Minimization principles and computation for the generalized linear response eigenvalue problem

Abstract

The minimization principle and Cauchy-like interlacing inequalities for the generalized linear response eigenvalue problem are presented. Based on these theoretical results, the best approximations through structure-preserving subspace projection and a locally optimal block conjugate gradient-like algorithm for simultaneously computing the first few smallest eigenvalues with the positive sign are proposed. Numerical results are presented to illustrate essential convergence behaviors of the proposed algorithm.

Appendix: Best approximations: the singular/unequal dimension case

This appendix continues the investigation in Sect. 4 to seek best approximate eigenpairs of $H-\lambda E$ for given $\{\mathcal{U}, \mathcal{V}\}$, a pair of approximate deflating subspaces of $H-\lambda E$ with $\dim (\mathcal{U})=\ell _1$ and $\dim (\mathcal{V})=\ell _2$. In Sect. 4, we have treated the case in which $\ell _1=\ell _2$ and $W \mathop {=}\limits ^{{\hbox {def}}}U^{{{\mathrm{T}}}}E_+V$ is nonsingular, where $U\in {\mathbb R}^{n\times \ell _1},\,V\in {\mathbb R}^{n\times \ell _2}$ are the basis matrices of $\mathcal{U}$ and $\mathcal{V}$, respectively. In what follows, we will focus on the general case: $\ell _1$ and $\ell _2$ are not necessarily equal or $W$ may be singular.

The case is much more complicated than the one in section 4, but it can be handled in the similar way as in [3] which is for $E=I_{2n}$. So we will simply summarize the results and the reader is referred to [1, Appendix A] for detail.

Factorize

$$\begin{aligned} W=W_1^{{{\mathrm{T}}}}W_2, \quad W_i\in {\mathbb R}^{r\times \ell _i},\quad r={{\mathrm{rank}}}(W)\le \min _i\ell _i. \end{aligned}$$

(8.1)

Both $W_i$ have full row rank. Factorize⁵

$$\begin{aligned} W_i^{{{\mathrm{T}}}}=Q_i\begin{bmatrix} R_i \\ 0 \end{bmatrix} \quad \hbox {for} i=1,2, \end{aligned}$$

(8.2)

where $R_i\in {\mathbb R}^{r\times r},\,Q_i\in {\mathbb R}^{\ell _i\times \ell _i}$ ($i=1,2$) are nonsingular. Partition

Set

$$\begin{aligned} \widehat{H}_{\hbox {SR}}=\begin{bmatrix} 0&R_1^{-1}\mathcal {K}_{11}R_1^{-{{\mathrm{T}}}} \\ R_2^{-1}\mathcal {M}_{11}R_2^{-{{\mathrm{T}}}}&0 \end{bmatrix}\in {\mathbb R}^{2r\times 2r}, \end{aligned}$$

(8.4)

where $K_{22}^{\dagger }$ and $M_{22}^{\dagger }$ are the Moore-Penrose inverses of $K_{22}$ and $M_{22}$, respectively, and

$$\begin{aligned} \mathcal {K}_{11} =K_{11}-K_{12}K_{22}^{\dagger }K_{12}^{{{\mathrm{H}}}}, \quad \mathcal {M}_{11} =M_{11}-M_{12}M_{22}^{\dagger }M_{12}^{{{\mathrm{H}}}}. \end{aligned}$$

(8.5)

Denote by $\mu _j$ for $j=1,\ldots ,r$ the eigenvalues with the positive sign of $\widehat{H}_{\hbox {SR}}$ in the ascending order and by $\hat{z}_j$ the associated eigenvectors:

$$\begin{aligned} \widehat{H}_{\hbox {SR}}\hat{z}_j=\mu _j\hat{z}_j, \quad \hat{z}_j=\begin{bmatrix} \hat{y}_j \\ \hat{x}_j \end{bmatrix}. \end{aligned}$$

(8.6)

It can be verified that $\rho (\tilde{x}_j,\tilde{y}_j)=\mu _j\quad \hbox {for} j=1,\ldots ,r,$ where

$$\begin{aligned} \tilde{x}_j=UQ_1^{-{{\mathrm{T}}}}\begin{bmatrix} R_1^{-{{\mathrm{T}}}}\hat{x}_j \\ u_j \end{bmatrix} , \quad \tilde{y}_j=VQ_2^{-{{\mathrm{T}}}}\begin{bmatrix} R_2^{-{{\mathrm{T}}}}\hat{y}_j \\ v_j \end{bmatrix} \end{aligned}$$

(8.7)

for any $u_j$ and $v_j$ satisfying

$$\begin{aligned} K_{22}u_j=-K_{12}^{{{\mathrm{T}}}}R_1^{-{{\mathrm{T}}}}\hat{x}_j, \quad M_{22}v_j=-M_{12}^{{{\mathrm{T}}}}R_2^{-{{\mathrm{T}}}}\hat{y}_j. \end{aligned}$$

(8.8)

