We propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair $(A,B)$ with the generalized singular values in a given interval. The solver is a highly nontrivial extension of the FEAST eigensolver for the (generalized) eigenvalue problem and CJ-FEAST SVDsolver for the SVD problem. For a partial GSVD problem, given three left and right searching subspaces, we propose a general projection method that works on $(A,B)$ {\em directly}, and computes approximations to the desired GSVD components. For the concerning GSVD problem, we exploit the Chebyshev--Jackson (CJ) series to construct an approximate spectral projector of the generalized eigenvalue problem of the matrix pair $(A^TA,B^TB)$ associated with the generalized singular values of interest, and use subspace iteration on it to generate a right subspace. Premultiplying it with $A$ and $B$ constructs two left subspaces. Applying the general projection method to the subspaces constructed leads to the CJ-FEAST GSVDsolver. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general-purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate the efficiency of the CJ-FEAST GSVDsolver.

翻译：我们提出了一种CJ-FEAST GSVD求解器，用于计算大矩阵对$(A,B)$的部分广义奇异值分解（GSVD），其广义奇异值位于给定区间内。该求解器是FEAST特征值求解器（针对（广义）特征值问题）和CJ-FEAST SVD求解器（针对SVD问题）的高度非平凡扩展。对于部分GSVD问题，给定三个左右搜索子空间，我们提出了一种通用投影方法，该方法直接作用于$(A,B)$，并计算所需GSVD分量的近似值。针对所关注的GSVD问题，我们利用Chebyshev-Jackson（CJ）级数构造与感兴趣广义奇异值相关的矩阵对$(A^TA,B^TB)$的广义特征值问题的近似谱投影算子，并通过子空间迭代生成右子空间。用$A$和$B$左乘该右子空间，构造两个左子空间。将通用投影方法应用于所构造的子空间，从而得到CJ-FEAST GSVD求解器。我们推导了近似谱投影算子及其特征值的精度估计，并建立了关于底层子空间以及CJ-FEAST GSVD求解器获得的近似GSVD分量的一系列收敛性结果。我们提出了级数阶数和子空间维度的通用选择策略。数值实验验证了CJ-FEAST GSVD求解器的效率。