PHSVDS

A preconditioned hybrid SVD method for accurately computing singular triplets of large matrices. The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but they are still challenged even for medium matrix sizes due to the difficulty of the problem. We propose a novel SVD approach that can take advantage of preconditioning and of any well-designed eigensolver to compute both largest and smallest singular triplets. Accuracy and efficiency is achieved through a hybrid, two-stage meta-method, PHSVDS. In the first stage, PHSVDS solves the normal equations up to the best achievable accuracy. If further accuracy is required, the method switches automatically to an eigenvalue problem with the augmented matrix. Thus it combines the advantages of the two stages, faster convergence and accuracy, respectively. For the augmented matrix, solving the interior eigenvalue is facilitated by proper use of good initial guesses from the first stage and an efficient implementation of the refined projection method. We also discuss how to precondition PHSVDS and to cope with some issues that arise. Numerical experiments illustrate the efficiency and robustness of the method.


References in zbMATH (referenced in 11 articles , 1 standard article )

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  1. Dax, Achiya: A cross-product approach for low-rank approximations of large matrices (2020)
  2. Avron, Haim; Druinsky, Alex; Toledo, Sivan: Spectral condition-number estimation of large sparse matrices. (2019)
  3. Dax, Achiya: Computing the smallest singular triplets of a large matrix (2019)
  4. Goldenberg, Steven; Stathopoulos, Andreas; Romero, Eloy: A Golub-Kahan Davidson method for accurately computing a few singular triplets of large sparse matrices (2019)
  5. Huang, Jinzhi; Jia, Zhongxiao: On inner iterations of Jacobi-Davidson type methods for large SVD computations (2019)
  6. Wu, Lingfei; Xue, Fei; Stathopoulos, Andreas: TRPL+K: thick-restart preconditioned Lanczos+K method for large symmetric eigenvalue problems (2019)
  7. Teng, Zhongming; Wang, Xuansheng: Majorization bounds for SVD (2018)
  8. Tsakiris, Manolis C.; Vidal, René: Filtrated algebraic subspace clustering (2017)
  9. Wu, Lingfei; Romero, Eloy; Stathopoulos, Andreas: PRIMME_SVDS: a high-performance preconditioned SVD solver for accurate large-scale computations (2017)
  10. Wu, Lingfei; Laeuchli, Jesse; Kalantzis, Vassilis; Stathopoulos, Andreas; Gallopoulos, Efstratios: Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse (2016)
  11. Wu, Lingfei; Stathopoulos, Andreas: A preconditioned hybrid SVD method for accurately computing singular triplets of large matrices (2015)