Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

Zhongming Teng; Hong-Xiu Zhong

doi:10.1515/math-2019-0052

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Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

Zhongming Teng und Hong-Xiu Zhong

Veröffentlicht/Copyright: 9. Juli 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

Keywords: linear response eigenvalue problem; Rayleigh-Ritz approximation; canonical angles; majorization; error bounds

MSC 2010: 65F15; 65L15

1 Introduction

In computational quantum chemistry and physics, the excitation states and absorption spectra for molecules or surface of solids are described by the random phase approximation (RPA) or the Bethe-Salpeter (BS) equation [1, 2]. One important question in RPA or BS equation is to compute a few eigenpairs associated with the smallest positive eigenvalues of the following eigenvalue problem:

Hw=AB−B−Auv=λuv=λw, (1.1)

where A, B ∈ ℝ^n×n are both symmetric matrices and ABBA is positive definite [3, 4, 5]. The matrix 𝓗 in (1.1) is a special Hamiltonian matrix whose eigenvalues are real and in pairs {λ, −λ}. Therefore, we can order the 2n eigenvalues of 𝓗 as

−λn≤⋯≤−λ1<λ1≤⋯≤λn. (1.2)

Through a similarity transformation, the eigenvalue problem (1.1) can be equivalently transformed into

Hz=0KM0yx=λyx=λz, (1.3)

where K = A − B and M = A + B are both n × n real symmetric positive definite matrices. But, in consistent with [3], throughout the rest of paper, we relax the condition on K and M to that they are symmetric positive semi-definite and one of them is definite, unless explicitly stated differently. This means that possibly λ₁ = 0. This eigenvalue problem was still referred to as the linear response eigenvalue problem (LREP) [3, 6] and will be so in this paper, too. There are immense recent interest in developing new theories, efficient numerical algorithms of LREP and the associated excitation response calculations of molecules for materials design in energy science [7, 8, 9, 10].

As the dimension n is usually very large, LREP (1.3) is generally solved by iterative methods, such as the Locally Optimal Block Preconditioned 4D Conjugate Gradient Method (LOBP4DCG) [6] and its space-extended variation [11], the block Chebyshev-Davidson method [12], and the generalized Lanczos type methods [13, 14]. These efficient numerical algorithms are all based on the concept of the so-called pair of deflating subspaces which is a generation of the invariant subspace in the standard eigenvalue problem. For given k-dimensional subspaces 𝓧 ⊆ ℝⁿ and 𝓨 ⊆ ℝⁿ, {𝓧, 𝓨} is called a pair of deflating subspaces if they satisfy

KX⊆YandMY⊆X.

Whenever such a pair of deflating subspaces is available, LREP (1.3) can be projected into a much smaller problem in the form of (1.3) whose spectrum is part of that of H. That means a pair of deflating subspaces can be used to extract the eigenvalues corresponding to the deflating subspace pair. In practice, such a pair of exact deflating subspaces is usually not available, only approximate one, then the projection method computes good the approximate eigenvalues of LREP (1.3). Quantifying how good the approximate eigenvalues is the objective of this paper.

For the standard symmetric eigenvalue problem, there are existing results to estimate how well of such approximate eigenvalues may be; see [15, 16, 17, 18, 19]. For LREP (1.3), in [20], residual-based error bounds for approximate eigenvalues computed through the pair of approximated deflating subspaces are obtained. These results bound eigenvalue approximations characterized by the certain residuals. Another set of results also bound the eigenvalue errors in [21] but in terms of the canonical angles between the approximate deflating subspace pair and the exact pair. In this paper, we put forward two improvements to the Rayleigh-Ritz approximation theories in [21] by using weak majorization. The Rayleigh-Ritz majorization type eigenvalue error bounds have been well established in the symmetric eigenvalue problem; see [15, 17]. Compared with typical inequality results, majorization type bounds provide a succinct way to express numerous useful inequalities involving two vectors. The major goals of this paper are two-fold: to establish Rayleigh-Ritz majorization error bounds for LREP, and to extend our results by considering the unequal dimension between the exact and approximate pair of deflating subspaces. These improvements are helpful to understand how approximate eigenvalues move towards the associated exact eigenvalues in iterative methods of LREP.

The rest of the paper is organized as follows. In Section 2, some notations and preliminaries including the concept of majorization and the canonical angles of two subspaces are collected for use later. Section 3 contains our main results on how the vector of differences between the exact eigenvalues and their approximations is weakly majorized by the canonical angles from the exact to approximate pair of deflating subspaces. In Section 4, some numerical examples are presented to support our analysis. Finally, Conclusions are given in Section 5.

2 Preliminaries

2.1 Basic definitions

ℝ^m×n is the set of all m × n real matrices, ℝⁿ = ℝ^n×1, and ℝ = ℝ¹. For 𝓧 ⊆ ℝⁿ, dim(𝓧) is the dimension of 𝓧. For X ∈ ℝ^m×n, X^T is its transpose, 𝓡(X) is the column space of X, and the submatrix X_(:,i:j) of X consists of column i to column j. I_n is the n × n identity matrix or simply I if its dimension is clear from the context. For x ∈ ℝⁿ and y ∈ ℝⁿ,

x(k:ℓ)=[xk,…,xn,0,…,0⏟ℓ−n]T,for k≤n<ℓ,[⁡xk,…,xℓ]T,for k≤ℓ≤n,

the notation x ≤ y is used to compare x with y componentwise, and x ∘ y = [x₁y₁, …, x_ny_n]^T denotes the Hadamard product of x and y, in particular, x^∘2 = x ∘ x. For scalars α_i, diag(α₁, α₂, …, α_k) ∈ ℝ^k×k is a diagonal matrix with diagonal entries α₁, α₂, …, α_k.

