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Structured Divide-and-Conquer for the Definite Generalized Eigenvalue Problem

James Demmel, Ioana Dumitriu, Ryan Schneider

TL;DR

This work develops a fast, randomized, inverse-free divide-and-conquer solver for definite matrix pencils $(A,B)$ with Hermitian $A,B$ and Crawford number $\gamma(A,B)>0$. By extending pseudospectral shattering to definite pencils under structured perturbations (GUE or diagonal) and enforcing a structure-preserving, symmetric divide-and-conquer (EIG-DWH), the method maintains definiteness and computes spectral projectors efficiently via an inverse-free dynamically weighted Halley iteration (IF-DWH). The approach yields provably faster complexity than general divide-and-conquer for arbitrary pencils, leveraging real eigenvalues and structured perturbations to aggressively prune the spectrum through real-line splits and projector computations. The resulting solver is highly parallelizable, inverse-free, and tailored to data-driven problems where definite pencils arise, with open directions for implementation, extensions to other ensembles, and sparse settings.

Abstract

This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils $(A,B)$ in which $A$ and $B$ are Hermitian and the Crawford number $γ(A,B) = \min_{||x||_2 = 1} |x^H(A+iB)x|$ is positive. Adapted from the fastest known method for diagonalizing arbitrary matrix pencils [Foundations of Computational Mathematics 2024], the algorithm is both inverse-free and highly parallel. As in the general case, randomization takes the form of perturbations applied to the input matrices, which regularize the problem for compatibility with fast, divide-and-conquer eigensolvers -- i.e., the now well-established phenomenon of pseudospectral shattering. We demonstrate that this high-level approach to diagonalization can be executed in a structure-aware fashion by (1) extending pseudospectral shattering to definite pencils under structured perturbations (either random diagonal or sampled from the Gaussian Unitary Ensemble) and (2) formulating the divide-and-conquer procedure in a way that maintains definiteness. The result is a specialized solver whose complexity, when applied to definite pencils, is provably lower than that of general divide-and-conquer.

Structured Divide-and-Conquer for the Definite Generalized Eigenvalue Problem

TL;DR

This work develops a fast, randomized, inverse-free divide-and-conquer solver for definite matrix pencils with Hermitian and Crawford number . By extending pseudospectral shattering to definite pencils under structured perturbations (GUE or diagonal) and enforcing a structure-preserving, symmetric divide-and-conquer (EIG-DWH), the method maintains definiteness and computes spectral projectors efficiently via an inverse-free dynamically weighted Halley iteration (IF-DWH). The approach yields provably faster complexity than general divide-and-conquer for arbitrary pencils, leveraging real eigenvalues and structured perturbations to aggressively prune the spectrum through real-line splits and projector computations. The resulting solver is highly parallelizable, inverse-free, and tailored to data-driven problems where definite pencils arise, with open directions for implementation, extensions to other ensembles, and sparse settings.

Abstract

This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils in which and are Hermitian and the Crawford number is positive. Adapted from the fastest known method for diagonalizing arbitrary matrix pencils [Foundations of Computational Mathematics 2024], the algorithm is both inverse-free and highly parallel. As in the general case, randomization takes the form of perturbations applied to the input matrices, which regularize the problem for compatibility with fast, divide-and-conquer eigensolvers -- i.e., the now well-established phenomenon of pseudospectral shattering. We demonstrate that this high-level approach to diagonalization can be executed in a structure-aware fashion by (1) extending pseudospectral shattering to definite pencils under structured perturbations (either random diagonal or sampled from the Gaussian Unitary Ensemble) and (2) formulating the divide-and-conquer procedure in a way that maintains definiteness. The result is a specialized solver whose complexity, when applied to definite pencils, is provably lower than that of general divide-and-conquer.

Paper Structure

This paper contains 15 sections, 20 theorems, 92 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Symmetric Pseudospectra
  5. The Crawford Number, Numerically
  6. Symmetric Pseudospectral Shattering
  7. Preliminaries
  8. Shattering Two Ways
  9. Additional Bounds for $\gamma(\widetilde{A},\widetilde{B})$
  10. Divide-and-Conquer for Definite Pencils
  11. Maintaining Definiteness
  12. Structured Divide-and-Conquer
  13. Conclusion
  14. Acknowledgements
  15. GRURV and IF-DWH

Key Result

Theorem 2.1

Let $(A,B)$ be an $n \times n$ definite pencil and suppose that the Hermitian matrices $E,F \in {\mathbb C}^{n \times n}$ satisfy Then the pencil $(\widetilde{A}, \widetilde{B}) = (A+E,B+F)$ is definite. Moreover, if $\lambda_1 \leq \cdots \leq \lambda_n$ and $\widetilde{\lambda}_1 \leq \cdots \leq \widetilde{\lambda}_n$ are the eigenvalues of $(A,B)$ and $(\widetilde{A},\widetilde{B})$, respect

Figures (3)

  • Figure 1: Pseudospectra of a $10 \times 10$ pencil $(A,B)$, constructed so that $B^{-1}A$ is a Jordan block, before and after perturbation. The black curves trace the boundaries of $\epsilon$-pseudospectra for two choices of $\epsilon$; the eigenvalues they contained are marked with dark circles. In the rightmost plot we include a random grid that shatters the tighter pseudospectrum of the perturbed problem, identifying in red a potential split for use in divide-and-conquer.
  • Figure 2: Spectra and symmetric $\epsilon$-pseudospectra of a $10 \times 10$ definite pencil $(A,B)$, constructed as in \ref{['eqn: shattering_examples']}, before and after perturbations. Eigenvalues are marked with ticks, while components of $\Lambda_{\epsilon}^{\text{sym}}(A,B)$ are plotted with filled circles. $(A,B)$ initially has a repeated eigenvalue at $+1$, which is highlighted in red, and each perturbation has size $\mu = 10^{-6}$. Initially, $\gamma(A,B) = 5.91 \times 10^{-8}$ while in the remaining subfigures (read clockwise) $\gamma(\widetilde{A},\widetilde{B})$ is $2.44 \times 10^{-7}$, $5.31 \times 10^{-8}$, and $1.04 \times 10^{-7}$, respectively.
  • Figure 3: Impact of perturbation size on $\gamma(\widetilde{A},\widetilde{B})$. Each plot presents a histogram of $\gamma(\widetilde{A},\widetilde{B})$ for $200$ GUE perturbations of a $500 \times 500$ definite pencil $(A,B)$, which is constructed so that $\gamma(A,B) = \sqrt{2} \times 10^{-7}$. $\mu$ increases from left to right; at its smallest, it is equal to the bound from \ref{['thm: no_scaling_shattering']}. We mark $\gamma(A,B)$ and the upper bound $\gamma(A,B)+2\mu\sqrt{\frac{\log(n)}{n}}$ with dashed black and red lines, respectively. The latter holds with probability at least $1-n^{-1}$ by \ref{['prop: probabilistic_upper_bound']}. In each case, Crawford numbers are computed via the algorithm of Kressner, Lu, and Vandereycken subspace_crawfno.

Theorems & Definitions (39)

  • Theorem 2.1: Stewart STEWART_DEFINITE
  • Theorem 2.2: Stewart STEWART_DEFINITE
  • Lemma 2.3: Elsner and Sun ELSNER1982341
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6: Definite Bauer-Fike
  • proof
  • Definition 3.1
  • Lemma 3.2
  • ...and 29 more