Naturally the approximate eigenvectors of $H-\lambda E$ should be taken as

$$\begin{aligned} \tilde{z}_j=\begin{bmatrix} \tilde{y}_j \\ \tilde{x}_j \end{bmatrix}\quad \hbox {for} j=1,\ldots ,r. \end{aligned}$$

(8.9)

Theorem 8.1

Let $\{\mathcal{U}, \mathcal{V}\}$ be a pair of approximate deflating subspaces of $H-\lambda E$ with $\dim (\mathcal{U})=\ell _1$ and $\dim (\mathcal{V})=\ell _2$, and let $U\in {\mathbb R}^{n\times \ell _1},\,V\in {\mathbb R}^{n\times \ell _2}$ be the basis matrices of $\mathcal{U}$ and $\mathcal{V}$, respectively. Let $\widehat{H}_{\hbox {SR}}$ be defined by (8.4). Then the best approximations to $\lambda _j$ for $1\le j\le k$ in the sense of (4.1) are the corresponding eigenvalues of $\widehat{H}_{\hbox {SR}}$, with the corresponding approximate eigenvectors given by (8.7)–(8.9).

Despite much more complicated appearance of $\widehat{H}_{\hbox {SR}}$ compared to $H_{\hbox {SR}}$ in Sect. 4, our next theorem surprisingly unifies both.

Theorem 8.2

The eigenvalues of $\widehat{H}_{\hbox {SR}}$ in (8.4) are the same as the finite eigenvalues of

$$\begin{aligned} \check{H}-\lambda \check{E}:&=\begin{bmatrix} U&0\\ 0&V \end{bmatrix}^{{{\mathrm{T}}}}(H-\lambda E)\begin{bmatrix} V&0\\ 0&U \end{bmatrix} \\&=\begin{bmatrix} 0&U^{{{\mathrm{T}}}}KU \\ V^{{{\mathrm{T}}}}MV&0 \end{bmatrix}-\lambda \begin{bmatrix} U^{{{\mathrm{T}}}}E_+V&\\&V^{{{\mathrm{T}}}}E_-U \end{bmatrix} \nonumber \end{aligned}$$

(8.10)

and the eigenvector $\hat{z}=\begin{bmatrix} \hat{y} \\ \hat{x} \end{bmatrix}$ of $\widehat{H}_{\hbox {SR}}$ and the eigenvector $\check{z}=\begin{bmatrix} \check{y} \\ \check{x} \end{bmatrix}$ of the pencil (8.10) associated with a finite eigenvalue are related by

$$\begin{aligned} \check{x}=Q_1^{-{{\mathrm{T}}}}\begin{bmatrix} R_1^{-{{\mathrm{T}}}}\hat{x} \\ -K_{22}^{\dagger }K_{12}^{{{\mathrm{T}}}}R_1^{-{{\mathrm{T}}}}\hat{x}+g \end{bmatrix}, \quad \check{y}=Q_2^{-{{\mathrm{T}}}}\begin{bmatrix} R_2^{-{{\mathrm{T}}}}\hat{y} \\ -M_{22}^{\dagger }M_{12}^{{{\mathrm{T}}}}R_2^{-{{\mathrm{T}}}}\hat{y}+h \end{bmatrix}, \end{aligned}$$

(8.11)

where $g$ is any vector in the kernel of $K_{22}$ and $h$ is any vector in the kernel of $M_{22}$. In particluar, if $\ell _1=\ell _2=r$, the relation in (8.11) is simplified to $\hat{z}=(W_2\oplus W_1)\check{z}$ as in Theorem 4.2.

Proof

Let $P_i=Q_i^{-{{\mathrm{T}}}}(R_i^{-{{\mathrm{T}}}}\oplus I_{\ell _i-r})$ for $i=1,2$ and both are nonsingular. It can be verified that

$$\begin{aligned} (P_1\oplus P_2)^{{{\mathrm{T}}}}(\check{H}-\lambda \check{E})(P_2\oplus P_1) =\begin{bmatrix} 0&\widehat{K} \\ \widehat{M}&0 \end{bmatrix}-\lambda \begin{bmatrix} \widehat{I}&\\ 0&\widehat{I}^{\,{{\mathrm{T}}}} \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned} \widehat{M}&=\begin{bmatrix} R_2^{-1}&\\&I_{\ell _2-r} \end{bmatrix} \begin{bmatrix} M_{11}&M_{12} \\ M_{12}^{{{\mathrm{T}}}}&M_{22} \end{bmatrix} \begin{bmatrix} R_2^{-{{\mathrm{T}}}}&\\&I_{\ell _2-r} \end{bmatrix}, \end{aligned}$$

(8.12)

$$\begin{aligned} \widehat{K}&=\begin{bmatrix} R_1^{-1}&\\&I_{\ell _1-r} \end{bmatrix} \begin{bmatrix} K_{11}&K_{12} \\ K_{12}^{{{\mathrm{T}}}}&K_{22} \end{bmatrix} \begin{bmatrix} R_1^{-{{\mathrm{T}}}}&\\&I_{\ell _1-r} \end{bmatrix}, \end{aligned}$$

