Fast and Inverse-Free Algorithms for Deflating Subspaces

TL;DR

This work tackles the problem of computing spectral projectors for deflating subspaces of a regular matrix pencil $(A,B)$ without matrix inversion or full Schur decompositions. It introduces a unified inverse-free framework that uses rational-function approximations of the indicator function and Benner–Byers matrix-relations arithmetic to map eigenvalues through $r(B\backslash A)$, enabling efficient construction of right/left deflating subspace projectors. The paper develops and analyzes multiple inverse-free methods—Implicit Repeated Squaring (IRS), and sign-function based approaches including IF-Newton, IF-Halley, and IF-DWH—providing convergence and floating-point stability results, and demonstrates their practical performance on large pencils. It also discusses adapting these inverse-free techniques to generalized Schur form via randomized divide-and-conquer, highlighting potential for near-matrix-multiplication-time computations on well-conditioned problems. Collectively, the results offer a versatile toolkit for fast, stable, inversion-free projector computation with applicability to advanced divide-and-conquer eigensolvers and structured pencils.

Abstract

This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil $(A,B)$, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to computing these projectors, which combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. In an appendix, we demonstrate that recent, randomized divide-and-conquer eigensolvers -- which are built on fast methods for individual projectors -- can be adapted to produce the generalized Schur form of any matrix pencil in nearly matrix multiplication time.

Figures (6)

• Figure 1: The circles of Apollonius corresponding to $\alpha = \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$, and $\frac{1}{16}$.
• Figure 2: Result of applying $f_{\alpha}$ to portions of $C_{\alpha}^+$ (see \ref{['defn: circles_of_Apollonius']}), where $f_{\alpha}$ is the rational function defined according to \ref{['eqn: f_alpha']} and \ref{['eqn: weighted_coefficients']} with $l = \frac{1-\alpha}{1+\alpha}$ and $\alpha = 0.7$. A subplot focused on the real axis verifies \ref{['prop: dwh_convergence_bound']} for the corresponding value of $\alpha'$ (in this case $\frac{1-\alpha'}{1+\alpha'} \approx 0.6$). For reference we also mark the regions $C_{\alpha^2}^+$ and $C_{\alpha^4}^+$.
• Figure 3: Forward error for IRS, IF-Newton, IF-Halley, and IF-DWH when used to compute a spectral projector of a $500 \times 500$ pencil $(A,B)$ constructed according to \ref{['eqn: example_1']} and \ref{['eqn: example_1_lambda']}. Each plot corresponds to a different combination of eigenvalues and eigenvectors. For context, evaluating \ref{['eqn: if_dwh_steps']} with $\delta = \varepsilon/\kappa_2(X)$ for $\varepsilon$ the error attained by QZ yields $5$ when eigenvalues are well separated from zero and $8$ when they are poorly separated. This provides an upper bound on the number of iterations required by IF-DWH to match QZ (in exact arithmetic).
• Figure 4: Forward projector error for methods based on the Halley iteration. The input pencil $(A,B)$ is again $500 \times 500$ and constructed according to \ref{['eqn: example_1']}, this time with a Haar unitary eigenvector matrix $X$ and eigenvalues $\pm 4/k^{3/2}$ (plot (a)) and $\pm 4/k^2$ (plot (b)) for $1 \leq k \leq 250$, which corresponds to the listed values of $l_0$. For each modified iteration, we note in parentheses the number of standard Halley steps applied before switching to the weighted version. For context, we also label the plots with the number of iterations required by IRS and IF-Newton; to be more efficient overall, a Halley-based method must converge in fewer than half as many.
• Figure 5: A repeat of \ref{['fig: proj_4x4']} for an indefinite pencil $(A,B)$ constructed according to \ref{['eqn: indefinite_example']} for the same choices of $\Lambda$ and $X$ as in \ref{['section: examples']}.

Theorems & Definitions (49)

• Remark 1.1
• Definition 1.2: Arithmetic for Matrix Relations
• Theorem 1.3: Benner and Byers Benner_Byers
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Theorem 2.6
• Definition 3.1
• Definition 3.2