Chapter 15 - Large-Scale Matrix Computations in Control: Krylov Subspace Methods

Abstract

This chapter presents large-scale matrix computations and a class of classical methods known as the Krylov subspace methods that have been found to be suitable for sparse matrix computations. The reason is that these methods can be implemented using matrix-vector multiplications only. Therefore, the sparsity in the original problem can be preserved. The examples are the Generalized Minimal Residual and the Quasi-Minimal Residual methods for linear systems problem; the Arnoldi, Lanezos, the Jaeobi-Davidson methods; and several variants of them such as the restarted and block Arnoldi methods and band Lanezos method for eigenvalue problems. It is only natural to develop algorithms for large-scale control problems using these effective large-scale techniques of matrix computations. The chapter briefly describes the basic Arnoldi and Lanczos methods to facilitate the understanding of how these methods are applied to solve large-scale control problems. The chapter aims to show the readers how these modem iterative numerical methods can be gainfully employed to solve some of the large and sparse matrix problems arising in control.