1. Introduction
We consider the following rational eigenvalue problem (REP)
where
is a matrix rational function with respect to the scalar parameter
and the details for the degree of
can be seen in [
1]. As we know,
is an eigenvalue of the problem (
1) if and only if it satisfies the characteristic equation
where
denotes the determinant of its following matrix. The nonzero vector
and the two-tuple
are called as the corresponding eigenvector of
and an eigenpair of the REP (
1), respectively. The REP arises in a wide variety of applications including vibration of fluid–solid structures [
2], optimization of acoustic emissions of high-speed trains [
3], free vibration of plates with elastically attached masses [
4], free vibrations of a structure with a viscoelastic constitutive relation describing the behavior of a material [
5,
6], electronic structure calculations of quantum dots [
7,
8], and so on.
More precisely, in this paper we consider that
is shown as follows:
where
is a matrix polynomial,
and
are coprime scalar polynomials of degrees
and
with
, respectively, and
are all constant matrices.
At present, there are mainly three types of numerical methods to compute the eigenvalues of the REP (
1). The first type of numerical method is to solve the REP via a brute-force approach. That is to say, multiply the both sides of (
2) by all scalar polynomials
, which results in a
order polynomial eigenvalue problem (PEP). Nevertheless, these methods are not as efficient as required for the large-scale problems, especially when the term
is not small enough. Moreover, the corresponding PEP would have more extra eigenvalues than the original REP (
1). The second type of numerical method is to linearize the REP into a PEP with some specific tricks. For example, Su and Bai [
9] presented a linearization-based method by converting the REP into a well-studied PEP and preserved the structures and properties of the original REP. Dopico and González-Pizarro [
10] proposed a compact rational Krylov method for the large-scale REP. The third type of numerical method is to treat the REP as the general nonlinear eigenvalue problems, and solve them via a nonlinear eigensolver, see, e.g., [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24]. Although the abovementioned methods can solve the REP well, they ignore some structures and properties of the original rational eigenvalue problems. To overcome this disadvantage, it is necessary to study spectral properties and distribution of the REP first, and then try to put forward some effective numerical methods according to these properties. As far as we know, there are engineering applications that require the computation of only some of the eigenvalues lying within an interval [
5]. Therefore, in this paper we focus on some numerical methods to compute eigenpairs of the REP in an interval.
The rest of the paper is organized as follows.
Section 2 briefly introduces some preliminary results.
Section 3 discusses the spectral properties and the distribution of the rational eigenvalues. In
Section 4, we develop two numerical methods for solving the symmetric REP based on the spectral properties. Some numerical examples are given to show the effectiveness of the proposed methods in
Section 5. Finally, we give some concluding remarks in
Section 6.
For convenience, we use the following notations: I denotes the identity matrix of suitable size. denotes the jth column of the identity matrix I. The superscript T denotes the transpose of a vector or a matrix, respectively. denotes the Euclidean norm of a vector or a matrix. denotes the Frobenius norm of a matrix. denotes the inner product of vector x and vector y.
2. Preliminaries
In this section, to facilitate the theoretical analysis and further obtain the main results for rational eigenvalue problems (
1) and (
2), the following useful assumptions are addressed.
Hypothesis 1 (
H1)
. Coefficient matrices are the symmetric positive definite matrices with and is symmetric.
Hypothesis 2 (
H2)
. Matrices are the low-rank symmetric positive semidefinite matrices with .
Hypothesis 3 (
H3)
. Rational functions are monotonically increasing functions with respect to parameter λ on the intervals separated by the zeros of polynomials where .
Remark 1. The following example provides an intuitive illustration for assumptionH3.
Example 1. Assume that , where and . We can easily get that where . Therefore, we have that are monotonically increasing functions with respect to parameter λ on the intervals where .
For the REP (
1) and (
2), in this paper we only discuss the eigenvalue distribution on the positive semi-real axis, namely
. Assume that all zeros of
with
on the real positive semiaxis are arranged in the following order:
Set where . Then we have and if .
