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The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A… (More)

This paper concerns the Rayleigh–Ritz method for computing an approximation to an eigenspace X of a general matrix A from a subspace W that contains an approximation to X. The method produces a pair (N, ˜ X) that purports to approximate a pair (L, X), where X is a basis for X and AX = XL. In this paper we consider the convergence of (N, ˜ X) as the sine of… (More)

This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (µ + τ, ˜ x) to (λ, x). Three convergence results are established as the deviation of x from W… (More)

It is well-known that many Krylov solvers for linear systems, eigenvalue problems, and singular value decomposition problems have very simple and elegant formulas for residual norms. These formulas not only allow us to further understand the methods theoretically but also can be used as cheap stopping criteria without forming approximate solutions and… (More)

We investigate the SPAI and PSAI preconditioning procedures and shed light on two important features of them: (i) For the large linear system Ax = b with A irregular sparse, i.e., with A having s relatively dense columns, SPAI may be very costly to implement, and the resulting sparse approximate inverses may be ineffective for preconditioning. PSAI can be… (More)

The SPAI algorithm, a sparse approximate inverse preconditioning technique for large sparse linear systems, proposed by Grote and Huckle [SIAM J. Sci. Com-put., 18 (1997), pp. 838–853.], is based on the F-norm minimization and computes a sparse approximate inverse M of a large sparse matrix A adaptively. However , SPAI may be costly to seek the most… (More)