Numerical methods for the solution of large and very large, sparse Lyapunov equations
In this dissertation the author considers the numerical solution of large (100 {le} n {le} 1000) and very large (n {ge} 1000), sparse Lyapunov equations AX-+ XA' + Q = 0. The author first presents a parallel version of the Hammarling algorithm for the solution of Lyapunov equations where the coefficient matrix A is large and dense. The author then presents a novel parallel algorithm for the solution of Lyapunov equations where A is large and banded. A detailed analysis of the computational requirements in tandem with the results of numerical experiments with these algorithms on an Alliant FX-8 multiprocessor is provided. In the second half of this dissertation, the author considers 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. The author presents in detail two methods for the low rank approximate solution of the Lyapunov equation. The first method, Trace Maximization, computes an orthogonal matrix V {element of}{Re}{sup n{times}k} that maximizes the trace of the solution {Sigma}{sub V} of the associated reduced order Lyapunov equation (V'AV){Sigma}{sub V} + {Sigma}{sub V}(V'A'V) + V'QV = 0. While Trace Maximization is an effective method for low rank approximation of explicitly specified Hermitian matrices, the author shows 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.
- Research Organization:
- Illinois Univ., Urbana, IL (United States)
- OSTI ID:
- 5585125
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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