Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh-Ritz procedure for generalized Hermitian eigenvalue problems

1 Introduction

We consider verifying the $m$ eigenvalues $λ_{i}$ , counting multiplicity, in a prescribed interval $Ω = [a, b] \subset R$ of the generalized Hermitian eigenvalue problem

A x_{i} = λ_{i} B x_{i}, x_{i} \in C^{n} ∖ {0}, λ_{1} \leq λ_{2} \leq \dots \leq λ_{m},

(1.1)

where $A = A^{H} \in C^{n \times n}$ , $B = B^{H} \in C^{n \times n}$ is positive semidefinite, and the matrix pencil $z B - A$ ( $z \in C$ ) is regular, i.e, $det (z B - A)$ is not identically equal to zero. We call $λ_{i}$ an eigenvalue and $x_{i}$ the corresponding eigenvector of the problem (1.1) or matrix pencil $z B - A$ , $z \in C$ interchangeably. Throughout, we assume that the number of eigenvalues in the interval $Ω$ is known to be $m$ and there do not exist eigenvalues of (1.1) at the end points $a$ , $b \in R$ . We also denote the eigenvalues of (1.1) outside $Ω$ by $λ_{i}$ ( $i = m + 1, m + 2, \dots, r$ ), where $r = r a n k B$ .

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

[12] 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 [13]. Behnke [1] uses a variational principle, and Yamamoto [20] 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 [6]. This method is based on an eigensolver [17], 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 $B$ 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 $B$ 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

[6]. We first review the Rayleigh–Ritz procedure approach of a projection method using complex moment [18, 3].

Define the $k$ th complex moment matrix by

M_{k} = \frac{1}{2 π i} \oint_{Γ} {(z - γ)}^{k} (z B - A)^{- 1} d z, k = 0, 1, 2, \dots, M - 1

(2.1)

on a positively oriented closed Jordan curve $Γ$ through the end points of the interval $Ω = [a, b]$ , where $i = \sqrt{- 1}$ is the imaginary unit, and $π$ is the circle ratio. Then, using the matrix

S = [S_{0}, S_{1}, \dots, S_{M - 1}], S_{k} = M_{k} B V, k = 0, 1, 2, \dots, M - 1,

(2.2)

we transform the eigenvalue problem (1.1) into a reduced eigenvalue problem

S^{H} (A - γ B) S y = (λ - γ) S^{H} B S y, x = S y, y \in C^{n} ∖ {0},

(2.3)

where $γ \in R$ 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 [17]. To this end, we rewrite the coefficient matrices of (2.3) below. Recall the Weierstrass canonical form of the matrix pencil $z B - A$ [2, Proposition 7.8.3]. There exists a nonsingular matrix $X \in C^{n \times n}$ such that

X^{H} (z B - A) X = z I_{0} - Λ,

(2.4)

where the $i$ th column of $X$ is the eigenvector $x_{i}$ corresponding to the eigenvalue $λ_{i}$ , $I_{o} = I_{r} \oplus O \in R^{n \times n}$ , and $Λ = d i a g (λ_{1}, λ_{2}, \dots, λ_{r}) \oplus I_{n - r} \in R^{n \times n}$ whose leading $r$ diagonal entries are the eigenvalues of (1.1). Here, $I_{m} \in R^{m \times m}$

is the identity matrix and

\oplus

denotes the direct sum of matrices. With this canonical form and the eigendecomposition

	$(z B - A)^{- 1}$	$= X (z I_{0} - Λ)^{- 1} X^{H}$		(2.5)
		$= r \sum i = 1 (z - λ_{i})^{- 1} x_{i} {x_{i}}^{H},$		(2.6)

Caucy’s integral formula gives the $k$ th order complex moment

$M_{k}$	$= r \sum i = 1 [\frac{1}{2 π i} \oint_{Γ} (z - γ)^{k} (z - λ_{i})^{- 1} d z] x_{i} {x_{i}}^{H}$	(2.7)
	$= m \sum i = 1 (λ_{i} - γ)^{k} x_{i} {x_{i}}^{H}$	(2.8)
	$= X_{Ω} {(Λ_{Ω} - γ I_{m})}^{k} X_{Ω}^{H}$	(2.9)

for $k = 0$ , $1$ , $\dots$ , $M - 1$ , where $X_{Ω} = [x_{1}, x_{2}, \dots, x_{m}]$ and $Λ_{Ω} = d i a g (λ_{1}, λ_{2}, \dots, λ_{m})$ . Hence, we rewrite the coefficient matrices of (2.3) as

