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A preconditioning technique of Gauss--Legendre quadrature for the logarithm of symmetric positive definite matrices

Fuminori Tatsuoka, Tomohiro Sogabe, Tomoya Kemmochi, Shao-Liang Zhang

TL;DR

The paper tackles efficient computation of the principal matrix logarithm for SPD matrices when the condition number is large. It introduces a preconditioning strategy that combines a scaling $\widetilde{A}=cA$ with a preconditioner $\widetilde{P}_s=(\widetilde{A}+sI)^{-1}$ to render two GL evaluations effective via $\log(\widetilde{A}\widetilde{P}_s)$ and $\log(\widetilde{P}_s)$, achieving a theoretical reduction of conditioning from $\kappa(A)$ to $\sqrt{\kappa(A)}$ (with optimal $s=1$). The authors provide error bounds showing accelerated convergence $\mathcal{O}(e^{-\rho(\sqrt{\kappa(\widetilde{A})}) m/2})$ and demonstrate, through numerical experiments, that the preconditioned GL (PGL) method outperforms GL and DE in the practical range $130 \lesssim \kappa(\widetilde{A}) \lesssim 3.0\times 10^5$. This yields faster computation of $\log(A)$ for high-condition SPD matrices and extends the applicability of quadrature-based matrix functions to challenging regimes.

Abstract

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between $130$ and $3.0\times 10^5$.

A preconditioning technique of Gauss--Legendre quadrature for the logarithm of symmetric positive definite matrices

TL;DR

The paper tackles efficient computation of the principal matrix logarithm for SPD matrices when the condition number is large. It introduces a preconditioning strategy that combines a scaling with a preconditioner to render two GL evaluations effective via and , achieving a theoretical reduction of conditioning from to (with optimal ). The authors provide error bounds showing accelerated convergence and demonstrate, through numerical experiments, that the preconditioned GL (PGL) method outperforms GL and DE in the practical range . This yields faster computation of for high-condition SPD matrices and extends the applicability of quadrature-based matrix functions to challenging regimes.

Abstract

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between and .

Paper Structure

This paper contains 5 sections, 2 theorems, 14 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Convergence of the GL quadrature for the logarithm of SPD matrices
  3. Preconditioning
  4. Numerical experiments
  5. Conclusion

Key Result

Theorem 1

Suppose that both $A,P \in {\mathbb{C}}^{n \times n}$ have no eigenvalues on $\{ z \in {\mathbb{C}}:z \notin ( - \infty,0\rbrack\}$ and that $AP = PA$. For each eigenvalue $\lambda$ of $A$ there is an eigenvalue $\mu$ of $P$ such that $\lambda + \mu$ is an eigenvalue of $A + P$, and the eigenvalue $

Figures (2)

  • Figure 1: Convergence speed of quadrature formulas for $\log(A)$, i.e., the convergence of a quadrature formula is fast if the values in the graph are large.
  • Figure 2: Convergence profiles of DE, GL, and PGL. We also plotted the error of the GL quadrature for $\log(c'\widetilde{A}\widetilde{P}_1)$ and $\log(c"\widetilde{P}_1)$ that are used in PGL.

Theorems & Definitions (3)

  • Theorem 1: see higham2008functions
  • Theorem 2
  • proof