用于大规模GSVD计算的改进联合双对角化方法及隐式重启算法
The Refined Joint Bidiagonalization Method and an Implicitly Restarted Algorithm for Large GSVD Computations
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中文总结 AI 辅助
本文针对大规模GSVD计算，分析联合双对角化（JBD）方法的收敛问题，提出改进JBD（RJBD）方法及隐式重启算法，实验表明该算法效率优于隐式重启JBD算法。
中文摘要 AI 辅助
本文对联合双对角化（JBD）方法进行了收敛性分析，该方法用于计算正则矩阵对$\boldsymbol{\{A,L\}}$的若干极端广义奇异值分解（GSVD）分量。研究表明，由该方法得到的左右里茨向量可能会无规律地收敛，甚至可能无法收敛，但里茨值会收敛。在期望右广义奇异向量与右子空间的偏差趋于零的假设下，这些收敛结果适用于GSVD问题的一类广义瑞利-里茨投影方法。本文证明了里茨值与广义奇异值的交错性质，并将其扩展到$\boldsymbol{\{A,L\}}$的广义奇异值及其列子集构成的矩阵对。为克服JBD方法的无规律收敛或可能的不收敛问题，本文将特征值问题的改进瑞利-里茨投影非平凡地扩展到GSVD问题，提出了改进JBD（RJBD）方法，该方法用满足一定残差最优性的新近似（称为右改进里茨向量）替代右里茨向量，同时定义了新的近似左广义奇异向量（称为左改进里茨向量）。本文证明，在相同假设下，左右改进里茨向量无条件收敛。本文将隐式重启方案扩展到RJBD方法，开发了具有改进位移的隐式重启RJBD算法。数值实验表明，新算法至少具有竞争力，且通常比隐式重启JBD算法效率高得多。
英文摘要
We make a convergence analysis on the joint bidiagonalization (JBD) method that computes several extreme generalized singular value decomposition (GSVD) components of a regular matrix pair $\{A,L\}$, and show that the right and left Ritz vectors obtained by it may converge erratically and even may fail to converge, while Ritz values converge. These convergence results hold for a class of general Rayleigh--Ritz projection methods for the GSVD problem under the hypothesis that the deviation of a desired right generalized singular vector from the right subspace tends to zero. We prove the interlacing property of Ritz values and generalized singular values, and extend it to the generalized singular values of $\{A,L\}$ and the matrix pairs consisting of subsets of its columns. To overcome the irregular convergence or possible non-convergence of the JBD method, we nontrivially extend the refined Rayleigh--Ritz projection for the eigenvalue problem to the GSVD problem, and propose a refined JBD (RJBD) method that replaces the right Ritz vectors by new approximations, called the right refined Ritz vectors, satisfying certain residual optimality; we define new approximate left generalized singular vectors, called the left refined Ritz vectors. We prove that the left and right refined Ritz vectors unconditionally converge under the same hypothesis. We extend the implicit restarting scheme to the RJBD method, and develop an implicitly restarted RJBD algorithm with the refined shifts proposed. Numerical experiments illustrate that the new algorithm is at least competitive and often considerably more efficient than the implicitly restarted JBD algorithm.
