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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)
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)
This paper concerns the Rayleigh{Ritz method for computing an approximation to an eigenpair (; x) of a non-Hermitian matrix A. Given a subspace W that contains an approximation to x, this method returns an approximation (; ~ x) to (; x). We establish four convergence results that hold as the deviation of x from W approaches zero. First, the Ritz value(More)