For x = [x₁,x₂, …, x_n]^T ∈ ℝⁿ, x↓=[x1↓,x2↓,…,xn↓]T is the rearrangement of x_i in descending order, i.e., x1↓≥x2↓≥⋯≥xn↓, and x↑=[x1↑,x2↑,…,xn↑]T represents x with its elements rearranged in ascending order. We use λ(A) = λ^↓(A) to denote the vector of eigenvalues of a symmetric matrix A arranged in descending order. σ(B) = σ^↓(B) = [σ₁(B), σ₂(B), …, σ_n(B)]^T is denoted as the vector of singular values of B ∈ ℝ^m×n arranged in descending order, where σ_i(B) ≥ 0 and m ≥ n.

For vectors x, y ∈ ℝⁿ, the usually inner product and its induced norm are defined by

〈x,y〉=xTy,∥x∥2=〈x,x〉.

Consider two subspaces 𝓧 and 𝓨 in ℝⁿ, and suppose

k=dim⁡(X)≤dim⁡(Y)=ℓ. (2.1)

Let X ∈ ℝ^n×k and Y ∈ ℝ^n×ℓ be orthonormal basis matrices of 𝓧 and 𝓨, respectively, i.e.,

XTX=Ik,R(X)=X,andYTY=Iℓ,R(Y)=Y.

The vector of cosines of canonical angels from^[1] the subspace 𝓧 to the subspace 𝓨 is defined by cosΘ(𝓧, 𝓨) = σ^↑(X^TY) with

Θ(X,Y)=Θ↓(X,Y)=[θ1(X,Y),…,θk(X,Y)]T, (2.2)

i.e., θ₁(𝓧, 𝓨) ≥ … ≥ θ_k(𝓧, 𝓨). It is clear that Θ(𝓧, 𝓨) so defined is invariant with respect to the different choice of the orthonormal basis matrices X and Y. Therefore, in what follows we sometimes place a matrix in one of or both arguments of θ_i(⋅, ⋅) and Θ(⋅, ⋅), with the understanding that it refers to the subspace spanned by the columns of the matrix argument.

Lemma 2.1

([22, Proposition 2.1 and Proposition 2.4]). Let 𝓧 and 𝓨 be two subspaces in ℝⁿ.

If dim(𝓧) = k ≤ dim(𝓨) = ℓ, for any 𝓨̂ ⊆ 𝓨 with dim(𝓨̂) = dim(𝓧) = k, we have Θ(𝓧, 𝓨̂) ≥ Θ(𝓧, 𝓨). In addition, there exists a k-dimensional subspace 𝓨̂₁ ⊆ 𝓨 such that Θ(𝓧, 𝓨̂₁) = Θ(𝓧, 𝓨).
If dim(𝓧) = dim(𝓨) = ℓ, for any 𝓧̂ ⊆ 𝓧 with dim(𝓧̂) = k ≤ ℓ, we have Θ(𝓧, 𝓨)_(1:k) ≥ Θ(𝓧̂, 𝓨) ≥ Θ(𝓧, 𝓨)_{(ℓ−k+1:ℓ)}.

Lemma 2.2

([23, Proposition 2.2]). Let 𝓧 and 𝓨 be two subspaces in ℝⁿ satisfying (2.1). Then

θi(X,Y)=0,for k−m0+1≤i≤k,

where m₀ = dim(𝓧 ∩ 𝓨).

Lemma 2.3

([17, Theorem 4.7]). Let 𝓧 and 𝓨 be two subspaces in ℝⁿ with dim(𝓧) = dim(𝓨) = k. Then, λ (Π_𝓧 − Π_𝓨)_(1:k) = sinΘ(𝓧, 𝓨), where Π_𝓧 and Π_𝓨 are orthogonal projectors onto subspaces 𝓧 and 𝓨, respectively.

For any given symmetric and positive definite matrix W ∈ ℝ^n×n, the W-inner product and its induced W-norm are defined by

〈x,y〉W=yTWx,∥x∥W=〈x,x〉W.

For two subspaces 𝓧, 𝓨 ⊆ ℝⁿ satisfying (2.1), let X and Y be the W-orthonormal basis matrices of subspaces 𝓧 and 𝓨, respectively, i.e.,

XTWX=Ik,R(X)=X,andYTWY=Iℓ,R(Y)=Y.

Similarly to the standard canonical angles (2.2), we define

cosΘW(X,Y)=σ↑(XTWY),

where

ΘW(X,Y)=ΘW↓(X,Y)=[θW(1)(X,Y),…,θW(k)(X,Y)]T.

denotes the vector of the W-canonical angles from 𝓧 to 𝓨 in descending order. Let W = CC^T with nonsingular C ∈ ℝ^n×n. Then, by [24, Theorem 4.2],

ΘW(X,Y)=Θ(CTX,CTY). (2.3)

Therefore, Lemmas 2.1 and 2.2 in which the standard canonical angles are simply replaced by W-canonical angles still hold.

2.2 Majorization and weak majorization

For x, y ∈ ℝⁿ, we say that x is weakly majorized by y, in symbols x ≺_w y, if

∑i=1kxi↓≤∑i=1kyi↓,for 1≤k≤n.

If in addition,

∑i=1nxi↓=∑i=1nyi↓,

we say that the vector x is (strongly) majorized by y, written x ≺ y.

To facilitate our discussion, we collect several simple and general properties of the majorization and weak majorization in Lemma 2.4, and the reader is referred to [25] for proofs and more.

Lemma 2.4

Let x, y ∈ ℝⁿ and x ≺_w y.

If y ≤ z, then x ≺_w z.
If u ≺_w v, then x + u ≺ x^↓ + u^↓≺_w y + v. This also holds with all ≺_w replaced by ≺.
If u ∈ R+n, then x^↓ ∘ u^↓≺_w y^↓ ∘ u^↓. In particular, if α is a positive real number, then x ≺_w y implies α x ≺_w α y.

The following lemmas on the majorization or weak majorization relations of eigenvalues and singular values are critical to our main results.

Lemma 2.5

([25, Theorem II.3.6]). Let A be an n × n matrix. We have ∣λ(A)∣≺_wσ(A). In particular, if A^T = A, we have ∣λ(A)∣ = σ(A).

