Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Forward Error-Oriented Iterative Refinement for Eigenvectors of a Real Symmetric Matrix

Takeshi Terao, Katsuhisa Ozaki

Abstract

In this paper, we discuss numerical methods for the eigenvalue decomposition of real symmetric matrices. While many existing methods can compute approximate eigenpairs with sufficiently small backward errors, the magnitude of the resulting forward errors is often unknown. Consequently, when high-precision numerical solutions are required, the computational cost tends to increase significantly because backward errors must be reduced to an excessive degree. To address this issue, we propose an efficient approximation algorithm that aims to achieve a prescribed forward error, together with a high-accuracy numerical algorithm based on the Ozaki scheme -- an emulation technique for matrix multiplication -- adapted to this problem. Since the proposed method is not primarily focused on reducing backward errors, the computational cost can be significantly reduced. Finally, we present numerical experiments to evaluate the efficiency of the proposed method.

Forward Error-Oriented Iterative Refinement for Eigenvectors of a Real Symmetric Matrix

Abstract

In this paper, we discuss numerical methods for the eigenvalue decomposition of real symmetric matrices. While many existing methods can compute approximate eigenpairs with sufficiently small backward errors, the magnitude of the resulting forward errors is often unknown. Consequently, when high-precision numerical solutions are required, the computational cost tends to increase significantly because backward errors must be reduced to an excessive degree. To address this issue, we propose an efficient approximation algorithm that aims to achieve a prescribed forward error, together with a high-accuracy numerical algorithm based on the Ozaki scheme -- an emulation technique for matrix multiplication -- adapted to this problem. Since the proposed method is not primarily focused on reducing backward errors, the computational cost can be significantly reduced. Finally, we present numerical experiments to evaluate the efficiency of the proposed method.
Paper Structure (15 sections, 1 theorem, 46 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 15 sections, 1 theorem, 46 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Previous work
  3. Notation
  4. Pairwise arithmetic
  5. Error-free transformation for matrix multiplication
  6. Iterative refinement for symmetric eigenvalue decomposition
  7. Main result
  8. Outline
  9. Definitions and assumptions
  10. Matrix splitting method
  11. Numerical experiments
  12. Implementation
  13. Test for random matrices
  14. Sparse matrices
  15. Summary

Key Result

Theorem 1

Let $A \in \mathbb{R}^{n \times n}$ be a real symmetric matrix with all distinct eigenvalues. Apply Algorithm 1 to $A$ and $\widehat{X} \in \mathbb{R}^{n \times n}$. Let $\epsilon := \|E\|$, and let $e_j$ be the $j$-th column of $E$. If then

Figures (4)

  • Figure 1: Comparison of forward error $\|X-\widehat{X}\|_2$, and backward errors $\|\widehat{X}^T\widehat{X}-I\|_2$ and $\|\widehat{X}^TA\widehat{X}-\widehat{D}\|_2$ for an eigenvector matrix ($n=100$). The condition number is $\|A\|_2\cdot\|A^{-1}\|_2$.
  • Figure 2: Goal results of proposed method $(n=100)$. Here, $\delta$ is the specifiable scalar, and the condition number is $\|A\|_2\cdot\|A^{-1}\|_2$.
  • Figure 3: Convergence history of forward error $\|X - \widehat{X}\|$ for proposed method ($\mathtt{cnd}=10^{10}$).
  • Figure 4: Convergence history of forward error $\|X - \widehat{X}\|$ for method in uchino2024high ($\mathtt{cnd}=10^{10}$).

Theorems & Definitions (1)

  • Theorem 1