We consider verifying the eigenvalues , counting multiplicity, in a prescribed interval of the generalized Hermitian eigenvalue problem
where , is positive semidefinite, and the matrix pencil () is regular, i.e, is not identically equal to zero. We call an eigenvalue and the corresponding eigenvector of the problem (1.1) or matrix pencil , interchangeably. Throughout, we assume that the number of eigenvalues in the interval is known to be and there do not exist eigenvalues of (1.1) at the end points , . We also denote the eigenvalues of (1.1) outside by (), where .
Previous studies of verified eigenvalue and eigenvector computations are classified into two categories: one is for the verification of specific eigenvalues and eigenvectors, and the other is for all the eigenvalues and eigenvectors at once. This study focuses on the former category for generalized Hermitian eigenvalue problems. For the purposes, different approaches have been taken. Rump regards a given eigenvalue problem as a system of nonlinear equations and uses Newton-like iterations for solving the equations to verify specific eigenpairs. See also . Behnke  uses a variational principle, and Yamamoto  uses Sylvester’s law of inertia. See [14, 7, 6]
for further studies and references therein. Verified eigenvalue computations arise in applications, e.g., from the numerical verification of a priori error estimations for finite element solutions[21, 19].
Our previous study proposes a verification method using complex moments . This method is based on an eigensolver , which reduces a given generalized Hermitian matrix eigenvalue problem into another generalized eigenvalue problem with block Hankel matrices, and evaluates all the errors in the reduction for verification. The errors are split into truncation errors in numerical quadrature and rounding errors. To evaluate the truncation error, an interval arithmetic-friendly formula is derived. This method is feasible even when is singular. Also, we develop an efficient technique to validate the solutions of linear systems of equations corresponding to each quadrature point. We call this method the Hankel matrix approach throughout.
This study improves its truncation error using the Rayleigh–Ritz procedure [18, 3] and halves the number of quadrature points required by the Hankel matrix approach to satisfy a prescribed quadrature error. This Rayleigh–Ritz procedure approach inherits features of the Hankel matrix approach, such as the efficient error evaluation technique for linear systems and the parameter tuning technique. This approach is also feasible for singular when verifying eigenvalues. Numerical experiments prove the feasibility of this concept and show the performance of the proposed method.
This paper is organized as follows. Section 2 presents the proposed method, derives computable error bounds for complex moments to justify it, and discusses implementation issues. Section 3 presents experimental results to illustrate the performance of the proposed method. Section 4 concludes the paper.
2 Rayleigh–Ritz procedure approach.
The Rayleigh–Ritz procedure projects a given eigenvalue problem into an (approximated) eigenspace of interest. We develop a Rayleigh–Ritz procedure version of the verified computation method for generalized Hermitian eigenvalue problems. We first review the Rayleigh–Ritz procedure approach of a projection method using complex moment [18, 3].
Define the th complex moment matrix by
on a positively oriented closed Jordan curve through the end points of the interval , where is the imaginary unit, and is the circle ratio. Then, using the matrix
we transform the eigenvalue problem (1.1) into a reduced eigenvalue problem
where is a shift parameter. By solving the transformed generalized eigenvalue problem (2.3), we obtain the eigenvalues of interest under certain conditions.
We then show the identity between the Rayleigh–Ritz procedure approach and the Hankel matrix approach . To this end, we rewrite the coefficient matrices of (2.3) below. Recall the Weierstrass canonical form of the matrix pencil [2, Proposition 7.8.3]. There exists a nonsingular matrix such that
where the th column of is the eigenvector corresponding to the eigenvalue , , and whose leading diagonal entries are the eigenvalues of (1.1). Here,
is the identity matrix anddenotes the direct sum of matrices. With this canonical form and the eigendecomposition
Caucy’s integral formula gives the th order complex moment
for , , , , where and . Hence, we rewrite the coefficient matrices of (2.3) as
for , , , . Here, we used the identity , in which the eigenvectors , , , are -orthonormal. Let be the reduced th complex moment given in [6, equation (2)]. Then, the identities
for , , , , , or
show that the Rayleigh–Ritz procedure and Hankel matrix approaches reduce the generalized eigenvalue problems (1.1) into the same eigenvalue problem with block Hankel matrices. The left-hand sides of (2.10) form the transformed matrices in the Rayleigh–Ritz procedure approach, whereas the right-hand sides of (2.10) form the transformed matrices in the Hankel matrix approach. We call these two approaches the complex moment approach throughout. Further, the following theorem justifies that these methods determine the eigenvalues and eigenvectors of (1.1).
Let be the number of eigenvalues of (1.1) in the region and be defined as in (2.2), and assume . Then, the eigenvalues of the regular part of the matrix pencil are the same as the eigenvalues of (1.1), , , , . Let be the eigenvector corresponding to the eigenvalue of . Then, is the eigenvector corresponding to the eigenvalue of (1.1).
