Jinzhi Huang

Jinzhi Huang, Zhongxiao Jia

We make a convergence analysis of the harmonic and refined harmonic extraction versions of Jacobi-Davidson SVD (JDSVD) type methods for computing one or more interior singular triplets of a large matrix $A$. At each outer iteration of these methods, a correction equation, i.e., inner linear system, is solved approximately by using iterative methods, which leads to two inexact JDSVD type methods, as opposed to the exact methods where correction equations are solved exactly. Accuracy of inner iterations critically affects the convergence and overall efficiency of the inexact JDSVD methods. A central problem is how accurately the correction equations should be solved so as to ensure that both of the inexact JDSVD methods can mimic their exact counterparts well, that is, they use almost the same outer iterations to achieve the convergence. In this paper, similar to the available results on the JD type methods for large matrix eigenvalue problems, we prove that each inexact JDSVD method behaves like its exact counterpart if all the correction equations are solved with $low\ or\ modest$ accuracy during outer iterations. Based on the theory, we propose practical stopping criteria for inner iterations. Numerical experiments confirm our theory and the effectiveness of the inexact algorithms.

Jinzhi Huang, Zhongxiao Jia

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.

Jinzhi Huang

A generalized skew-symmetric Lanczos bidiagonalization (GSSLBD) method is proposed to compute several extreme eigenpairs of a large matrix pair $(A,B)$, where $A$ is skew-symmetric and $B$ is symmetric positive definite. The underlying GSSLBD process produces two sets of $B$-orthonormal generalized Lanczos basis vectors that are also $B$-biorthogonal and a series of bidiagonal matrices whose singular values are taken as the approximations to the imaginary parts of the eigenvalues of $(A,B)$ and the corresponding left and right singular vectors premultiplied with the left and right generalized Lanczos basis matrices form the real and imaginary parts of the associated approximate eigenvectors. A rigorous convergence analysis is made on the desired eigenspaces approaching the Krylov subspaces generated by the GSSLBD process and accuracy estimates are made for the approximate eigenpairs. In finite precision arithmetic, it is shown that the semi-$B$-orthogonality and semi-$B$-biorthogonality of the computed left and right generalized Lanczos vectors suffice to compute the eigenvalues accurately. An efficient partial reorthogonalization strategy is adapted to GSSLBD in order to maintain the desired semi-$B$-orthogonality and semi-$B$-biorthogonality. To be practical, an implicitly restarted GSSLBD algorithm, abbreviated as IRGSSLBD, is developed with partial $B$-reorthogonalizations. Numerical experiments illustrate the robustness and overall efficiency of the IRGSSLBD algorithm.