Computations and Analysis with Non-normal Matrices

Abstract

The goal of this dissertation is to introduce some new iterative methods for solving problems involving non-Hermitian matrices and to add to our understanding of the convergence of these algorithms when applied to highly non-normal matrices. In the first part, the Arnoldi-OR algorithm [23] is introduced. Given an n by n matrix A and a rational function N (z)/D(z), where D(A) is nonsingular, this algorithm finds the approximations x_k, k = 1, 2, . . ., from successive Krylov subspaces, span{b, Ab, . . . , A^(k−1)b}, that minimize the 2-norm of the residual, ∥D(A)x_k −N(A)b∥_2 . Convergence of the Arnoldi-OR algorithm can be bounded based on the eigenvalues of A and the condition number of the best-conditioned matrix of eigenvectors, assuming that A is diagonalizable. This may be a large overestimate, however, if the best-conditioned matrix of eigenvectors is still very ill-conditioned. Starting in Chapter 4, the second part of this dissertation explores bounds on the norm of a function of A that can be applied when A is highly non-normal; i.e., either A is not diagonalizable or it is diagonalizable but the eigenvalues are very ill-conditioned. These bounds involve the ∞-norm of the function, not just on the eigenvalues of A, but on a larger set in the complex plane containing the eigenvalues. In the second part of the dissertation, we expand on some known results from Crouzeix and Palencia (2017) and Crouzeix and Greenbaum(2019) about K-spectral sets—sets Ω ⊂ C satisfying ∥f(A)∥ 2 ≤ K sup z∈Ω |f (z)| for all functions f analytic in Ω, (i.e. functions that can be arbitrarily well-approximated on Ω by rational functions with no poles in Ω). This work is described in Greenbaum and Wellen (2025). Here, we use it to give alternative bounds on the 2-norm of the residual in the Arnoldi-OR algorithm. We also consider a different way of solving non-Hermitian linear systems, which is to convert the problem to a Hermitian one, in Section 2.2. If one can find Hermitian positive definite matrices M and Y such that A = M −1 Y, then one can solve Ax = b using the conjugate gradient method (CG) applied to the system Yx = Mb with M as a preconditioner. We apply this technique to a problem involving the graph Laplacian that is of interest to Sandia National Laboratory, where I worked with Richard Lehoucq over the summer of 2023.