一种用于大规模GSVD计算的隐式重启联合双对角化算法
An Implicitly Restarted Joint Bidiagonalization Algorithm for Large GSVD Computations
AI总结：
针对大规模GSVD计算问题，本文将隐式重启技术扩展到联合双对角化过程，提出IRJBD算法，建立残差范数上界以设计停止准则，实验证明其性能良好且比厚重启JBD算法更高效。
摘要：
正则矩阵对$\{A,L\}$的联合双对角化（JBD）过程在数学上等价于堆叠矩阵$(A^{\text{T}},\text{ }L^{\text{T}})^{\text{T}}$的QR分解的Q因子上下两部分的两个同步Lanczos双对角化过程，前提是它们的起始向量以特定方式紧密相关。由此得到的用于计算$\{A,L\}$的极端广义奇异值及对应左右奇异向量的JBD方法，实现了$\{A,L\}$的广义奇异值分解（GSVD）问题在JBD过程生成的两个左子空间和一个右子空间上的标准Rayleigh–Ritz投影。本文将隐式重启技术非平凡且巧妙地扩展到JBD过程，开发了隐式重启JBD（IRJBD）算法，提出了关键位移的合理选取方法，并解决了有限精度运算中的几个关键实现细节。本文建立了精确运算和有限精度运算下近似GSVD分量残差范数的紧致上界，用于设计高效可靠的停止准则，并避免近似右广义奇异向量的昂贵计算。数值实验表明，IRJBD性能良好，且比厚重启JBD算法效率更高。
英文摘要：
The joint bidiagonalization (JBD) process of a regular matrix pair $\{A,L\}$ is mathematically equivalent to two simultaneous Lanczos bidiagonalization processes of the upper and lower parts of the Q-factor of QR factorization of the stacked matrix $(A^{\mathrm T},\,L^{\mathrm T})^{\mathrm T}$ when their starting vectors are closely related in a specific way. The resulting JBD method for computing extreme generalized singular values and corresponding left and right singular vectors of $\{A,L\}$ realizes the standard Rayleigh--Ritz projection of the generalized singular value decomposition (GSVD) problem of $\{A,L\}$ onto the two left and one right subspaces generated by the JBD process. In this paper, the implicit restarting technique is nontrivially and skillfully extended to the JBD process, and an implicitly restarted JBD (IRJBD) algorithm is developed with proper selection of crucial shifts proposed and a few key implementation details addressed in finite precision arithmetic. Compact upper bounds are established for the residual norm of an approximate GSVD component in both exact and finite precision arithmetic, which are used to design efficient and reliable stopping criteria and avoid the expensive computation of approximate right generalized singular vectors. Numerical experiments illustrate that IRJBD performs well and is more efficient than the thick-restart JBD algorithm.
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