Lemma 2.6

([25, Exercise II.1.14]). Let A and B be symmetric matrices. The eigenvalues of A, B and A + B satisfy λ(A + B)≺_wλ(A) + λ(B).

Lemma 2.7

([25, Corollary III.4.2]). Let A and B be symmetric matrices. The eigenvalues of A, B and A − B satisfy λ(A)−λ(B)≺λ(A − B).

Lemma 2.8

([26, Theorem 1]). For B, D, E ∈ ℝ^n×n, we have

σ(DBET)≺wσ(B)∘δ↓,

where δ is any vector which weakly majorizes σ(D) ∘ σ(E).

Lemma 2.8 implies that if B ∈ ℝ^n×m, D ∈ ℝ^k×n and E ∈ ℝ^ℓ×m where k ≤ ℓ ≤ m ≤ n, then

σ(DBET)≺wσ(B)(1:k)∘σ(D)(1:k)∘σ(E)(1:k). (2.4)

The weakly majorization relationship (2.4) is obtained by extending B, D and E with zero blocks to n × n matrices. The extension with zero blocks only appends zero singular values and does not change the ranks.

3 Main results

Many theoretical properties of LREP (1.3) have been established in [3]. In particular, the following theorem presents decompositions on K and M, which is necessary for our later developments.

Lemma 3.1

([, Theorem 2.3]). The following statements hold for any symmetric matrices K, M ∈ ℝ^n×n with M being positive definite.

There exists a nonsingular Ψ ∈ ℝ^n×n such that

K=ΨΛ2ΨTandM=ΦΦT, (3.1)

where Λ = diag(λ₁, λ₂, …, λ_n), λ12≤λ22≤⋯≤λn2 and Φ = Ψ^−T.
If K is also definite, then all λ_i > 0 and H is diagonalizable:

HΨΛΨΛΦ−Φ=ΨΛΨΛΦ−ΦΛΛ.

The eigen-decomposition of KM and MK are

(KM)Ψ=ΨΛ2and(MK)Φ=ΦΛ2,

respectively.

As we have introduced in Section 1, for two k-dimensional subspaces 𝓧 and 𝓨 in ℝⁿ, we call {𝓧, 𝓨} a pair of deflating subspaces if

KX⊆YandMY⊆X. (3.2)

Let X, Y ∈ ℝ^n×k be the basis matrices for 𝓧 and 𝓨, respectively. Then (3.2) implies that there exist K_R ∈ ℝ^k×k and M_R ∈ ℝ^k×k such that

KX=YKRandMY=XMR,

or equivalently,

HYX=YXHRwithHR=KRMR.

Furthermore, if

HRz^=KRMRy^x^=λ^y^x^=λ^z^,

then (λ^,Yy^Xx^) is an eigenpair of H.

Roughly speaking, most of efficient algorithms for LREP usually generate a sequence of approximate deflating subspace pairs that hopefully converge to or contain subspaces near the pair of deflating subspaces associated with the first few smallest λ_i. Therefore, in the rest of the paper, we focus on the pair of deflating subspaces {𝓡(Φ₁), 𝓡(Ψ₁)} where Φ₁ = Φ_(:,1:k) and Ψ₁ = Ψ_(:,1:k), and let {𝓤, 𝓥} satisfying

dim⁡(U)=dim⁡(V)=ℓ≥kandθ1(U,V)<π2 (3.3)

be a pair of approximate deflating subspaces of {𝓡(Φ₁), 𝓡(Ψ₁)} if ℓ = k, or, if ℓ > k, contain a k-dimensional subspace pair approximating {𝓡(Φ₁), 𝓡(Ψ₁)}. Let U, V ∈ ℝ^n×ℓ be any basis matrices for 𝓤 and 𝓥, respectively. Then, the condition θ₁(𝓤, 𝓥) < π/2 implies U^TV being nonsingular. By [6], the best approximation eigenpairs in sense of the trace minimization principle are obtained via computing the eigenpairs of

HSR=KSRMSR,

where KSR=W1−TUTKUW1−1,MSR=W2−TVTMVW2−1, and W₁, W₂ ∈ ℝ^ℓ×ℓ are from factorizing U^TV = W1T W₂ and nonsingular. In particular, if U^TV = I_ℓ, then H_SR becomes

HSR=UTKUVTMV.

Notice that H_SR so defined is of LREP type since both K_SR and M_SR are symmetric and have the same definiteness property as their corresponding K and M. Therefore, we can denote its eigenvalues by

−μℓ≤⋯≤−μ1≤μ1≤⋯≤μℓ,

in which some of μ_i should be good approximate eigenvalues of H. Moreover, by [3, Theorem 2.9], these eigenvalues of H_SR are independence of the basis matrices U and V, and factorization U^TV = W1T W₂ which is not unique. Now, we would like to bound the errors in μ_i as approximations to λ_i for 1 ≤ i ≤ k in terms of the distances from {𝓡(Φ₁), 𝓡(Ψ₁)} to {𝓤, 𝓥}. For the purpose, Zhang, Xue and Li in [21] established an inequality in the case ℓ = k, i.e.,

∑i=1k(μi2−λi2)≤tan2⁡θM−1(1)(U,MV)∑i=1kλi2+λn2−λ12cos2⁡θM−1(1)(U,MV)∥sinΘM−1(U,Φ1)∥22. (3.4)

We first continue the effort to extend the result by using the weak majorization replacing the traditional inequality in (3.4). To prove our main results, we also need the following lemma to provide special basis matrices for the pair {𝓤, 𝓥} which are used in the proof of [21, Theorem 3.1].

Lemma 3.2

Let 𝓤, 𝓥 ⊆ ℝⁿ satisfy (3.3) and ℓ = k. These exist basis matrices U, V ∈ ℝ^n×k of 𝓤 and 𝓥, respectively, and an orthonormal matrix P ∈ ℝ^n×n such that

UTV=Ik,U~=ΨTU=PTΓ−10,V~=ΦTV=PTΓΣ, (3.5)

where Γ = diag(γ₁, …, γ_k) with γi=cos⁡θM−1(i)(U,MV) for 1 ≤ i ≤ k and

for 2k ≤ n,

Σ=n−2kkΣ~0.k.=diag(sin⁡θM−1(1)(U,MV),…,sin⁡θM−1(k)(U,MV)),
for 2k > n,

Σ=n−kΣ~0n−k2k−n.,Σ~=diag(sin⁡θM−1(1)(U,MV),…,sin⁡θM−1(n−k)(U,MV)).