	${S_{k}}^{H} (A - γ B) S_{ℓ}$	$= V^{H} B X_{Ω} (Λ_{Ω} - γ I_{m})^{k} [X_{Ω}^{H} (A - γ B) X_{Ω}] {(Λ_{Ω} - γ I_{m})}^{ℓ} X_{Ω}^{H} B V$
		$= V^{H} B X_{Ω} (Λ_{Ω} - γ I_{m})^{k + ℓ + 1} X_{Ω}^{H} B V$

and

	${S_{k}}^{H} B S_{ℓ}$	$= V^{H} B X_{Ω} (Λ_{Ω} - γ I_{m})^{k} ({X_{Ω}}^{H} B X_{Ω}) (Λ_{Ω} - γ I_{m})^{ℓ} X_{Ω} B V$
		$= V^{H} B X_{Ω} (Λ_{Ω} - γ I_{m})^{k + ℓ} X_{Ω} B V$

for $k, ℓ = 0$ , $1$ , $\dots$ , $M - 1$ . Here, we used the identity ${X_{Ω}}^{H} B X_{Ω} = I_{m}$ , in which the eigenvectors $x_{1}$ , $x_{2}$ , $\dots$ , $x_{m}$ are $B$ -orthonormal. Let $M_{k} = V^{H} B M_{k} B V$ be the reduced $k$ th complex moment given in [6, equation (2)]. Then, the identities

{S_{k}}^{H} (A - γ B) S_{ℓ} = M_{k + ℓ + 1}, {S_{k}}^{H} B S_{ℓ} = M_{k + ℓ}

(2.10)

for $k$ , $ℓ = 0$ , $1$ , $\dots$ , $M - 1$ , or

	$S^{H} (A - γ B) S$	$= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} M_{1} & M_{2} & \dots & M_{M} M_{2} & M_{3} & M_{M + 1} ⋮ & ⋱ & ⋮ M_{M} & M_{M + 1} & \dots & M_{2 M - 1} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦,$		(2.11)
	$S^{H} B S$	$= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} M_{0} & M_{1} & \dots & M_{M - 1} M_{1} & M_{2} & M_{M} ⋮ & ⋱ & ⋮ S_{M - 1}^{H} B S_{0} & S_{M - 1}^{H} B S_{1} & \dots & M_{2 M - 2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$		(2.12)

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).

Theorem 2.1 ([4, Theorem 7], [5, Theorem 3]).

Let $m$ be the number of eigenvalues of (1.1) in the region $Ω$ and $S \in C^{n \times L}$ be defined as in (2.2), and assume $r a n k S = m$ . Then, the eigenvalues of the regular part of the matrix pencil $S^{H} (A - z B) S$ are the same as the eigenvalues $λ_{i}$ of (1.1), $i = 1$ , $2$ , $\dots$ , $m$ . Let $u_{i}$ be the eigenvector corresponding to the eigenvalue $λ_{i}$ of $S^{H} (A - z B) S$ . Then, $x_{i} = S u_{i}$ is the eigenvector corresponding to the eigenvalue $λ_{i}$ 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 $N$ -point quadrature rule.

The complex moment (2.1) is approximated by using the $N$ -point trapezoidal rule, taking a circle with center $γ$ and radius $ρ$ in the complex plane

Γ = {z \in C | z = γ + ρ e x p (i θ), θ \in R}, γ = \frac{b + a}{2}, ρ = \frac{b - a}{2}

