AI 中文总结
本文提出基于柯西积分的预处理 $f(A)b$ 的统一框架，通过两种不同方向的方法解决相关稳定性与效率问题，经数值实验验证其在两类算子计算中的有效性。
AI 中文摘要
矩阵函数与向量的乘积 $f(A)b$ 的计算是大型稀疏矩阵的主要计算瓶颈，尤其当不利的谱分布导致标准Krylov子空间方法停滞时。本文基于矩阵函数的柯西积分表示，提出了一种预处理 $f(A)b$ 的统一框架。利用平移不变性，我们将预条件子的计算与Krylov子空间的生成解耦，从两个方向发展该框架：其一，针对有理平移逆预条件子，我们解决了最优谱压缩与有限精度不稳定性之间的基本权衡，通过构造经双修正Gram-Schmidt正交化稳定的闭式提取方案，消除了伪幻影极点的形成；其二，我们提出了一种无矩阵多项式方法，为确保数值稳定性，我们利用投影Hessenberg矩阵的舒尔分解分离连续数值求积步骤，为进一步稳定轮廓奇点附近的积分并加速整体收敛，我们引入了针对预条件算子关键低模的精确LR消去方案。我们分析了这些方法的渐近稳定性和与奇点的接近程度，并通过数值实验展示了其在两类问题上的效率：二维拉普拉斯算子（$f=\textrm{exp}$）和格点量子色动力学中高度病态的Wilson-Dirac算子（$f=\textrm{sign}$），尽管该框架原则上可与任意$f$结合使用，且在对多个不同向量$b_i$应用$f(A)b_i$时尤其有益。
英文摘要
The computation of the action of a matrix function on a vector, $f(A)b$, is a major computational bottleneck for large, sparse matrices, particularly when unfavorable spectral distributions cause standard Krylov subspace methods to stagnate. In this work, we propose a unified framework for preconditioning $f(A)b$ based on the Cauchy integral representation of the matrix function. By exploiting shift-invariance properties, we decouple the preconditioner evaluation from the Krylov subspace generation. We develop this framework in two distinct directions. First, for rational shift-and-invert preconditioning, we resolve a fundamental trade-off between optimal spectral compression and finite-precision instability. We achieve this by formulating a closed-form extraction stabilized via Double Modified Gram-Schmidt reorthogonalization, which eliminates the formation of spurious phantom poles. Second, we present a matrix-free polynomial approach. To ensure numerical stability, we isolate the continuous numerical quadrature step using a Schur decomposition of the projected Hessenberg matrix. To further stabilize the integration near contour singularities and accelerate overall convergence, we incorporate an exact LR-deflation scheme targeting the critical low modes of the preconditioned operator. We analyze the asymptotic stability and proximity to singularity of these methods, and present numerical experiments demonstrating their efficiency on the 2D Laplacian with $f=\textrm{exp}$, and a highly ill-conditioned Wilson-Dirac operator from lattice quantum chromodynamics with $f=\textrm{sign}$, although the framework can be in principle used with any $f$ and it is particularly beneficial when applying $f(A)b_{i}$ with many different vectors $b_{i}$.
