Parallel algorithms for algebraic Riccati equations.

*(English)*Zbl 0763.93036Algebraic Riccati equations (AREs) arise naturally in control design problems. Their importance has led to a good deal of research concerning their efficient numerical solution. The authors of the paper under consideration offer a parallel algorithm for solving the generalized ARE
\[
A^ T XE+E^ T XA-E^ T XWXE+Q=0,
\]
where the matrices \(A\), \(E\), \(W\), and \(Q\) are all \(n\times n\), and the unknown \(X\) is also \(n\times n\). The given solution method is based on the matrix sign function algorithm. This is chosen primarily for its ease of parallelization. Three variations of the algorithm are given. The first uses pivoting in the matrix factorizations required; the second uses orthogonal transformations to put the matrix system into a tridiagonal form, which is then solved; and the third uses an updating strategy. The authors discuss implementation of the variations on a hypercube machine with distributed memory. They offer computational results, including speedups and efficiencies, for the resulting computer program on an Intel iPSC/d5 hypercube. Comparisons with sequential computations on a VAX 11/780 are also given. All parallel algorithms turn out to run faster than a standard scalar program when \(40\leq n\leq 100\), with the variation that uses pivoting performing best.

Reviewer: K.Cooper

##### MSC:

93C15 | Control/observation systems governed by ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

93-04 | Software, source code, etc. for problems pertaining to systems and control theory |

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\textit{J. D. Gardiner} and \textit{A. J. Laub}, Int. J. Control 54, No. 6, 1317--1333 (1991; Zbl 0763.93036)

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