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Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

James Demmel, Ioana Dumitriu, Ryan Schneider

TL;DR

This work develops a randomized, inverse-free framework to approximately diagonalize any $n\times n$ matrix pencil $(A,B)$. Central to the approach is a generalized pseudospectral shattering result that regularizes the pencil via Ginibre perturbations, enabling a divide-and-conquer eigensolver to succeed with high probability. The inverse-free pipeline combines Implicit Repeated Squaring, GRURV/RURV rank-revealing factorizations, and DEFLATE to compute spectral projectors and recursively solve subproblems, yielding a diagonalization with backward-error guarantees in near $\,T_{MM}(n)$ time. A final randomized pencil diagonalization (RPD) routine wraps the divide-and-conquer solver to produce matrices $S,T$ and diagonal $D$ such that $A\approx SDT^{-1}$ and $B\approx ST^{-1}$, with high probability and in exact arithmetic. The results imply highly parallelizable, near-optimal complexity for inverse-free generalized eigenvalue computations and provide a robust extension of pseudospectral techniques to matrix pencils, with promising finite-precision prospects and several directions for future work.

Abstract

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling $(A,B)$ regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of $(A,B)$ in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible $S,T$ and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - ST^{-1}||_2 \leq \varepsilon$ in at most $O \left(\log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where $T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact arithmetic matrix pencil diagonalization.

Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

TL;DR

This work develops a randomized, inverse-free framework to approximately diagonalize any matrix pencil . Central to the approach is a generalized pseudospectral shattering result that regularizes the pencil via Ginibre perturbations, enabling a divide-and-conquer eigensolver to succeed with high probability. The inverse-free pipeline combines Implicit Repeated Squaring, GRURV/RURV rank-revealing factorizations, and DEFLATE to compute spectral projectors and recursively solve subproblems, yielding a diagonalization with backward-error guarantees in near time. A final randomized pencil diagonalization (RPD) routine wraps the divide-and-conquer solver to produce matrices and diagonal such that and , with high probability and in exact arithmetic. The results imply highly parallelizable, near-optimal complexity for inverse-free generalized eigenvalue computations and provide a robust extension of pseudospectral techniques to matrix pencils, with promising finite-precision prospects and several directions for future work.

Abstract

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any matrix pencil . The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible and diagonal such that and in at most operations, where is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact arithmetic matrix pencil diagonalization.
Paper Structure (35 sections, 33 theorems, 163 equations, 9 figures, 2 tables, 6 algorithms)

This paper contains 35 sections, 33 theorems, 163 equations, 9 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. The Generalized Eigenvalue Problem
  3. Divide-and-Conquer Eigensolvers
  4. Pseudospectra and Random Matrix Theory
  5. Matrix Pencil Diagonalization
  6. Complexity and Numerical Stability
  7. Outline
  8. Background
  9. Matrix Pencils, Eigenvalues, and Eigenvectors
  10. Pseudospectra
  11. Pseudospectral Shattering for Matrix Pencils
  12. Preliminaries
  13. Multi-Parameter Tail Bound
  14. Shattering
  15. Inverse-Free Divide-and-Conquer
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $(A,B) \in {\mathbb C}^{n \times n} \times {\mathbb C}^{n \times n}$ be any matrix pencil and let $\varepsilon < 1$ be any desired (backward) diagonalization accuracy. There exists an exact-arithmetic, inverse-free, and randomized divide-and-conquer algorithm that takes as inputs $(A,B)$ and $\v with probability at least $1 - O(\frac{1}{n})$. This algorithm requires at most $O(\log^2(\frac{n}{

Figures (9)

  • Figure 1: Pseudospectra of $(A,B)$ and $B^{-1}A$ for Gaussian $A,B \in {\mathbb C}^{10 \times 10}$ following \ref{['defn:pseudospectrum']} and \ref{['defn: matrix pseudospectrum']}, respectively. Eigenvalues are plotted with open circles. Pseudospectra were obtained by graphing the level curves of$\log_{10}\left[(1 + |z|)||(A - zB)^{-1}||_2\right]$ and $\log_{10}\left[||(B^{-1}A - zI)^{-1}||_2\right]$ in Matlab R2023a.
  • Figure 2: Pseudospectra of a pencil $(A,B)$ before and after perturbation. Here, $A$ is a $10 \times 10$ Jordan block, $B$ is the identity matrix, and the perturbed matrices are $\widetilde{A} = A + 10^{-7}G_1$ and $\widetilde{B} = B + 10^{-7}G_2$ for $G_1$ and $G_2$ two independent, complex Gaussian matrices. Once again eigenvalues are plotted with open circles. A grid that shatters the $10^{-8}$-pseudospectrum of $(\widetilde{A}, \widetilde{B})$ is superimposed over figure (b).
  • Figure 3: Pseudospectra of a $10 \times 10$ pencil $(\widetilde{A}, \widetilde{B})$ and its corresponding product matrix $\widetilde{B}^{-1}\widetilde{A}$. Here, $\widetilde{A} = A + 10^{-7}G_1$ and $\widetilde{B} = B + 10^{-7}G_2$ for $A$ and $B$ drawn randomly and $B$ modified to be singular (without changing its remaining singular values). In both plots, $\widetilde{B}$ is scaled so that the maximum eigenvalue has modulus one. In addition, we provide a subplot focused around the origin to examine the pseudospectra around the small eigenvalues of $(\widetilde{A}, \widetilde{B})$, which correspond to finite eigenvalues of $(A,B)$. In plot (a), we omit the pseudospectra for $\epsilon = 10^{-2}$ and $\epsilon = 10^{-3}$ since they are too close to the eigenvalues to be visible.
  • Figure 4: A diagram of RPD (\ref{['alg: RPD']}) for producing an approximate diagonalization of $(A,B)$. As in the pseudocode, we assume for simplicity that each split in EIG is made by a vertical grid line.
  • Figure 5: Performance data for RPD on the $50 \times 50$ planted-spectrum example with decreasing values of $\varepsilon$. Each plot corresponds to 500 runs of RPD.
  • ...and 4 more figures

Theorems & Definitions (75)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 65 more