Volume 47, pp. 153-178, 2017.

Weighted Golub-Kahan-Lanczos bidiagonalization algorithms

Hong-Xiu Zhong and Hongguo Xu

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.

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Key words

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

AMS subject classifications

65F15, 15A18

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