Theorem 3.1

Let {𝓤, 𝓥} satisfying (3.3) with ℓ = k be an approximation to {𝓡(Φ₁), 𝓡(Ψ₁)}, μ = [μ_k, …, μ₁]^T and λ = [λ_k, …, λ₁]^T. Assume that M is definite. We have

0≤μ∘2−λ∘2≺wδ∘sin∘2ΘM−1(U,Φ1)+tan2⁡θM−1(1)(U,MV)λ∘2, (3.6)

where δ=λn2cos2⁡θM−1(1)(U,MV)−λ12cos2⁡θM−1(1)(U,MV),…,λn−k+12cos2⁡θM−1(k)(U,MV)−λ12cos2⁡θM−1(1)(U,MV)T.

Proof

Since the eigenvalues of H_SR are unchanged with different choices of the basis matrices for 𝓤 and 𝓥, we choose the basis matrices U and V as mentioned in Lemma 3.2. Let U͠, V͠, P, and Γ be defined in Lemma 3.2. Partition P, U͠ and Λ as

P=n−kkP11P12P21P22kn−k.U~=n−kkU~1U~2k,Λ=n−kkΛ1Λ2kn−k. (3.7)

Then, it follows by (3.5) that,

U~1=P11TΓ−1,U~2=P12TΓ−1,U~TV~=Ik,V~TV~=Ik. (3.8)

If 2k > n, we caution the reader that here γ_i = cos θM−1(i) (𝓤, M𝓥) = 1 for n − k + 1 ≤ i ≤ k by Lemma 2.2 are used to prove V͠^TV͠ = I_k. Notice that M⁻¹ = ΨΨ^T, RU~=RP11TP12T and P12P12T=Ik−P11P11T. Thus, by (2.3), we have

cosΘM−1(U,Φ1)=cosΘ(U~,I(:,1:k))=cosΘP11TP12T,Ik0=σ↑(P11), (3.9)

sinΘM−1(U,Φ1)=e−σ↑(P11P11T)=λ(Ik−P11P11T)=λ(P12P12T), (3.10)

where e=[1,…,1⏟k]T.

By Lemma 3.1, K = ΨΛ²Ψ^T, M = ΦΦ^T, μ^∘2 = λ ((U^TKU)(V^TMV)) and λ^∘2 = λ(Λ12). Therefore, by λ ≤ μ known in [3, Theorem 4.1], we have

0≤μ∘2−λ∘2=λ(UTKU)(VTMV)−λ(Λ12)=λU~TΛ2U~V~TV~−λ(Λ12)=λU~TΛ2U~−λ(Λ12)(by (3.8))=λU~1TΛ12U~1+U~2TΛ22U~2−λ(Λ12)=λU~1TΛ12U~1+U~2TΛ22U~2−λ(U~1TΛ12U~1)+λ(U~1TΛ12U~1)−λ(Λ12)≺λ(U~2TΛ22U~2)+λ(U~1TΛ12U~1)−λ(Λ12)↓. (3.11)

The last line holds because of Lemmas 2.4(b) and 2.7. Now, we bound separately the two terms in the sum on the right-hand side of the last line. We start with the first term. By (3.10), and Lemmas 2.4(c), 2.5 and 2.8, we get

λ(U~2TΛ22U~2)=λ(Λ22U~2U~2T)(1:k)≺wσ(Λ22U~2U~2T)(1:k)≺wσ(Λ22)(1:k)∘σ(U~2U~2T)(1:k)=σ(Λ22)(1:k)∘σ(P12TΓ−1Γ−TP12)(1:k)≺wσ(Λ22)(1:k)∘σ(P12T)(1:k)∘σ(Γ−1Γ−T)∘σ(P12)(1:k)=σ(Λ22)(1:k)∘σ(Γ−2)∘sin∘2ΘM−1(U,Φ1). (3.12)

Considering the second term, by (3.8), and Lemmas 2.4 and 2.7, we have

λ(U~1TΛ12U~1)−λ(Λ12)=λ(Λ1U~1U~1TΛ1)−λ(Λ12)=λ(Λ1P11TΓ−1Γ−TP11Λ1)−λ(Λ12)≤1γ12λ(Λ1P11TP11Λ1)−λ(Λ12)=1γ12λ(Λ1P11TP11Λ1)−1γ12λ(Λ12)+1−γ12γ12λ(Λ12)≺1γ12λΛ1(P11TP11−Ik)Λ1+1−γ12γ12λ(Λ12)=1γ12λ(Ik−P11TP11)1/2(−Λ12)(Ik−P11TP11)1/2+1−γ12γ12λ(Λ12)≤−λ12γ12λ(Ik−P11TP11)+1−γ12γ12λ(Λ12)=−λ12γ12e−λ↑(P11TP11)+1−γ12γ12λ(Λ12)=−λ12γ12e−cos∘2ΘM−1(U,Φ1)+1−γ12γ12λ(Λ12)=−λ12γ12sin∘2ΘM−1(U,Φ1)+1−γ12γ12λ(Λ12). (3.13)

At last, (3.11), (3.12) and (3.13) together give

μ2−λ2=λ(UTKU)(VTMV)−λ(Λ12)≺wσ(Λ22)(1:k)∘σ(Γ−2)∘sin∘2ΘM−1(U,Φ1)−λ12γ12sin∘2ΘM−1(U,Φ1)+1−γ12γ12λ(Λ12)=σ(Λ22)(1:k)∘σ(Γ−2)−λ12γ12esin∘2ΘM−1(U,Φ1)+tan2θM−1(1)(U,MV)λ(Λ12), (3.14)

which leads to (3.6).□

Remark 3.1

Listed below are some comments for Theorem 3.1.

