Numerical methods for the solution of large and very large, sparse Lyapunov equations

Abstract

In this dissertation we consider the numerical solution of large $(100 \leq n \leq 1000)$ and very large $(n \geq 1000)$, sparse Lyapunov equations $AX + XA\sp\prime + Q = 0$. We first present a parallel version of the Hammarling algorithm for the solution of Lyapunov equations where the coefficient matrix A is large and dense. We then present a novel parallel algorithm for the solution of Lyapunov equations where A is large and banded. We provide a detailed analysis of the computational requirements in tandem with the results of numerical experiments with these algorithms on an Alliant FX-8 multiprocessor.

In the second half of this dissertation, we consider the numerical solution of Lyapunov equations where the coefficient matrix A is very large and sparse. Under these conditions, the solution X of the Lyapunov equation is typically full rank and dense. The associated excessive storage requirements compel us to compute low rank approximations of the solution X of the Lyapunov equation. We present in detail two methods for the low rank approximate solution of the Lyapunov equation. The first method, Trace Maximization, computes an orthogonal matrix $V \in \Re\sp{n\times k}$ that maximizes the trace of the solution $\Sigma\sb{V}$ of the associated reduced order Lyapunov equation $(V\sp\prime AV)\Sigma\sb{V}$ + $\Sigma\sb{V}(V\sp\prime A\sp\prime V)$ + $V\sp\prime QV = 0$. While Trace Maximization is an effective method for low rank approximation of explicitly specified Hermitian matrices, we show that Trace Maximization is not an effective strategy for low rank approximation of positive semidefinite Hermitian matrices X that are implicitly specified as the solution of a Lyapunov equation. Our second algorithm for low rank approximate solution of Lyapunov equations, Approximate Power Iteration, attempts to directly compute an orthogonal basis of the dominant eigenspace of the solution X. We are able to show that if the dominant eigenvalues $\lambda\sb1$ and $\lambda\sb2$ of X are sufficiently well separated $(\lambda\sb1 \gg \lambda\sb2)$, then a special case of the Approximate Power Iteration algorithm has at least one fixed point v that is near to the dominant eigenvector $u\sb1$ of X, and that there is a small attractive region in $\Re\sp{n}$ containing both $u\sb1$ and v.