(2.13)

as the domain of integration $Γ$ . It follows from the error analysis in [8] that the $N$ -point trapezoidal rule with the equi-distributed quadrature points

z_{j} = γ + ρ e x p (i θ_{j}), θ_{j} = \frac{2 j - 1}{N} π, j = 1, 2, \dots, N

(2.14)

approximates the complex moment $M_{k}$ as

M_{k} ≃ M_{k}^{(N)} = r \sum i = 1 (λ_{i} - γ)^{k} d_{i}^{(N)} x_{i} x_{i}^{H},

(2.15)

where

d_{i}^{(N)} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1}{1 - {(\frac{λ_{i} - γ}{ρ})}^{N}}, & i = 1, 2, \dots, m, \frac{- {(\frac{ρ}{λ_{i} - γ})}^{N}}{1 - {(\frac{ρ}{λ_{i} - γ})}^{N}}, & i = m + 1, m + 2, \dots, r . \end{matrix}

(2.16)

The approximation $M_{k} ≃ M_{k}^{(N)}$ is confirmed as $d_{i}^{(N)} \to 1$ for $i = 1$ , $2$ , $\dots$ , $m$ and $d_{i}^{(N)} \to 0$ for $i = m + 1$ , $m + 2$ , $\dots$ , $r$ for $N \to \infty$ .

2.2 Effect of eigenvalues inside and outside $Ω$

To see the effect of the eigenvalues inside and outside the interval $Ω$ on the quadrature errors and for notational convenience, we split the complex moment into two

M_{k}^{(N)} = M_{k, i n}^{(N)} + M_{k, o u t}^{(N)}

(2.17)

where

	$M_{k, i n}^{(N)}$	$= X_{Ω} (Λ_{Ω} - γ I_{m})^{k} D_{Ω}^{(N)} {X_{Ω}}^{H},$		(2.18)
	$M_{k, o u t}^{(N)}$	$= X_{Ω^{c}} (Λ_{Ω^{c}} - γ I_{r - m})^{k} D_{Ω^{c}}^{(N)} {X_{Ω^{c}}}^{H}$		(2.19)

are associated with the eigenvalues inside and outside the interval $Ω$ , respectively, for $k = 0$ , $1$ , $\dots$ , $M - 1$ . Here, we used the notations

$D_{Ω}^{(N)}$	$= d i a g (d_{1}^{(N)}, d_{2}^{(N)}, \dots, d_{m}^{(N)}),$	(2.20)
$D_{Ω^{c}}^{(N)}$	$= d i a g (d_{m + 1}^{(N)}, d_{m + 2}^{(N)}, \dots, d_{r}^{(N)}),$	(2.21)
$X_{Ω^{c}}$	$= [x_{m + 1}, x_{m + 1}, \dots, x_{r}],$	(2.22)
$Λ_{Ω^{c}}$	$= d i a g (λ_{m + 1}, λ_{m + 2}, \dots, λ_{r}) .$	(2.23)

With the above approximation $M_{k} ≃ M_{k}^{(N)}$ , we obtain the approximated transformation matrix

S_{k} ≃ S_{k}^{(N)} = M_{k}^{(N)} B V

(2.24)

and split it into two $S_{k}^{(N)} = S_{k, i n}^{(N)} + S_{k, o u t}^{(N)}$ , where

	$S_{k, i n}^{(N)}$	$= M_{k, i n}^{(N)} B V,$		(2.25)
	$S_{k, o u t}^{(N)}$	$= M_{k, o u t}^{(N)} B V$		(2.26)

are associated with the eigenvalues inside and outside the region $Ω$ , respectively. With this approximated transformation matrix $S_{k}^{(N)}$ , the reduced complex moment $M_{k + ℓ + 1}$ is approximated as

	$M_{k + ℓ + 1}$	$≃ M_{k + ℓ + 1}^{(N)}$		(2.27)
		$= (S_{k}^{(N)})^{H} (A - γ B) S_{ℓ}^{(N)} .$		(2.28)

The approximated reduced complex moment is split into two

M_{k + ℓ + 1}^{(N)} = M_{k + ℓ + 1, i n}^{(N)} + M_{k + ℓ + 1, o u t}^{(N)},

(2.29)

where

	$M_{k + ℓ + 1, i n}^{(N)}$	$= (S_{k, i n}^{(N)})^{H} (A - γ B) S_{ℓ, i n}^{(N)},$		(2.30)
	$M_{k + ℓ + 1, o u t}^{(N)}$	$= (S_{k, o u t}^{(N)})^{H} (A - γ B) S_{ℓ, o u t}^{(N)}$		(2.31)

are associated with the eigenvalues inside and outside the region $Ω$ , respectively, for $k$ , $ℓ = 0$ , $1$ , $\dots$ , $M - 1$ .

