Parallel adaptive wavefront algorithms solving Lyapunov equations for the Cholesky factor on message passing multiprocessors

Parallel Solvers for Sylvester-Type Matrix Equations with Applications in Condition Estimation, Part I

“…In Benner and QuintanaOrtí [1999] and Benner et al [2002;2004] fully iterative methods for solving matrix equations are considered, an approach that is highly scalable but may fail to converge for ill-posed problems, which was demonstrated along with a performance comparison with the classical Bartels-Stewart method for SYCT in Granat and Kâgstrom [2006]. Parallel element-wise wavefront algorithms for solving Lyapunov equations using Hammarling's method were presented in Claver [1999]. Jonsson and Kâgstrom presented fast recursive blocked algorithms for solving matrix equations in Jonsson and Kâgstrom [2002a;2002b] and developed the software library RECSY [Jonsson and Kâgstrom 2003;RECSY].…”

“…The subsolutions X,j (and Yij) in the blocked algorithms for the reduced matrix equations are now obtained by 32: 10 • R. Granat and B. Kâgstrom a wavefront traversai of the block diagonals or block antidiagonals of the righthand side matrix (or matrices). Indeed, the wavefront block approach can be seen as a generalization of both the dataflow approach, illustrated in O'Leary and Stewart [1985] and applied to a triangular standard Sylvester equation, and the element-wise wavefront algorithms presented in Claver [1999]. We remark that the blocked formulations of the triangular matrix equations in Section 2 reveal the data dependencies which in turn control the dataflow at a block level of our distributed algorithms.…”

Block variants of Hammarling's method for solving Lyapunov equations

“…However, in view of the results in Section 5.2 such further improvements may only have a limited impact on the overall execution time required for solving an unreduced Lyapunov equation. Another direction of future research is to investigate whether the ideas from this paper lead to parallel distributed memory algorithms competitive with existing parallel variants of Hammarling's method, see, e.g., [10,11].…”

Algoritmos paralelos para la reducción de sistemas lineales de control estables

“…Pueden consultarse algunas implementaciones paralelas básicas de la resolución de la ecuación de Lyapunov generalizada mediante el método de Hammarling en [GHRV97a] y [GHRV97b]. En [Cla99,CH99] se analiza una implementación para resolver el caso discreto de la ecuación estándar mediante paso de mensajes usando una distribución de datos que maximice el grado de paralelismo (se va operando por bloques antidiagonales de la matriz, distribuida cíclicamente por columnas). En [SZ03] se proponen algunas optimizaciones del método básico.…”