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Weighted Golub-Kahan-Lanczos bidiagonalization algorithms

    Hong-Xiu Zhong, Hongguo Xu

ETNA - Electronic Transactions on Numerical Analysis, pp. 153-178, 2017/11/15

doi: 10.1553/etna_vol47s153

doi: 10.1553/etna_vol47s153


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doi:10.1553/etna_vol47s153



doi:10.1553/etna_vol47s153

Abstract

We present weighted Golub-Kahan-Lanczos algorithms. We demonstrate their applications to the eigenvalue problem of a product of two symmetric positive definite matrices and an eigenvalue problem for the linear response problem. A convergence analysis is provided and numerical test results are reported. As another application we make a connection between the proposed algorithms and the preconditioned conjugate gradient (PCG) method.

Keywords: weighted Golub-Kahan-Lanczos bidiagonalization algorithm, eigenvalue, eigenvector, Ritz value, Ritz vector, linear response eigenvalue problem, Krylov subspace, bidiagonal matrices