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Iterative Refinement for a Subset of Eigenvectors of Symmetric Matrices via Matrix Multiplications

Takeshi Terao, Katsuhisa Ozaki, Toshiyuki Imamura, Takeshi Ogita

Abstract

We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method expresses the eigenvector error through a correction matrix that can be approximated efficiently from Rayleigh quotients and residuals. Unlike refinement methods for a single eigenpair or for a full eigenbasis, the proposed method refines only the selected $k$ eigenvectors using $\mathcal{O}(nk)$ additional storage, and its dominant work can be organized as matrix--matrix multiplications. Under an eigenvalue separation condition, the refinement converges linearly; we also provide a conservative sufficient condition. Practical variants of the separation condition (e.g., via shifting) enable targeting other extremal parts of the spectrum. For tightly clustered eigenvalues, we discuss limitations and show that preprocessing can restore convergence in a representative sparse example. Numerical experiments on dense test matrices and sparse matrices from the SuiteSparse Matrix Collection illustrate attainable accuracy and problem-dependent convergence.

Iterative Refinement for a Subset of Eigenvectors of Symmetric Matrices via Matrix Multiplications

Abstract

We develop an iterative refinement method that improves the accuracy of a user-chosen subset of eigenvectors () of an real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method expresses the eigenvector error through a correction matrix that can be approximated efficiently from Rayleigh quotients and residuals. Unlike refinement methods for a single eigenpair or for a full eigenbasis, the proposed method refines only the selected eigenvectors using additional storage, and its dominant work can be organized as matrix--matrix multiplications. Under an eigenvalue separation condition, the refinement converges linearly; we also provide a conservative sufficient condition. Practical variants of the separation condition (e.g., via shifting) enable targeting other extremal parts of the spectrum. For tightly clustered eigenvalues, we discuss limitations and show that preprocessing can restore convergence in a representative sparse example. Numerical experiments on dense test matrices and sparse matrices from the SuiteSparse Matrix Collection illustrate attainable accuracy and problem-dependent convergence.
Paper Structure (24 sections, 7 theorems, 83 equations, 3 figures, 6 tables, 4 algorithms)

This paper contains 24 sections, 7 theorems, 83 equations, 3 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Results
  4. Algorithm
  5. Construction of the orthogonal matrix H
  6. Iterative refinement for a subset of eigenvectors
  7. Implementation details
  8. Stopping criteria
  9. Precision and orthogonality
  10. Choice of $\delta$
  11. Convergence Analysis
  12. Previous work
  13. Analysis of the proposed method
  14. Sufficient Conditions
  15. Handling Clustered Eigenvalues
  16. ...and 9 more sections

Key Result

Lemma 1

For eq:HhatHhat and eq:HhatA, if $\epsilon=\|E\|<1$, then hold.

Figures (3)

  • Figure 1: Convergence history of the relative residual norm $\|A\widehat{X}-\widehat{X}\widetilde{D}_1\|_F/\|A\|$ with $k=1, 25$, and $50$ for n=100 (with $K=k$). Subfigures (a) and (b) correspond to mode=3 and mode=4, respectively.
  • Figure 2: Convergence history of the relative eigenpair residual $\|A\widehat{x}^{(p_j)}-\widehat{\lambda}_{p_j}\widehat{x}^{(p_j)}\|_2/\|A\|$ for $j=1,\dots,5$ (n=100). Each panel is labeled with $(K,\texttt{mode})$.
  • Figure 3: Convergence history of the relative residual norm $\|A\widehat{X}-\widehat{X}\widetilde{D}_1\|_F/\|A\|$ with $k=5$ and $K=10$. Dashed horizontal lines indicate the residual levels of the initial single-precision and double-precision eigs outputs.

Theorems & Definitions (11)

  • Lemma 1: Ogita--Aishima ogita2018iterative
  • Lemma 2: Ogita--Aishima ogita2018iterative
  • Lemma 3
  • Proof 1
  • Lemma 4
  • Proof 2
  • Corollary 5
  • Theorem 6
  • Proof 3
  • Theorem 7
  • ...and 1 more