Three refined and refined harmonic extraction-based Jacobi--Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition (GSVD) components of a large regular matrix pair. The new methods are called refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, abbreviated as RCPF-JDGSVD, RCPF-HJDGSVD and RIF-HJDGSVD, respectively. The new JDGSVD methods are more efficient than the corresponding standard and harmonic extraction-based JDSVD methods proposed previously by the authors, and can overcome the erratic behavior and intrinsic possible non-convergence of the latter ones. Numerical experiments illustrate that RCPF-JDGSVD performs better for the computation of extreme GSVD components while RCPF-HJDGSVD and RIF-HJDGSVD suit better for that of interior GSVD components.

翻译：提出了三种基于改进提取与调和提取的雅可比-戴维森（JD）类型方法，并开发了带紧缩和清除的厚重启算法，用于计算大型正则矩阵对的若干广义奇异值分解（GSVD）分量。这些新方法分别称为改进型无叉积（RCPF）、改进型无叉积调和（RCPF-harmonic）以及改进型无逆调和（RIF-harmonic）JDGSVD算法，简记为RCPF-JDGSVD、RCPF-HJDGSVD和RIF-HJDGSVD。新的JDGSVD方法比作者先前提出的相应标准提取与调和提取JDSVD方法更高效，并能克服后者的不规则行为及固有潜在不收敛问题。数值实验表明，RCPF-JDGSVD在计算极端GSVD分量时表现更优，而RCPF-HJDGSVD和RIF-HJDGSVD则更适合计算内部GSVD分量。