Recall (3.6) and (3.14). It is noted that, if 2k > n, σ(Λ22)(1:k)=[λn2,…,λk+12,0,…,0⏟2k−n]T in (3.14) is not equal to [λn2,…,λk+12,λk2,…,λn−k+12⏟2k−n]T in δ of (3.6). In fact, in such a case, the last 2k − n entries of Θ_M⁻¹(𝓤,Φ₁) are all zero by Lemma 2.2. Thus, the weak majorization relationship (3.6) still holds for the case 2k > n.
The weak majorization bound (3.6) directly implies that, for j = 1, …, k,

∑i=1j(ui2−λi2)↓≤∑i=1jλn−i+12cos2⁡θM−1(i)(U,MV)−λ12cos2⁡θM−1(1)(U,MV)sin2⁡θM−1(i)(U,Φ1)+tan2⁡θM−1(1)(U,MV)∑i=1jλi2≤λn2−λ12cos2⁡θM−1(1)(U,MV)∑i=1jsin2⁡θM−1(i)(U,Φ1)+tan2⁡θM−1(1)(U,MV)∑i=1jλi2. (3.15)

Two implied inequalities by (3.15) that are often sufficient for numerical purposes are (3.4) and

max1≤i≤k(ui2−λi2)≤λn2−λ12cos2⁡θM−1(1)(U,MV)sin2⁡θM−1(1)(U,Φ1)+λ12tan2⁡θM−1(1)(U,MV).
In some numerical methods for LREP, such as the first Lanczos method [13, 27] and Chebyshev-Davidson method [12], the subspace 𝓤 is chosen such that 𝓤 = M𝓥, which leads to cos θM−1(i) (𝓤, M𝓥) = 1 for 1 ≤ i ≤ k and tan θM−1(1) (𝓤, M𝓥) = 0. Then, the majorization relationship (3.6) reduces to

0≤μ∘2−λ∘2≺w[(λn2−λ12),…,(λn−k+12−λ12)]T∘sin∘2ΘM−1(U,Φ1).
Suppose that K is definite. Then, a simple modification by exchanging the roles of K and M in the above proof leads to the following majorization relationship

0≤μ∘2−λ∘2≺wδK∘sin∘2ΘK−1(V,Ψ^1)+tan2⁡θK−1(1)(V,KU)λ∘2, (3.16)

where δK=λn2cos2⁡θK−1(1)(V,KU)−λ12cos2⁡θK−1(1)(V,KU),…,λn−k+12cos2⁡θK−1(k)(V,KU)−λ12cos2⁡θK−1(1)(V,KU)T and Ψ͡₁ = Ψ͡_(:,1:k) coming from the new decompositions M = Φ͡Λ²Φ͡^T and K = Ψ͡Ψ͡^T instead of (3.1). In particular, if M is also definite, there is no need to distinguish Ψ from Ψ͡ in (3.16) because of 𝓡(Ψ) = 𝓡(Ψ͡).

As stated in [21, Example 3.1], if Θ_M⁻¹(𝓤,Φ₁) = 0, then μ²−λ² = 0. However, Θ_M⁻¹(𝓤, M𝓥) is not necessary being 0 because of the indefiniteness of K. This phenomenon is overcome under the assumption that K and M are both definite, and in the following theorem, we will establish the associated majorization upper bounds for sinΘ_M⁻¹(𝓤, M𝓥) and sinΘ_K⁻¹(𝓥, K𝓤), respectively, by using sinΘ_M⁻¹(𝓤,Φ₁) and sinΘ_K⁻¹(𝓥,Ψ₁).

Theorem 3.2

Let {𝓤, 𝓥} satisfy (3.3) and ℓ = k. Suppose both K and M being definite. We have

sinΘM−1(U,MV)≺wsinΘM−1(U,Φ1)+κ∘sinΘK−1(V,Ψ1), (3.17)

sinΘK−1(V,KU)≺wsinΘK−1(V,Ψ1)+κ∘sinΘM−1(U,Φ1), (3.18)

where κ=λnλ1,…,λn−k+1λkT.

Proof

Use the same notations and basis matrices U and V for 𝓤 and 𝓥, respectively, as the proof of Theorem 3.1. Partition U͠, P and Λ as in (3.7) and V͠ as

V~=n−kkV~1V~2k.

Thus, by (2.3) and (3.9),

sinΘM−1(U,Φ1)=sinΘ(U~,I(:,1:k))=sinΘP11TP12T,Ik0=σ(P12)(1:k), (3.19)

sinΘK−1(V,Ψ1)=sinΘ(Λ−1V~,I(:,1:k))=σΛ2−1V~2(V~TΛ−2V~)−1/2(1:k). (3.20)

The first equality in (3.20) holds because 𝓡(I_(:,1:k)) = 𝓡(Λ⁻¹I_(:,1:k)). By Lemmas 2.3 and 2.6,

sinΘM−1(U,MV)=sinΘ(U~,V~)=λ(ΠR(U~)−ΠR(V~))(1:k)=λΠR(U~)−I(:,1:k)I(:,1:k)T+I(:,1:k)I(:,1:k)T−ΠR(V~)(1:k)≺wλΠR(U~)−I(:,1:k)I(:,1:k)T(1:k)+λI(:,1:k)I(:,1:k)T−ΠR(V~)(1:k)=sinΘ(U~,I(:,1:k))+sinΘ(I(:,1:k),V~). (3.21)

Now, we need to relate sinΘ(I_(:,1:k), V͠) to sinΘ(Λ⁻¹V͠, I_(:,1:k)). It follows by V͠^TV͠ = I_k and Lemma 2.8 that

sinΘ(I(:,1:k),V~)=σ(V~2)(1:k)=σΛ2Λ2−1V~2(V~TΛ−2V~)−1/2(V~TΛ−2V~)1/2(1:k)≺wσ(Λ2)(1:k)∘σΛ2−1V~2(V~TΛ−2V~)−1/2(1:k)∘σ(V~TΛ−2V~)1/2≤σ(Λ2)(1:k)∘σΛ2−1V~2(V~TΛ−2V~)−1/2(1:k)∘σ(Λ−1)(1:k)=κ∘sinΘ(Λ−1V~,I(:,1:k)). (3.22)

