计算大型矩阵对的部分广义奇异值分解（GSVD）的无向量积Jacobi-Davidson型方法
A cross-product free Jacobi-Davidson type method for computing a partial generalized singular value decomposition (GSVD) of a large matrix pair
摘要：
提出了一种计算大型正则矩阵对$(A,B)$的部分广义奇异值分解（GSVD）的无向量积（CPF）Jacobi-Davidson（JD）方法.它隐式地解决了$(A^TA,B^TB)$的数学等价的广义本征值问题，但没有显式地形成向量积矩阵，从而避免了计算广义奇异值和广义奇异向量可能带来的精度损失.该方法是一种内-外迭代方法，其中右搜索子空间的扩展形成近似求解所涉及的校正方程的内迭代，而外迭代提取关于子空间的近似GSVD分量。建立了内迭代和外迭代的收敛性结果，并根据这些结果设计了内迭代的实用停止准则。提出了一种带收缩的厚重启动CPF-JDGSVD算法，用于计算多个GSVD分量。数值实验验证了算法的有效性。
英文摘要：
A Cross-Product Free (CPF) Jacobi-Davidson (JD) type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair $(A,B)$. It implicitly solves the mathematically equivalent generalized eigenvalue problem of $(A^TA,B^TB)$ but does not explicitly form the cross-product matrices and thus avoids the possible accuracy loss of the computed generalized singular values and generalized singular vectors. The method is an inner-outer iteration method, where the expansion of the right searching subspace forms the inner iterations that approximately solve the correction equations involved and the outer iterations extract approximate GSVD components with respect to the subspaces. Some convergence results are established for the inner and outer iterations, based on some of which practical stopping criteria are designed for the inner iterations. A thick-restart CPF-JDGSVD algorithm with deflation is developed to compute several GSVD components. Numerical experiments illustrate the efficiency of the algorithm.
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