As long as the matrices
and
are symmetrical for all
and
, we can define the Rayleigh functional
for the REP. That is to say, if
satisfies the following equation
is the Rayleigh functional of
. Notice that in the linear case
, it is exactly the Rayleigh quotient. Let
, then
is a root of
= 0. Because
is positive definite with
,
is positive semidefinite and
is monotonically increasing function, it is easy to verify that
is a monotone increasing function on the intervals separated by the zeros of polynomials
. Therefore,
where
.
For each fixed
, we consider the following standard eigenvalue problem (SEP):
We can easily see that if
is an eigenvalue of the REP (
1) and (
2),
is an eigenvalue of the above SEP (
5). Conversely, it is also true. Therefore, the eigenvalue of the REP (
1) and (
2) can be characterized by the zero eigenvalue of the SEP (
5). For the standard eigenvalue problem, we have the minmax principle
where
represents the set of Hilbert subspaces with dimension
j of
. Similarly, we have the minmax principle of the REP
where
denotes the domain of the Rayleigh functional
which satisfies (
3). For a more detailed discussion, see, e.g., [
17,
18].
3. Spectral Properties and Distribution of the REPs
According to the assumption (
H2), we know that the matrices
are low-rank. Hence, the REP (
2) can be regarded as a low-rank perturbation of the following PEP [
25]
where
with
and
.
For any
and any value
in the interval
, the REP (
1) and (
2) has the corresponding PEP (
7). Assume that the PEP (
7) has an eigenvalue
in the interval
. Then we will prove that the eigenvalue
of the REP (
1) and (
2) is between
and
. Before giving this theorem, we first show some lemmas.
Lemma 1. Assume that and represent any pair of scalar polynomials and with , respectively. For any with , we havewhere with and standing for the coefficients of and , respectively. Proof. Without losing generality, we let
and
. Hence,
Let
and we have
.
Because , then . Based on the assumption we have . Thus, the conclusion holds true. □
For the PEP (
7), the Rayleigh functional
should satisfy
, namely,
Lemma 2. Assume that and there exists a vector , such that with where . Then andwhere is the domain of the Rayleigh functional which satisfies (3) in . Proof. With the fact that
and the relation (
8), we have
It follows from Lemma 1 and the assumption (H2) that if and if . Similarly, it can be proved that if and if . Finally, because is continuous, we have and , which completes the proof. □
Theorem 1. Assume that (H1)–(H3) hold, and let with . Assume that and contains the j-th eigenvalue of the PEP (7). Then the REP (1) and (2) has a corresponding eigenvalue and . Proof. We first show that there exists a subspace
, such that
In fact, suppose that there exist
and
, such that
Because
and
, we have
from Lemma 2. Then
. For any
, we have
. Therefore,
. That is,
Set
. Then it is easy to obtain that
. That is,
. It follows from (
4) that
for any
. Hence,
.
In the following, we show that for any
, if
, we have
We prove this result by reduction to absurdity. Suppose that there exists such that , but . Let such that . Then we have . Actually, if , it is easy to get and .
That is,
contradicting the fact that
. Hence,
.
Set
. Then we have
. Moreover, because
, it is easy to obtain
For any
, we let
and
. Then
. Because
, it follows from (
4) that
. There exists
such that
. Thus,
and
which conflicts with the assumption.
To summarise, we have
which completes the proof. □
Actually, eigenvalue
of the REP (
1) and (
2) is the function with respect to
, which implies that
. The following theorem will elaborate the continuity of the function
.
Theorem 2. Assume that H1–H3 hold, then is a continuous decreasing function with respect to κ.
Proof. Let
. For any
, there exists
such that
when
with
small enough. Because the eigenvalue is a continuous function with respect to the elements of its matrix [
26], we have that
is a continuous function with respect to
.
It follows from (
8) that
is the root of the following polynomial equation
where
with
and
With fixed
x,
remains unchanged where
. Moreover, from the assumption (
H3), we know that
is an increasing function of
. Then we can easily prove that
is a decreasing function of
. Finally, through the minmax principle (
6), we can conclude that
is a continuous decreasing function of
, which completes the proof. □
Theorems 1 and 2 show that if the REP (
1) and (
2) have an eigenvalue
, there must be such a value
and one eigenvalue
of the PEP (
7) in the interval
. Conversely, if the REP (
1) and (
2) have no eigenvalues in the interval
, the PEP (
7) will not have any eigenvalues in this interval even if the values
in the interval
are taken all over. Therefore, there exists a one-to-many relationship between
and
.
