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

A Preconditioned Hybrid SVD Method for Computing Accurately Singular Triplets of Large Matrices