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Local Convergence Behavior of Extended LOBPCG for Computing Eigenvalues of Hermitian Matrices

Zhechen Shen, Xin Liang

TL;DR

This paper analyzes the local convergence of an extended LOBPCG method for computing extreme eigenvalues of Hermitian matrices, establishing sharper asymptotic rates than previous work. It develops a Chebyshev-polynomial framework that links iterate errors to polynomial actions of the matrix, and applies this to several LOCG variants: basic, larger Krylov subspaces, historical-term extensions, and multi-eigenvalue blocks. The authors provide explicit rate bounds and show empirical validation across representative problems, demonstrating practical improvements from larger subspaces and historical terms. The results generalize to generalized Hermitian problems and remain robust under preconditioning, offering a quantitative tool for analyzing gradient-type eigensolvers beyond LOBPCG.

Abstract

This paper provides a comprehensive and detailed analysis of the local convergence behavior of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for computing the extreme eigenvalue of a Hermitian matrix. The convergence rates derived in this work are either obtained for the first time or sharper than those previously established, including those in Ovtchinnikov's work ({\em SIAM J. Numer. Anal.}, 46(5):2567--2592, 2008). The study also extends to generalized problems, including Hermitian matrix polynomials that admit an extended form of the Rayleigh quotient. The new approach used to obtain these rates may also serve as a valuable tool for the convergence analysis of other gradient-type optimization methods.

Local Convergence Behavior of Extended LOBPCG for Computing Eigenvalues of Hermitian Matrices

TL;DR

This paper analyzes the local convergence of an extended LOBPCG method for computing extreme eigenvalues of Hermitian matrices, establishing sharper asymptotic rates than previous work. It develops a Chebyshev-polynomial framework that links iterate errors to polynomial actions of the matrix, and applies this to several LOCG variants: basic, larger Krylov subspaces, historical-term extensions, and multi-eigenvalue blocks. The authors provide explicit rate bounds and show empirical validation across representative problems, demonstrating practical improvements from larger subspaces and historical terms. The results generalize to generalized Hermitian problems and remain robust under preconditioning, offering a quantitative tool for analyzing gradient-type eigensolvers beyond LOBPCG.

Abstract

This paper provides a comprehensive and detailed analysis of the local convergence behavior of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for computing the extreme eigenvalue of a Hermitian matrix. The convergence rates derived in this work are either obtained for the first time or sharper than those previously established, including those in Ovtchinnikov's work ({\em SIAM J. Numer. Anal.}, 46(5):2567--2592, 2008). The study also extends to generalized problems, including Hermitian matrix polynomials that admit an extended form of the Rayleigh quotient. The new approach used to obtain these rates may also serve as a valuable tool for the convergence analysis of other gradient-type optimization methods.
Paper Structure (12 sections, 9 theorems, 91 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 9 theorems, 91 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Main results
  4. The simplest case $\mathop{\mathrm{LOCG}}\nolimits(1,1,1)$
  5. Larger Krylov subspace: $\mathop{\mathrm{LOCG}}\nolimits(1,m_e,1)$
  6. More historical terms: $\mathop{\mathrm{LOCG}}\nolimits(1,1,m_h)$
  7. More eigenvalues: $\mathop{\mathrm{LOCG}}\nolimits(n_b,1,1)$
  8. Numerical Experiments
  9. Concluding remarks
  10. Acknowledgement
  11. Exact line search in the block version
  12. Proofs

Key Result

Theorem 2.1

For the sequences $\{*\}{\rho_{k;j}}$, $\{*\}{x_{k;j}}$ (the $j$th column of $X_k$) generated by $\mathop{\mathrm{LOCG}}\nolimits(n_b,m_e,m_h)$, only one of the following two mutually exclusive situations can occur: Moreover, if $X_*^{\mathop{\mathrm{H}}\nolimits}X_0$ is nonsingular, then $\widehat{\rho}_j=\lambda_j$ and $\widehat{x}_j$ are the associated eigenvectors.

Figures (3)

  • Figure 1: Laplacian 50
  • Figure 2: Cluster-Outlier
  • Figure 3: Outlier-Cluster

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Lemma 2.1: bennerL2022convergence
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Remark 3.1
  • Theorem 3.6
  • ...and 11 more