Then, (3.19), (3.20), (3.21) and (3.22) together yield (3.17). Similarly, to prove (3.18), we consider

sinΘK−1(V,KU)=sinΘ(Λ−1ΦTV,Λ−1ΦTKU)=sinΘ(Λ−1V~,ΛU~)=λ(ΠR(Λ−1V~)−ΠR(ΛU~))(1:k)=λΠR(Λ−1V~)−I(:,1:k)I(:,1:k)T+I(:,1:k)I(:,1:k)T−ΠR(ΛU~)(1:k)≺wλΠR(Λ−1V~)−I(:,1:k)I(:,1:k)T(1:k)+λI(:,1:k)I(:,1:k)T−ΠR(ΛU~)(1:k)=sinΘ(Λ−1V~,I(:,1:k))+sinΘ(I(:,1:k),ΛU~), (3.23)

and clarify the majorization relationship between sinΘ(I_(:,1:k), ΛU͠) with sinΘ(U͠, I_(:,1:k)), i.e.,

sinΘ(I(:,1:k),ΛU~)=sinΘ(I(:,1:k),ΛP1)=σΛ2P12T(P1TΛ2P1)−1/2(1:k)≺wσ(Λ2)(1:k)∘σ(P12T)(1:k)∘σ(P1TΛ2P1)−1/2≤σ(Λ2)(1:k)∘σ(P12)(1:k)∘σ(Λ−1)(1:k)=κ∘sinΘM−1(U,Φ1), (3.24)

where P₁ = [P₁₁, P₁₂]^T. At last, (3.18) is obtained by combining (3.19), (3.20), (3.23) and (3.24).□

Theorems 3.1 and 3.2 are established in assuming ℓ = k. But, the case ℓ > k is more common in practical eigenvalue computations of LREP, e.g., in the first Lanczos type method for LREP, the subspaces 𝓤 and 𝓥 are the Krylov subspaces generated by initial vectors v₀ ∈ ℝⁿ and u₀ = Mv₀, i.e., 𝓤 = 𝓚_ℓ(MK, u₀) and 𝓥 = 𝓚_ℓ(KM, v₀), and usually the pair of Krylov subspaces {𝓤, 𝓥} as a whole is not close to any pair of deflating subspaces but more likely it contains a subspace pair of lower dimension being a good approximation of {𝓡(Φ₁), 𝓡(Ψ₁)}. Thus, it is natural to generalize Theorems 3.1 and 3.2 for the case ℓ > k. This gives Theorem 3.3 and 3.4 below. Similar comments we made in Remark 3.1 are also valid for Theorem 3.3.

Theorem 3.3

Let {𝓤, 𝓥} satisfying (3.3) contain a pair of k-dimensional subspaces approximating {𝓡(Φ₁), 𝓡(Ψ₁)}. Assume that M is definite. Using the notations of Theorem 3.1, we have

0≤μ2−λ2≺wωcos2⁡θM−1(1)(U,MV)∘sin∘2ΘM−1(U,Φ1)+tan2⁡θM−1(1)(U,MV)λ∘2, (3.25)

where ω=(λn2−λ12),…,(λn−k+12−λ12)T.

Proof

By Lemma 2.1(a), we choose k-dimensional subspaces 𝓤₁ ⊂ 𝓤 satisfying

ΘM−1(U1,Φ1)=ΘM−1(U,Φ1), (3.26)

and 𝓥₁ ⊂ 𝓥 such that Θ_M⁻¹(𝓤₁,M𝓥₁) = Θ_M⁻¹(𝓤₁,M𝓥). Since θ₁(𝓤, 𝓥) < π/2, by Lemma 2.1(b), we have

ΘM−1(U1,MV1)=ΘM−1(U1,MV)≤ΘM−1(U,MV)(1:k)<π2. (3.27)

Let U₁, V₁ ∈ ℝ^n×k be basis matrices of 𝓤₁ and 𝓥₁, respectively. Then, (3.27) means U1TV1 being nonsingular. We consider the matrix

H~SR=0K~SRM~SR0,

where K~SR=(V1TU1)−TU1TKU1(V1TU1)−1 and M~SR=V1TMV1. Since 𝓤₁ ⊂ 𝓤 and 𝓥₁ ⊂ 𝓥, we have λ(K͠_SRM͠_SR) ≥ μ^∘2 by [3, Theorem 4.1]. By Theorem 3.1,

0≤μ∘2−λ∘2≤λ(K~SRM~SR)−λ∘2≺wωcos2⁡θM−1(1)(U1,MV1)∘sin∘2ΘM−1(U1,Φ1)+tan2⁡θM−1(1)(U1,MV1)λ∘2. (3.28)

At last, combine (3.26), (3.27) and (3.28) to give (3.25).□

Theorem 3.4

Let {𝓤, 𝓥} satisfy (3.3), and both K and M be definite. Using the notations of Theorem 3.2, we have

sinΘM−1(U,MV)(ℓ−k+1:ℓ)≺wsinΘM−1(U,Φ1)+κ∘sinΘK−1(V,Ψ1), (3.29)

sinΘK−1(V,KU)(ℓ−k+1:ℓ)≺wsinΘK−1(V,Ψ1)+κ∘sinΘM−1(U,Φ1). (3.30)

Proof

Similarly to the proof of Theorem 3.3, let k-dimensional subspaces 𝓤₂ ⊂ 𝓤 and 𝓥₂ ⊂ 𝓥 be chosen such that

ΘM−1(U2,Φ1)=ΘM−1(U,Φ1)andΘK−1(V2,Ψ1)=ΘK−1(V,Ψ1).

