Recent ScaLAPACK-style implementations of the Bartels-Stewart
method [1, 4, 5] and a parallel implementation of an iterative matrix-sign-
function-based method [2, 3] for solving continuous-time Sylvester
matrix equations are evaluated with respect to generality of use,
execution time and accuracy of computed results.
The test problems include well-conditioned as well as ill-conditioned
Sylvester equations. Experiments carried out on two different distributed
memory machines show that the parallel explicitly blocked Bartels-Stewart
algorithm can solve more general problems and delivers far more accuracy for
ill-conditioned problems. It is also up to four times faster for large
enough
problems on the most balanced parallel platform (IBM SP), while the
parallel
iterative algorithm is almost always the fastest of the two on the less
balanced
platform (HPC2N Linux Super Cluster).
References:
[1]R.H. Bartels and G.W. Stewart,
"Algorithm 432: Solution of the Equation 
",
Comm. ACM, 15(9):820-826.
[2]P. Benner, E.S. Quintana-Orti,
"Solving Stable Generalized Lyapunov Equations with the matrix sign
functions", Numerical Algorithms, 20 (1), pp. 75-100, 1999.
[3]P. Benner, E.S. Quitana-Orti, G. Quintana-Orti,
"Numerical Solution of Discrete Schur Stable Linear Matrix Equations on
Multicomputers", Parallel Alg. Appl, Vol. 17, No. 1, pp. 127-146, 2002.
[4]R. Granat, "A Parallel ScaLAPACK-style Sylvester Solver",
Master Thesis, UMNAD 435/03, Dept. Computing Science,
Umeå University, Sweden, January, 2003.
[5]R. Granat, B. Kågström, P. Poromaa,
"Parallel ScaLAPACK-style Algorithms for Solving Continous-Time
Sylvester Equationsr", In H. Kosch et al., Euro-Par 2003 Parallel Processing.
Lecture Notes in Computer Science, Vol. 2790, pp. 800-809, 2003.
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2004-02-06