In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is approximately solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, a preconditioned correction equation is derived that extracts useful information from current searching subspaces to construct effective preconditioners for the correction equation and is proved to retain the same convergence of outer iterations of JDSVD. The resulting method is called inner preconditioned JDSVD (IPJDSVD) method. Convergence results show that MINRES for the preconditioned correction equation can converge much faster when there is a cluster of singular values closest to a given target, so that IPJDSVD is more efficient than JDSVD. A new thick-restart IPJDSVD algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets of a large matrix. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD.

翻译：在求解奇异值分解（SVD）问题的雅可比-戴维森（JD）型方法（称为JDSVD）中，每次外层迭代需近似迭代求解一个大型对称且通常不定的修正方程，该过程构成内层迭代并主导JDSVD的整体效率。本文对修正方程的最小残量（MINRES）法进行了收敛性分析。基于所得结果，推导出一种预处理修正方程，该方程从当前搜索子空间中提取有效信息以构建修正方程的高效预处理器，并证明能保持JDSVD外层迭代的相同收敛性。由此产生的方法称为内预处理JDSVD（IPJDSVD）方法。收敛结果表明，当存在最接近给定目标的奇异值簇时，预处理修正方程的MINRES法可显著加速收敛，因此IPJDSVD比JDSVD更高效。本文提出一种新的带紧缩重启、压缩与净化技术的IPJDSVD算法，该算法同时加速标准紧缩重启JDSVD的外层与内层收敛，并计算大型矩阵的多个奇异三联体。数值实验验证了理论分析，并展示了IPJDSVD相较于JDSVD的显著优越性。