Let $H_{M}^{<} = S^{H} (A - γ B) S$ and $H_{M} = S^{H} B S$ be the block Hankel matrices in (2.12). Then, in the Rayleigh–Ritz procedure approach, they are approximated as

	$H_{M}^{<}$	$≃ H_{M}^{<, (N)} = (S^{(N)})^{H} (A - γ B) S^{(N)},$		(2.32)
	$H_{M}$	$≃ H_{M}^{(N)} = (S^{(N)})^{H} B S^{(N)} .$		(2.33)

For convenience, we split the approximated block Hankel matrices into two

H_{M}^{<, (N)} = H_{M, i n}^{<, (N)} + H_{M, o u t}^{<, (N)}, H_{M}^{(N)} = H_{M, i n}^{(N)} + H_{M, o u t}^{(N)},

(2.34)

where

H_{M, i n}^{<, (N)} = (S_{i n}^{(N)})^{H} (A - γ B) S_{i n}^{(N)}, H_{M, o u t}^{<, (N)} = (S_{o u t}^{(N)})^{H} (A - γ B) S_{o u t}^{(N)}

(2.35)

and

H_{M, i n}^{(N)} = (S_{i n}^{(N)})^{H} B S_{i n}^{(N)}, H_{M, o u t}^{(N)} = (S_{o u t}^{(N)})^{H} B S_{o u t}^{(N)} .

(2.36)

are associated with the eigenvalues inside and outside the region $Ω$ , respectively,

2.3 Verification of eigenvalues.

To validate the eigenvalues of (2.3), it is straightforward to enclose the coefficient matrices of (2.3). Nevertheless, we exploit alternative quantities. To this end, we prepare the following lemma.

Lemma 2.1.

Let $D = D_{1} \oplus D_{2} \in R^{n \times n}$ be a diagonal matrix with $D_{1} \in R^{m \times m}$

and the column vectors of

X \in C^{n \times n}

and

X_{Ω} \in C^{n \times m}

be the eigenvectors

x_{1}

x_{2}

\dots

x_{n}

and

x_{1}

x_{2}

\dots

x_{m}

of (1.1), respectively. Then, we have

D_{1} {X_{Ω}}^{H} B X = {X_{Ω}}^{H} B X D .

(2.37)

Proof.

As ${X_{Ω}}^{H} B X = [I_{m}, O]$ holds for the $B$ -orthonormality of the eigenvectors, we have

$D_{1} {X_{Ω}}^{H} B X$	$= D_{1} [I_{m}, O]$	(2.38)
	$= [I_{m}, O] D$	(2.39)
	$= {X_{Ω}}^{H} B X D .$	(2.40)

∎

We now give a link between the coefficient matrices of (2.3) and their splittings.

Theorem 2.2.

Let $B$ be a Hermitian positive semidefinite matrix and $S$ be defined as in (2.2) and

(2.41)

where $S_{k, i n}^{(N)}$ is as defined in (2.25). Assume $r a n k S = m$ . Then, the matrix pencils $S^{H} (A - z B) S$ and $(S_{i n}^{(N)})^{H} (A - z B) S_{i n}^{(N)}$ have the same eigenvalues.

Proof.

Let $D = d i a g (d_{1}, d_{2}, \dots, d_{n})$ with $d_{i} \in C$ and $d_{i} = 1$ for $i = r + 1$ , $r + 2$ , $\dots$ , $n$ , and $X \in C^{n \times n}$ be defined as in Lemma 2.1. Denote the $j$ th column vector of $V = X C \in C^{n \times L}$ and $V^{'} = X D C \in C^{n \times L}$ by $v_{j} = \sum_{i = 1}^{n} c_{i j} x_{i}$ and $v_{i}^{'} = \sum_{i = 1}^{n} c_{i j} d_{i} x_{i}$ , respectively, i.e., an expansion of the $j$ th column of $V$ by the eigenvectors, for $j = 1$ , $2$ , $\dots$ , $L$ , where $C = (c_{i j}) \in C^{n \times L}$ . Then, we have

