Zhongxiao Jia

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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, X̃) 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, X̃) 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, x̃) to (λ, x). Three convergence results are established as the deviation of x from W(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)
To implicitly restart the second-order Arnoldi (SOAR) method proposed by Bai and Su for the quadratic eigenvalue problem (QEP), it appears that the SOAR procedure must be replaced by a modified SOAR (MSOAR) one. However, implicit restarts fails to work provided that deflation takes place in the MSOAR procedure. In this paper, we first propose a Refined(More)
The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the eeciency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modiied(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)