\lambda )=\sum _{k=0}^{\ell }x_{k}(\lambda -\lambda _{0})^{k}.}
Mathematical software
The eigenvalue solver package SLEPc contains C-implementations of many numerical methods for nonlinear eigenvalue problems.
The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties.
The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics combined with contour integration techniques.
The MATLAB toolbox NLEIGS contains an implementation of fully rational Krylov with a dynamically constructed rational interpolant.
The MATLAB toolbox CORK contains an implementation of the compact rational Krylov algorithm that exploits the Kronecker structure of the linearization pencils.
The MATLAB toolbox AAA-EIGS contains an implementation of CORK with rational approximation by set-valued AAA.
The MATLAB toolbox RKToolbox (Rational Krylov Toolbox) contains implementations of the rational Krylov method for nonlinear eigenvalue problems as well as features for rational approximation.
The Julia package NEP-PACK contains many implementations of various numerical methods for nonlinear eigenvalue problems, as well as many benchmark problems.
The review paper of Güttel & Tisseur contains MATLAB code snippets implementing basic Newton-type methods and contour integration methods for nonlinear eigenproblems.
Eigenvector nonlinearity
Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function
M
{\displaystyle M}
maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices.
References
Further reading
Françoise Tisseur and Karl Meerbergen, "The quadratic eigenvalue problem," SIAM Review 43 (2), 235–286 (2001) (link).
Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," Journal of Computational and Applied Mathematics 123, 35–65 (2000).
Philippe Guillaume, "Nonlinear eigenproblems," SIAM Journal on Matrix Analysis and Applications 20 (3), 575–595 (1999) (link).
Cedric Effenberger, "Robust solution methods fornonlinear eigenvalue problems", PhD thesis EPFL (2013) (link)
Roel Van Beeumen, "Rational Krylov methods fornonlinear eigenvalue problems", PhD thesis KU Leuven (2015) (link)