Accuracy of the Lanczos Process for the Eigenproblem and Solution of Equations
Christopher C. Paige · SIAM Journal on Matrix Analysis and Applications · 2019
In [ SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2347--2359] it was shown that $k$ steps of the finite precision Lanczos process for tridiagonalizing an $n\times n$ Hermitian matrix $A$ could be viewed as an exact Lanczos process for a $(k+n)\times (k+n)$ augmented Hermitian matrix, producing exactly orthogonal vectors. Here we use this and related results to prove the highly accurate behavior of the finite precision Lanczos process when used for finding the eigensystem of $A$, or for solving linear systems $Ax=b$. It turns out that the finite precision process mimics the exact process in iterative rather than $n$-step ways and makes available backward stable results. These results are also complete, such as making available the complete eigensystem of an $A$ with distinct eigenvalues. The matrix $S_k$ used to obtain these results is shown to provide valuable theoretical information on the loss of orthogonality between any $k$ vectors, among its many other useful properties.