The difference between the Rayleigh–Ritz and Hankel matrix approaches arises when approximating the integral (2.1) by using a numerical quadrature. Next, we evaluate the error in the Rayleigh–Ritz procedure approach, similarly to the previous study for the Hankel matrix approach [6, sections 2, 3].
2.1 -point quadrature rule.
The complex moment (2.1) is approximated by using the -point trapezoidal rule, taking a circle with center and radius in the complex plane
as the domain of integration . It follows from the error analysis in  that the -point trapezoidal rule with the equi-distributed quadrature points
approximates the complex moment as
The approximation is confirmed as for , , , and for , , , for .
2.2 Effect of eigenvalues inside and outside
To see the effect of the eigenvalues inside and outside the interval on the quadrature errors and for notational convenience, we split the complex moment into two
are associated with the eigenvalues inside and outside the interval , respectively, for , , , . Here, we used the notations
With the above approximation , we obtain the approximated transformation matrix
and split it into two , where
are associated with the eigenvalues inside and outside the region , respectively. With this approximated transformation matrix , the reduced complex moment is approximated as
The approximated reduced complex moment is split into two
are associated with the eigenvalues inside and outside the region , respectively, for , , , , .
Let and be the block Hankel matrices in (2.12). Then, in the Rayleigh–Ritz procedure approach, they are approximated as
For convenience, we split the approximated block Hankel matrices into two
are associated with the eigenvalues inside and outside the region , respectively,
2.3 Verification of eigenvalues.
As holds for the -orthonormality of the eigenvectors, we have
We now give a link between the coefficient matrices of (2.3) and their splittings.
Let with and for , , , , and be defined as in Lemma 2.1. Denote the th column vector of and by and , respectively, i.e., an expansion of the th column of by the eigenvectors, for , , , , where . Then, we have
Thanks to Theorem 2.2, we enclose instead of for , , , , . From the splitting (2.29), can be regarded as the truncated error for quadrature. Denote the quantity obtained by numerically computing by . Hereafter, we denote a numerically computed quantity that may suffer from rounding errors with a tilde.
Denote the interval matrix with radius and center at by . Then, the enclosure of is given by
for , , , .
The first enclosure of is obtained by the equality . The second enclosure is obtained by this equality and the inequality
Theorem 2.3 implies that to enclose , we can use and the truncated complex moment computed by using standard verification methods using interval arithmetic to obtain an enclosure of the truncation error . Theorem 2.3 readily gives the following enclosure:
An enclosure of is obtained as follows.
Let be a Hermitian positive semidefinite definite matrix. Assume and that satisfies . Then, in (2.19) is bounded by
for , , , .
Let . Then, applying the triangular inequality, we have
for . Noting the geometric series and applying the triangular inequality, we obtain
for , , , . Multiplied by the factor , we obtain
for , , , and . Here, the assumption ensures . Noting that the last expression is independent of the index , we have
The bound (2.50) for the proposed Rayleigh-Ritz procedure approach is twice sharper than the one for the Hankel matrix approach [6, Theorem 3.3] [6, Theorem 3.3], i.e., the proposed method requires half the number of quadrature points required by the Hankel matrix approach to allow the same amount of truncation errors. This observation is demonstrated in section 3.
2.4 Verification of eigenvectors.
To verify the eigenvectors of (1.1) via the Rayleigh–Ritz procedure approach as well as the Hankel matrix approach, we show the identity of the eigenvectors given by and .
Let . Then, from Lemma 2.1, it follows that
Because each eigencomponent of each column vector of is a scalar multiple of that of , ∎
Motivated by this theorem, we focus on verifying , instead of .
Then, we have the following enclosure of the approximated transformation matrix:
The proof is given similarly to that of Theorem 2.3. ∎
Assume that is a Hermitian and positive definite matrix. Assume and that satisfies . Then, defined in (2.26) is bounded as
for , , , .
Similarly to the proof of Theorem 2.4, we have
for , , , . Here, we used the -orthonormality of the eigenvectors . ∎
The evaluation of the numerical error in (2.63), i.e., for each , , , , involves the error evaluation of the solution
of the linear system of equations with multiple right-hand sides associated with
We present implementation issues of the proposed method. We assume that the numbers of and satisfy . Also, the proposed method needs to determine the number of the parameter . Each quadrature point gives rise to a linear system to solve. The evaluation of a solution for each linear system is the most expensive part, whereas the quadrature errors and reduces as the number of quadrature points increases (see Theorems 2.4 and 2.7). To achieve efficient verification, it is favorable to evaluate solutions of the linear systems as few as possible. Hence, there is a trade-off between the computational cost and quadrature error. The number of quadrature points
has been heuristically determined in the complex moment eivensolvers for numerical computations. For numerical verification, a reasonable numbercan be determined according to the quadrature error. The error bounds (2.50) and (2.64) can be used to determine a reasonable number of quadrature points. The least number of such that