Based on the results in Lemma 2.1 and Theorem 3.2, we have

sinΘM−1(U,MV)(ℓ−k+1:ℓ)≤sinΘM−1(U2,MV)≤sinΘM−1(U2,MV2)≺wsinΘM−1(U2,Φ1)+κ∘sinΘK−1(V2,Ψ1)=sinΘM−1(U,Φ1)+κ∘sinΘK−1(V,Ψ1),

which gives (3.29). Similarly we can prove (3.30).□

Theorems 3.2 and 3.4 can be regarded as an extension of [21, Lemma 3.3] in which κ is a scalar λ_n/λ₁. Comparing to the results of Theorem 3.2, one may argue that an unsatisfactory part in Theorem 3.4 is the majorization upper bounds on the first smallest k components of the vectors sinΘ_M⁻¹(𝓤, M𝓥) and sinΘ_K⁻¹(𝓥, K𝓤), not for the first largest k components. In fact, if ℓ > k, sinΘ_M⁻¹(𝓤,Φ₁) = sinΘ_K⁻¹(𝓥,Ψ₁) = 0 fails to yield sinΘ_M⁻¹(𝓤, M𝓥)_(1:ℓ−k) = 0 or sinΘ_K⁻¹(𝓥, K𝓤)_(1:ℓ−k) = 0 even if both K and M are definite. For example, we consider an LREP with K = M⁻¹ = ΨΨ^T, and let 𝓤 and 𝓥 be spanned by the columns of M⁻¹-orthonormal and K⁻¹-orthonormal basis matrices U = [Φ₁, u_ℓ] ∈ ℝ^n×ℓ and V = [Ψ₁, v_ℓ] ∈ ℝ^n×ℓ, respectively, where ℓ = k + 1,

uℓ=Φ(:,k+1:k+2)×10andvℓ=Ψ(:,k+1:k+2)×2222.

Notice that σ(UTV)=σIk0022 satisfies the condition of Theorem 3.4, and

cosΘM−1(U,Φ1)=σ↑(Φ1TM−1U)=σ↑([Ik,0]).

Therefore, sinΘ_M⁻¹(𝓤,Φ₁) = 0. Similarly, we can check sinΘ_K⁻¹(𝓥, Ψ₁) = 0. However, in this example,

cosΘM−1(U,MV)=cosΘK−1(V,KU)=σ↑(UTV)=σ↑Ik0022,

which leads to sin θM−1(1)(U,MV)=sin⁡θK−1(1)(V,KU)=2/2.

4 Numerical examples

In this section, we present some numerical examples to illustrate our results in Theorems 3.1 and 3.3. In particular, we will demonstrate the terms associated with Θ_M⁻¹(𝓤, M𝓥) in majorization upper bounds of (3.6) and (3.25) are not able to be neglected.

Example 4.1

We first examine majorization upper bounds of Theorem 3.1. For simplicity, we consider diagonal matrices for K and M in this example. Take K = M = diag(λ₁, …, λ_n) with n = 100 and λ_i = i/n for 1 ≤ i ≤ n. In such a case, Φ = M^1/2 and Ψ = M^−1/2. Let k = ℓ = 3 and

E=1000100014n+4sin⁡4tan⁡4⋮⋮⋮nn+nsin⁡(n)tan⁡(n).

Consider two pairs of approximate deflating subspaces {𝓤₁, 𝓥₁} and {𝓤₂, 𝓥₂} which are spanned by the basis matrices U₁ = Φ₁ + η × E with η = 10⁻⁵, V₁ = U₁, and U₂ = U₁, V₂ = M⁻¹U₂, respectively. In such a way, the pairs {𝓤₁, 𝓥₁} and {𝓤₂, 𝓥₂} satisfy the condition (3.3). We are interested in bounding the differences between the eigenvalues λ_i for 1 ≤ i ≤ k and their approximations μ_i by (3.6). We measure the following errors, for 1 ≤ j ≤ k,

ε1,j=∑i=1j(μi2−λi2)↓,ε2,j=∑i=1j(λn−i+12cos2⁡θM−1(i)(U,MV)−λ12cos2⁡θM−1(1)(U,MV))sin2⁡θM−1(i)(U,Φ1) (4.1)

+tan2⁡θM−1(1)(U,MV)∑i=1jλi2, (4.2)

ε3,j=∑i=1j(λn−i+12−λ12)sin2⁡θM−1(i)(U,Φ1). (4.3)

Since 𝓤₂ = M𝓥₂, by Remark 3.1(c), then ε_2,j = ε_3,j for 1 ≤ j ≤ k are the upper bounds on the approximate eigenvalue errors ε_1,j associated with {𝓤₂, 𝓥₂}.

Table 4.1 reports the errors ε_2,j and ε_3,j as defined in (4.2) and (4.3) on the eigenvalue approximations ε_1,j as defined in (4.1) corresponding to {𝓤₁, 𝓥₁} and {𝓤₂, 𝓥₂}, respectively. It follows from Table 4.1 that our upper bounds provided by Theorem 3.1 are rather sharp in this example, and they are comparable to the observed errors ε_1,j. In particular, for the pair of approximate deflating subspaces {𝓤₁, 𝓥₁}, though ε_3,2 > ε_1,2 and ε_3,3 > ε_1,3, ε_3,1 in absence of the terms cos² θM−1(1) (𝓤, M𝓥) and tan² θM−1(1) (𝓤, M𝓥) is a little smaller than ε_1,1 in this example. However, for the pair {𝓤₂, 𝓥₂}, ε_3,j for 1 ≤ j ≤ k are always available to bound ε_1,j.

Table 4.1

Eigenvalue approximations ε_1,j corresponding to the approximate deflating subspace pairs {𝓤₁, 𝓥₁} and {𝓤₂, 𝓥₂} together with their associated ε_2,j and ε_3,j in Example 4.1.