		$= V^{H} B X_{Ω} D_{Ω} (Λ_{Ω} - z I_{m})^{k + ℓ + 1} D_{Ω} {X_{Ω}}^{H} B V$		(2.42)
		$= {V^{'}}^{H} B X_{Ω} (Λ_{Ω} - z I_{m})^{k + ℓ + 1} {X_{Ω}}^{H} B V^{'}$		(2.43)

for $k, ℓ = 0, 1, \dots, M - 1$ . Because Theorem 2.1 holds irrespective of the scalar multiples of the eigenvectors involved in the columns of $V$ , (2.41) holds. ∎

Thanks to Theorem 2.2, we enclose $M_{k + ℓ + 1, i n}^{(N)}$ instead of $M_{k + ℓ + 1}$ for $k$ , $ℓ = 0$ , $1$ , $\dots$ , $M - 1$ . From the splitting (2.29), $M_{k + ℓ + 1, o u t}^{(N)}$ can be regarded as the truncated error for quadrature. Denote the quantity obtained by numerically computing $M_{k}^{(N)}$ by ${~ M}_{k}^{(N)}$ . Hereafter, we denote a numerically computed quantity that may suffer from rounding errors with a tilde.

Theorem 2.3.

Denote the interval matrix with radius $R \in R_{+}^{L \times L}$ and center at $C \in R^{L \times L}$ by $⟨ C, R ⟩$ . Then, the enclosure of $M_{k, i n}^{(N)}$ is given by

	$M_{k, i n}^{(N)}$	$\in ⟨ M_{k}^{(N)}, ∣ ∣ M_{k, o u t}^{(N)} ∣ ∣ ⟩$		(2.44)
		$\subset ⟨ {~ M}_{k}^{(N)}, ∣ ∣ M_{k, o u t}^{(N)} ∣ ∣ + ∣ ∣ M_{k}^{(N)} - {~ M}_{k}^{(N)} ∣ ∣ ⟩$		(2.45)

for $k = 0$ , $1$ , $\dots$ , $2 M - 1$ .

Proof.

The first enclosure of $M_{k, i n}^{(N)}$ is obtained by the equality $M_{k}^{(N)} - M_{k, i n} = M_{k, o u t}^{(N)}$ . The second enclosure is obtained by this equality and the inequality

	$∣ ∣ M_{k, i n}^{(N)} - {~ M}_{k}^{(N)} ∣ ∣$	$\leq ∣ ∣ M_{k, i n}^{(N)} - M_{k}^{(N)} ∣ ∣ + ∣ ∣ {~ M}_{k}^{(N)} - M_{k}^{(N)} ∣ ∣$		(2.46)
		$= ∣ ∣ M_{p, o u t}^{(N)} ∣ ∣ + ∣ ∣ {~ M}_{k}^{(N)} - M_{k}^{(N)} ∣ ∣ .$		(2.47)

∎

Theorem 2.3 implies that to enclose $M_{k, i n}^{(N)}$ , we can use $| M_{k, o u t}^{(N)} |$ and the truncated complex moment $M_{k}^{(N)}$ computed by using standard verification methods using interval arithmetic to obtain an enclosure of the truncation error $| M_{k}^{(N)} - {~ M}_{k}^{(N)} |$ . Theorem 2.3 readily gives the following enclosure:

	$H_{M, i n}^{<, (N)}$	$\subset ⟨ {~ H}_{M}^{<, (N)}, ∣ ∣ H_{M, o u t}^{<, (N)} ∣ ∣ + ∣ ∣ H_{M}^{<, (N)} - {~ H}_{M}^{<, (N)} ∣ ∣ ⟩,$		(2.48)
	$H_{M, i n}^{(N)}$	$\subset ⟨ {~ H}_{M}^{(N)}, ∣ ∣ H_{M, o u t}^{(N)} ∣ ∣ + ∣ ∣ H_{M}^{(N)} - {~ H}_{M}^{(N)} ∣ ∣ ⟩ .$		(2.49)

An enclosure of $| M_{k, o u t}^{(N)} |$ is obtained as follows.

Theorem 2.4.