(8.13)

$$\begin{aligned} \widehat{I}&=\begin{bmatrix} I_r&\\&0 \end{bmatrix}\in {\mathbb R}^{\ell _1\times \ell _2}, \end{aligned}$$

(8.14)

and $K_{ij}$ and $M_{ij}$ are defined by 8.3. Since $K$ and $M$ are positive (semi)definite, we have ${{\mathrm{span}}}(K_{12}^{{{\mathrm{T}}}})\subseteq {{\mathrm{span}}}(K_{22})$ and ${{\mathrm{span}}}(M_{12}^{{{\mathrm{T}}}})\subseteq {{\mathrm{span}}}(M_{22})$ and consequently

$$\begin{aligned} K_{22}K_{22}^{\dagger }K_{12}^{{{\mathrm{T}}}}=K_{12}^{{{\mathrm{T}}}}, \quad M_{22}M_{22}^{\dagger }M_{12}^{{{\mathrm{T}}}}=M_{12}^{{{\mathrm{T}}}}. \end{aligned}$$

(8.15)

Let

$$\begin{aligned} Z_1=\begin{bmatrix} I_r&0 \\ -K_{22}^{\dagger }K_{12}^{{{\mathrm{T}}}}R_1^{-{{\mathrm{T}}}}&I_{\ell _1-r} \end{bmatrix}, \quad Z_2=\begin{bmatrix} I_r&0 \\ -M_{22}^{\dagger }M_{12}^{{{\mathrm{T}}}}R_2^{-{{\mathrm{T}}}}&I_{\ell _2-r} \end{bmatrix}. \end{aligned}$$

It can be verified that $Z_1^{{{\mathrm{T}}}}\widehat{I} Z_2=\widehat{I}$ and, after using (8.15),

$$\begin{aligned} Z_1^{{{\mathrm{T}}}}\widehat{K} Z_1=\begin{bmatrix} R_1^{-1}\mathcal {K}_{11} R_1^{-{{\mathrm{T}}}}&0 \\ 0&K_{22} \end{bmatrix}, \quad Z_2^{{{\mathrm{T}}}}\widehat{M} Z_2=\begin{bmatrix} R_2^{-1}\mathcal {M}_{11} R_2^{-{{\mathrm{T}}}}&0 \\ 0&M_{22} \end{bmatrix}, \end{aligned}$$

where $\mathcal {K}_{11}$ and $\mathcal {M}_{11}$ are defined in (8.5). Hence $(P_1Z_1\oplus P_2Z_2)^{{{\mathrm{T}}}}(\check{H}-\lambda \check{E})(P_2Z_2\oplus P_1Z_1)$ is

whose finite eigenvalues are the eigenvalues of

$$\begin{aligned} \begin{bmatrix} 0&R_1^{-1}\mathcal {K}_{11} R_1^{-{{\mathrm{T}}}} \\ R_2^{-1}\mathcal {M}_{11} R_2^{-{{\mathrm{T}}}}&0 \end{bmatrix}-\lambda I_{2r}=\widehat{H}_{\hbox {SR}}-\lambda I_{2r}. \end{aligned}$$

(8.17)

Now we turn to look for the eigenvector relation. Given an eigenvector $\hat{z}=\begin{bmatrix} \hat{y} \\ \hat{x} \end{bmatrix}$ of $\widehat{H}_{\hbox {SR}}$, we conclude by comparing (8.16) and (8.17) that the corresponding eigenvector of the matrix pencil (8.16) is

$$\begin{aligned} \begin{bmatrix} \hat{y} \\ h \\ \hat{x} \\ g \end{bmatrix}, \end{aligned}$$

where $g$ is any vector in the kernel of $K_{22}$ and $h$ is any vector in the kernel of $M_{22}$. Therefore the corresponding eigenvector $\check{z}=\begin{bmatrix} \check{y} \\ \check{x} \end{bmatrix}$ of $\check{H}-\lambda \check{E}$ is given by

$$\begin{aligned} \check{x}=P_1Z_1\begin{bmatrix} \hat{x} \\ g \end{bmatrix}, \quad \check{y}=P_2Z_2\begin{bmatrix} \hat{y} \\ h \end{bmatrix} \end{aligned}$$

which, after simplification, yields (8.11). $\square$

The next theorem says that there are Cauchy-like interlacing inequalities for $\widehat{H}_{\hbox {SR}}$, too. We omit its proof because its similarity to [3, Theorem 8.3] (see also [1, Appendix A]).

Theorem 8.3

Assume the conditions of Theorem 8.1. Then

$$\begin{aligned} \lambda _i\le \mu _i\le \,\lambda _{i+2n-(\ell _1+\ell _2)}\quad \hbox {for} 1\le i\le r, \end{aligned}$$

(8.18)

where $\lambda _{i+2n-(\ell _1+\ell _2)}=\infty$ if $i+2n-(\ell _1+\ell _2)>n$.