Sensitivity analysis of the Lanczos reduction (Q2760338)

The authors investigate for a given real \(n \times n \) matrix \(A\) and initial vectors \(V\) and \(W\) the sensitivity of the tridiagonal matrix \(T\), where \( T= W^T A V \), as well as the biorthogonal sets of vectors of the Lanczos reduction to small changes in \(A\), \(V\) and \(W\). They show why the choice of normalization is important for sensitivity results by discussing some optimal properties of normalizations. Further a quick summary of the unsymmetric Lanczos algorithm as background for the sensitivity analysis is given. The inverses of two matrices which are critical to the sensitivity are derived, as well as bounds for the general case including the well known Krylov subspaces and condition numbers for the symmetric case. NEWLINENEWLINENEWLINEAt the end of the paper a brief summarization and relation to other similar works by \textit{B. N. Parlet} and \textit{J. Le} [SIAM J. Matrix Anal. Appl. 14, No. 1, 279-316 (1993; Zbl 0767.65031)] and \textit{J.-F. Carpraux, S. K. Godunov} and \textit{S. V. Kuznetsov} [Linear Algebra Appl. 248, 137-160 (1996; Zbl 0861.65042)] is presented.