Let $B$ be a Hermitian positive semidefinite definite matrix. Assume $2 M - 1 < N$ and that $^λ \in R$ satisfies $|^λ - γ | = min$ . Then, $| M_{k, o u t}^{(N)} |$ in (2.19) is bounded by

∣ ∣ M_{k, o u t}^{(N)} ∣ ∣ \leq (r - m) {|^λ - γ |}^{k} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{{(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}}{1 - {(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ {∥ ∥ V^{H} B V ∥ ∥}_{F}^{H}

(2.50)

for $k = 0$ , $1$ , $\dots$ , $2 M - 1$ .

Proof.

Let $V_{i} = V^{H} B x_{i} x_{i}^{H} B V$ . Then, applying the triangular inequality, we have

	$∣ ∣ M_{k, o u t}^{(N)} ∣ ∣$	$= ∣ ∣ ∣ ∣ r \sum i = m + 1 (λ_{i} - γ)^{k} {d_{i}}^{2} V_{i} ∣ ∣ ∣ ∣$		(2.51)
		$\leq r \sum i = m + 1 {\| λ_{i} - γ \|}_{i}^{k} {d_{i}}^{2} \| V_{i} \|$		(2.52)

for $k = 0, 1, \dots, 2 M - 1$ . Noting the geometric series and applying the triangular inequality, we obtain

	${d_{i}}^{2}$	$= {[\infty \sum j = 1 {(\frac{ρ}{λ_{i} - γ})}^{j N}]}^{2}$		(2.53)
		$\leq \infty \sum j = 1 {∣ ∣ ∣ \frac{ρ}{λ_{i} - γ} ∣ ∣ ∣}^{2 j N} .$		(2.54)

for $i = m + 1$ , $m + 2$ , $\dots$ , $r$ . Multiplied by the factor $| λ_{i} - γ |^{k}$ , we obtain

${\| λ_{i} - γ \|}_{i}^{k} {d_{i}}^{2}$	$\leq \infty \sum j = 1 ρ^{2 j N} {\| λ_{i} - γ \|}_{i}^{- (2 j N - k)}$	(2.55)
	$\leq \infty \sum j = 1 ρ^{2 j N} \|^λ - γ \|^{- (2 j N - k)}$	(2.56)
	$= \|^λ - γ \|^{k} \frac{{(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}}{1 - {(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}}$	(2.57)

for $i = m + 1$ , $m + 2$ , $\dots$ , $r$ and $k = 0, 1, \dots, 2 M - 1$ . Here, the assumption $2 M - 1 < N$ ensures $k < N$ . Noting that the last expression is independent of the index $i$ , we have

∣ ∣ M_{k, o u t}^{(N)} ∣ ∣ \leq |^λ - γ |^{k} \frac{{(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}}{1 - {(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}} r \sum i = m + 1 | V_{i} | .

(2.58)

The bound $| V_{i} | \leq ∥ V^{H} B V ∥_{F}$ follows from the latter half of the proof of [6, Theorem 3.3]. Therefore, we obtain (2.50). ∎

Remark 2.1.

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 $x_{i}$ 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 $S$ and $S_{i n}^{(N)}$ .

Theorem 2.5.

Assume that $B$ is a Hermitian and positive definite matrix. Let $S$ and $S_{i n}^{(N)}$ be defined as in (2.2) and (2.41), respectively, and $y \in C^{L M}$ be an eigenvector of ${S_{i n}^{(N)}}^{H} A S_{i n}^{(N)} y = λ {S_{i n}^{(N)}}^{H} B S_{i n}^{(N)} y$ . If $S y$ is an eigenvector of (1.1), then $S_{i n}^{(N)} y$ is also an eigenvector of (1.1).

Proof.

Let $V^{'} = X D C$ . Then, from Lemma 2.1, it follows that

	$S_{k, i n}^{(N)}$	$= X_{Ω} (Λ_{Ω} - γ I_{m})^{k} D_{Ω} {X_{Ω}}^{H} B V$		(2.59)
		$= X_{Ω} (Λ_{Ω} - γ I_{m})^{k} {X_{Ω}}^{H} B V^{'} .$		(2.60)

Because each eigencomponent of each column vector of $V^{'}$ is a scalar multiple of that of $V$ , $R (S_{k}^{(N)}) = R (S_{k, i n}^{(N)})$ ∎

Motivated by this theorem, we focus on verifying $S_{i n}^{(N)}$ , instead of $S$ .