On the other hand, suppose that there exists
, then the PEP (
7) has two unequal eigenvalues
. If
and
are fixed points of
and
, respectively, we have
. In fact, because the eigenvalue is a continuous function with respect to the elements of its matrix, the multiplicity of the original eigenvalue will not change with the change of
. That is,
holds. Note that if
, we have
. Therefore, there is one-to-one correlation between
and
.
To summarise, we can obtain the following theorems.
Theorem 3. There is a one-to-one relationship between the eigenvalue of the REP (1) and (2) and the eigenvalue of the PEP (7) in with . Theorem 4. Assume that . Let , then there are exactly eigenvalues in the semi-interval .
4. Numerical Methods for Solving the REPs
In this section, based on the above spectral distribution of the REP we discuss the numerical methods for solving the REP (
1) and (
2). Given a
, we can find an eigenvalue
of the PEP (
7). Here, how to select the next new value
is the key to propose the novel numerical algorithms. Because
is a continuous decreasing function of
and
is a fixed point of
, we can choose the newest
by a certain fixed-point algorithm. For simplicity, we first consider to choose
via dichotomy as follows:
Therefore, we derive the following numerical method (Algorithm 1) for solving the REP (
1) and (
2).
Algorithm 1: Dichotomy iteration method for the REP (1) and (2) |
Input: rational matrix function , the target point and the tolerance . Output: the approximate eigenvalue closest to .
|
Remark 2. In actual computation, for the small-scale PEP (7) the classical approach is to turn it into a generalized eigenvalue problem (GEP) via linearization, or solve it directly by the in-built function of Matlab. For the large-scale ones, we can adopt the partially orthogonal projection method [27] to solve it. Remark 3. Assume that x is an eigenvector of μ for the PEP (7). If , x is also the eigenvector corresponding to λ of the REP (1) and (2). Therefore, we can use the Rayleigh functional to accelerate κ as follows: Thus another numerical method (Algorithm 2) for solving the REP (
1) and (
2) can be summarized as follows.
Algorithm 2: Rayleigh functional iteration method for the REP (1) and (2) |
Input: rational matrix function , the target point and the tolerance . Output: the approximate eigenvalue closest to .
|
5. Numerical Results
In this section, we report some numerical examples to show the effectiveness of the proposed Algorithms 1 and 2. All computations are performed under Matlab (version R2019a). In our examples,
is an exact eigenvalue of the REP (
1) and (
2), and
is an approximate eigenvalue computed by Algorithm 1 or Algorithm 2. CPU denotes the CPU time (in seconds) for computing an approximate solution, and Iter denotes the number of iteration steps. The stopping tolerance for the residual norm is chosen to be
.
Example 2 ([
9])
. We consider the following REP: where A and B are the positive definite tridiagonal matrices defined as with and . We can easily check that the above eigenvalue problem meets the assumptions
H1–
H3 in
Section 2. Let
and
. Here we are interested in computing all eigenvalues of the REP (
11) in the interval
, and we can divide it into two small intervals such as
and
.
It is easy to verify that
,
,
and
are the eigenvalues of the REP (
11) in the interval
because the corresponding smallest singular values of
are less than
.
Through Theorem 4, we have that the numbers of eigenvalues of the REP (
11) in
and
are 1, and 3, respectively. The above result is completely consistent with the actual distribution of eigenvalues for the REP (
11).
In the following, we choose different
in
and
such as
and apply Algorithms 1 and 2 to compute all eigenvalues of the (
11) in
and
.
The numerical results for Algorithms 1 and 2 are reported in
Table 1, which shows that the proposed methods are very useful and efficient to solve rational eigenvalue problems in one interval. Moreover, Algorithm 2 requires less CPU and iteration steps than Algorithm 1. Moreover, the numerical results remain the same when
n and
take the other different values.