The block Lanczos algorithm for linear ill-posed problems

Abstract

In the present paper, we propose a new method to inexpensively determine a suitable value of the regularization parameter and an associated approximate solution, when solving ill-conditioned linear system of equations with multiple right-hand sides contaminated by errors. The proposed method is based on the symmetric block Lanczos algorithm, in connection with block Gauss quadrature rules to inexpensively approximate matrix-valued function of the form \(W^Tf(A)W\), where \(W\in {\mathbb {R}}^{n\times k}\), \(k\ll n\), and \(A\in {\mathbb {R}}^{n\times n}\) is a symmetric matrix.