Theorem 2.6.

Let

S_{o u t}^{(N)} = [S_{0, o u t}^{(N)}, S_{1, o u t}^{(N)}, S_{M - 1, o u t}^{(N)}] .

(2.61)

Then, we have the following enclosure of the approximated transformation matrix:

	$S_{i n}^{(N)}$	$\in ⟨ S^{(N)}, ∣ ∣ S_{o u t}^{(N)} ∣ ∣ ⟩$		(2.62)
		$\subset ⟨ {~ S}^{(N)}, ∣ ∣ S_{o u t}^{(N)} ∣ ∣ + ∣ ∣ {~ S}^{(N)} - S^{(N)} ∣ ∣ ⟩ .$		(2.63)

Proof.

The proof is given similarly to that of Theorem 2.3. ∎

Theorem 2.7.

Assume that $B$ is a Hermitian and positive definite matrix. Assume $2 M - 1 < N$ and that $^λ \in R$ satisfies $|^λ - γ | = {min}_{i = m + 1, m + 2, \dots, r} | λ_{k} - γ |$ . Then, $S_{k, o u t}^{(N)}$ defined in (2.26) is bounded as

(2.64)

for $k = 0$ , $1$ , $\dots$ , $M - 1$ .

Proof.

Similarly to the proof of Theorem 2.4, we have

$∣ ∣ S_{k, o u t}^{(N)} ∣ ∣$	$= ∣ ∣ ∣ ∣ r \sum i = m + 1 (λ_{i} - γ)^{k} d_{i} B^{- 1 / 2} B^{1 / 2} x_{i} x_{i}^{H} B V ∣ ∣ ∣ ∣$	(2.65)
		(2.66)
	$\leq ∥ B^{- 1 / 2} ∥_{2} ∥ B^{1 / 2} V ∥_{2} r \sum i = m + 1 \| λ_{i} - γ \|^{k} \infty \sum p = 1 {(\frac{ρ}{\| λ_{i} - γ \|})}^{p N} {∥ B^{1 / 2} x_{i} ∥_{2}}^{2}$	(2.67)
	$= ∥ B^{- 1} ∥_{2}^{1 / 2} {∥ V^{H} B V ∥_{2}}^{1 / 2} r \sum i = m + 1 \infty \sum p = 1 ρ^{p N} \| λ_{i} - γ \|^{- (p N - k)}$	(2.68)
	$\leq {(∥ B^{- 1} ∥_{2} ∥ V^{H} B V ∥_{2})}_{2}^{1 / 2} r \sum i = m + 1 \infty \sum p = 1 ρ^{p N} \|^λ - γ \|^{- (p N - k)}$	(2.69)
	$= {(λ_{m i n} (B)^{- 1} ∥ V^{H} B V ∥_{F})}_{m i n}^{1 / 2} (r - m) \|^λ - γ \|^{k} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{{(\frac{}{ρ} \|^λ - γ \|)}^{N}}{1 - {(\frac{ρ}{\|^λ - γ \|})}^{N}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$	(2.70)

for $i = m + 1$ , $m + 2$ , $\dots$ , $r$ . Here, we used the $B$ -orthonormality of the eigenvectors ${∥ B^{1 / 2} x_{i} ∥_{2}}^{2} = {x_{i}}^{H} B x_{i} = 1$ . ∎

Remark 2.2.

The evaluations (2.63), (2.64) can also be used for the Hankel matrix approach [6] for the evaluation of eigenvectors.

Remark 2.3.

In Theorem 2.7, a Hermitian matrix $B$ is required to be positive definite for the verification of eigenvectors, contrarily to the verification of eigenvalues, cf. Theorem 2.4.

