A posteriori error bounds for the block-Lanczos method for matrix function approximation

TL;DR

This work develops a posteriori error bounds for block-Lanczos-FA to approximate $f(\mathbf{H})\mathbf{V}$ with Hermitian $\mathbf{H}$, extending previous single-vector results. The main bound expresses the error as the product of a computable contour-integral term and the linear-system error $\|\textup{err}_k(w)\|$, with the integral computable from the block-Lanczos data and $f$ analytic on a contour $\Gamma$ enclosing the spectra. The theory incorporates a spectral-shape parameter $Q_S(w,z)$ and remains applicable in finite precision, where a perturbation term is added to the recurrence; numerical experiments show robustness to block size and effective use as a stopping criterion, with contour choices like Pac-Man guiding parameter selection. The results enable practical, spectrally aware error control for block-Lanczos-FA in applications such as quadratic form estimation, stochastic trace estimation, and quantum-physics related computations. Overall, the paper advances reliable, scalable error control for block-based matrix function approximations and provides actionable guidance for hyperparameter choices and stopping criteria. In addition, supporting data are made available, supporting reproducibility.}

Abstract

We extend the error bounds from [SIMAX, Vol. 43, Iss. 2, pp. 787-811 (2022)] for the Lanczos method for matrix function approximation to the block algorithm. Numerical experiments suggest that our bounds are fairly robust to changing block size and have the potential for use as a practical stopping criteria. Further experiments work towards a better understanding of how certain hyperparameters should be chosen in order to maximize the quality of the error bounds, even in the previously studied block-size one case.

Figures (9)

• Figure 1: Imaginary part of the top-left entry of $f(z)\textup{err}_k(z)$ near the spectrum of $\mathbf H$ for block size $b = 1, 2, 3, 4, 8, 16$. Note that larger oscillations will result in more slack in the triangle inequality used between \ref{['eqn:error_CIF']} and \ref{['eqn:triangle_ineq']}.
• Figure 2: Ratio $T(z)$ defined in \ref{['eqn:slack ratio']} (with $w = \lambda_{\textup{min}}/100$) near the spectrum of $\mathbf H$ for block sizes $b = 1, 2, 3, 4, 8, 16$. Note that larger values of $T(z)$ result in more slack between \ref{['eqn:triangle_ineq']} and the result of \ref{['thm:main']}.
• Figure 3: Pac-Man contour with parameters $R$, $\Theta$, $O$
• Figure 4: Performance of the Pac-Man contour across various parameters $R$ and $\Theta$ for $f(x) = \sqrt{x}$. The block-Lanczos-FA error is measured with $w = 0$. The bound "triangle inequality" is \ref{['eqn:triangle_ineq']} and the bound "computable bound" is \ref{['thm:main']}, both evaluated numerically. When $R$ and $\Theta$ are both reasonably large, the quality of the bounds is extremely good, but for some parameter choices, the bounds may have significant slack.
• Figure 5: Bound divided by block-Lanczos-FA for the Pac-Man contour across various parameters $R$ and $\Theta$ for $f(x) = \sqrt{x}$ at $k=30$. The block-Lanczos-FA error is measured with $w = 0$. The bound "triangle inequality" is \ref{['eqn:triangle_ineq']} and the bound "computable bound" is \ref{['thm:main']} evaluated numerically.

Theorems & Definitions (12)

• Definition 1.1
• Definition 1.2
• Theorem 2.1
• Lemma 2.1
• Lemma 2.2
• Definition 2.1
• Corollary 2.1
• proof : Proof of \ref{['thm:main']}
• Lemma 2.3
• proof : Proof of \ref{['eqn:unif_bound']}