Results of {𝓤₁, 𝓥₁}				Results of {𝓤₂, 𝓥₂}
j	ε_1,j	ε_2,j	ε_3,j	ε_1,j	ε_2,j = ε_3,j
1	5.7693 × 10⁻⁵	2.9445 × 10⁻⁴	5.4394 × 10⁻⁵	2.6003 × 10⁻⁵	5.4394 × 10⁻⁵
2	5.9402 × 10⁻⁵	4.4015 × 10⁻⁴	9.9443 × 10⁻⁵	2.6995 × 10⁻⁵	9.9443 × 10⁻⁵
3	5.9407 × 10⁻⁵	4.6531 × 10⁻⁴	9.9459 × 10⁻⁵	2.6997 × 10⁻⁵	9.9459 × 10⁻⁵

Example 4.2

In this example, we use a pair of matrices K and M from the linear response analysis of sodium dimer Na₂ [12] with order n = 1862. Let k = 3 < ℓ = 6, U₁ = [ϕ₁, E]+η × F, V₁ = U₁, U₂ = U₁ and V₂ = M⁻¹U₂ where η and E as defined in Example 4.1, and

F=100000⋮⋮⋮⋮⋮⋮0000016n6n+6sin⁡6cos⁡6tan⁡6cot⁡6⋮⋮⋮⋮⋮⋮nnnn+nsin⁡(n)cos⁡(n)tan⁡(n)cot⁡(n).

As Example 4.1, we also consider to bound the eigenvalue approximations based on the approximate deflating subspace pairs {𝓤₁, 𝓥₁} and {𝓤₂, 𝓥₂}, respectively, where 𝓤₁ = 𝓡(U₁), 𝓥₁ = 𝓡(V₁), 𝓤₂ = 𝓡(U₂) and 𝓥₂ = 𝓡(V₂). Since ℓ > k here, by Theorem 3.3, we compute ε_1,j, ε̃_2,j and ε_3,j for 1 ≤ j ≤ k in Table 4.2 where

ε~2,j=∑i=1jλn−i+12−λ12cos2⁡θM−1(1)(U,MV)sin2⁡θM−1(i)(U,Φ1)+tan2⁡θM−1(1)(U,MV)∑i=1jλi2,

and ε_1,j, ε_3,j as defined in (4.1) and (4.3), respectively. Table 4.2 suggests that our bounds ε̃_2,j for 1 ≤ j ≤ k are also very sharp in the case ℓ > k, while ε_3,j for 1 ≤ j ≤ k in the numerical results of {𝓤₁, 𝓥₁} underestimate ε_1,j too much in this example.

Table 4.2

Approximate eigenvalue errors ε_1,j associated with the pairs {𝓤₁, 𝓥₁} and {𝓤₂, 𝓥₂} together with their corresponding ε̃_2,j and ε_3,j of Example 4.2.

Results of {𝓤₁, 𝓥₁}				Results of {𝓤₂, 𝓥₂}
j	ε_1,j	ε̃_2,j (bound for ε_1,j)	ε_3,j	ε_1,j	ε̃_2,j = ε_3,j
1	0.7247	0.7799	1.0741 × 10⁻⁷	2.0972 × 10⁻⁹	1.0741 × 10⁻⁷
2	1.2544	1.5431	1.1451 × 10⁻⁷	3.0659 × 10⁻⁹	1.1451 × 10⁻⁷
3	1.5944	2.1888	1.1892 × 10⁻⁷	3.5445 × 10⁻⁹	1.1892 × 10⁻⁷

5 Conclusion

The pair of k-dimensional deflating subspaces {𝓡(Φ₁), 𝓡(Ψ₁)} due to its ability in recovering the eigenvalues of interest plays a vital role in some efficient numerical methods of LREP. Given a pair of ℓ-dimensional subspaces {𝓤, 𝓥} with ℓ ≥ k which approximates or contains a pair of k-dimensional subspaces approximating {𝓡(Φ₁), 𝓡(Ψ₁)}, Zhang, Xue and Li [21] established the Rayleigh-Ritz error bounds of LREP on the differences between the approximate eigenvalues and the eigenvalues of interest by the canonical angles between the exact and approximate pair of deflating subspaces in the case ℓ = k. There are two major contributions in this paper to improve the exist results in [21]. One is to obtain the Rayleigh-Ritz majorization type upper bounds for the approximate eigenvalue errors of LREP, i.e., Theorem 3.1. The other one is to extend Theorem 3.1 to Theorem 3.3 by considering ℓ > k. Numerical examples are presented to confirm the sharpness of these upper bounds, and to demonstrate the necessity of the terms on Θ_M⁻¹(𝓤, M𝓥) in Theorems 3.1 and 3.3.

From the point of view that the generalized linear response eigenvalue problem

Hz=0KM0yx=λE+E−yx=λEz

where K and M are n × n symmetric positive semi-definite and one of them is definite, E_± are n × n nonsingular matrices and E+T = E₋, can be equivalently transformed to the standard LREP by decomposing E₊ = CD^T [11, 28]. The development of Theorems 3.1, 3.2, 3.3 and 3.4 can be made to work for the generalized linear response eigenvalue problem by simple modifications: replacing Θ_M⁻¹(𝓤, M𝓥) by Θ_M⁻¹(E₋𝓤, M𝓥), Θ_M⁻¹(𝓤, Φ₁) by Θ_M⁻¹(E₋𝓤, E₋Φ₁), Θ_K⁻¹(𝓥,K𝓤) by Θ_K⁻¹(E₊𝓥,K𝓤) and Θ_K⁻¹(𝓥, Ψ₁) by Θ_K⁻¹(E₊𝓥, E₊Ψ₁). We omit the details.

Acknowledgement

The work of the first author is supported in part by National Natural Science Foundation of China NSFC-11601081 and the research fund for distinguished young scholars of Fujian Agriculture and Forestry University No. xjq201727. The work of the second author is supported in part by National Natural Science Foundation of China NSFC-11701225 and NSFC-11471122, the Fundamental Research Funds for the Central Universities No. JUSRP11719, and the Natural Science Foundation of Jiangsu Province No. BK20170173.

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Received: 2018-10-06

Accepted: 2019-05-05

Published Online: 2019-07-09

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0052

Schlagwörter für diesen Artikel

linear response eigenvalue problem; Rayleigh-Ritz approximation; canonical angles; majorization; error bounds

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