The evaluation of the numerical error $| {~ S}^{(N)} - S^{(N)} |$ in (2.63), i.e., $| {~ S}_{k}^{(N)} - S_{k}^{(N)} |$ for each $k = 0$ , $1$ , $\dots$ , $M - 1$ , involves the error evaluation of the solution

Y_{j} = (z_{j} B - A)^{- 1} B V

(2.71)

of the linear system of equations with multiple right-hand sides $(z_{j} B - A) Y_{j} = B V$ associated with

(2.72)

for $k = 0$ , $1$ , $\dots$ , $M - 1$ . The enclosure of $Y_{j}$ can be obtained by using standard verification methods, e.g. [15, 16]. For efficiency, the technique based on [6, Theorem 4.1] can be also used.

2.5 Implementation.

We present implementation issues of the proposed method. We assume that the numbers of $L$ and $M$ satisfy $L M = m$ . Also, the proposed method needs to determine the number of the parameter $N$ . Each quadrature point $z_{j}$ gives rise to a linear system $(z_{j} B - A) Y_{j} = B V$ to solve. The evaluation of a solution for each linear system is the most expensive part, whereas the quadrature errors $| M_{o u t}^{(N)} |$ and $| S_{o u t}^{(N)} |$ reduces as the number of quadrature points $N$ 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 $N$

has been heuristically determined in the complex moment eivensolvers for numerical computations. For numerical verification, a reasonable number

N

can 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

N

such that

\frac{1}{2} {(log \frac{ρ}{|^λ - γ |})}^{- 1} log (\frac{δ}{c_{1} (r - m) + δ})

${\| λ_{i} - γ \|}_{i}^{k} {d_{i}}^{2}$	$\leq \infty \sum j = 1 ρ^{2 j N} {\| λ_{i} - γ \|}_{i}^{- (2 j N - k)}$	(2.55)
	$\leq \infty \sum j = 1 ρ^{2 j N} \|^λ - γ \|^{- (2 j N - k)}$	(2.56)
	$= \|^λ - γ \|^{k} \frac{{(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}}{1 - {(\frac{ρ}{∣ ∣^λ - γ ∣ ∣})}^{2 N}}$	(2.57)

Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh-Ritz procedure for generalized Hermitian eigenvalue problems

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About the foundation of the Kubo Generalized Cumulants theory. A revisited and corrected approach

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Complex moment-based method with nonlinear transformation for computing large and sparse interior singular triplets

Adaptive MCMC for Generalized Method of Moments with Many Moment Conditions

1 Introduction

2 Rayleigh–Ritz procedure approach.

Theorem 2.1 ([4, Theorem 7], [5, Theorem 3]).

2.1 $N$ -point quadrature rule.

2.2 Effect of eigenvalues inside and outside $Ω$

2.3 Verification of eigenvalues.

Lemma 2.1.

Proof.

Theorem 2.2.

Proof.

Theorem 2.3.

Proof.

Theorem 2.4.

Proof.

Remark 2.1.

2.4 Verification of eigenvectors.

Theorem 2.5.

Proof.

Theorem 2.6.

Proof.

Theorem 2.7.

Proof.

Remark 2.2.

Remark 2.3.

2.5 Implementation.

Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh-Ritz procedure for generalized Hermitian eigenvalue problems

Related Research

Projection method for partial eigenproblems of linear matrix pencils

Complex moment-based methods for differential eigenvalue problems

About the foundation of the Kubo Generalized Cumulants theory. A revisited and corrected approach

Efficient method for calculating the eigenvalue of the Zakharov-Shabat system

On pole-swapping algorithms for the eigenvalue problem

Complex moment-based method with nonlinear transformation for computing large and sparse interior singular triplets

Adaptive MCMC for Generalized Method of Moments with Many Moment Conditions

1 Introduction

2 Rayleigh–Ritz procedure approach.

Theorem 2.1 ([4, Theorem 7], [5, Theorem 3]).

2.1 N-point quadrature rule.

2.2 Effect of eigenvalues inside and outside Ω

2.3 Verification of eigenvalues.

Lemma 2.1.

Proof.

Theorem 2.2.

Proof.

Theorem 2.3.

Proof.

Theorem 2.4.

Proof.

Remark 2.1.

2.4 Verification of eigenvectors.

Theorem 2.5.

Proof.

Theorem 2.6.

Proof.

Theorem 2.7.

Proof.

Remark 2.2.

Remark 2.3.

2.5 Implementation.

2.1 $N$ -point quadrature rule.

2.2 Effect of eigenvalues inside and outside $Ω$