Contribution
Parallel algorithms for certain matrix computations

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Abstract

The complexity of performing matrix computations, such as solving a linear system, inverting a nonsingular matrix or computing its rank, has received a lot of attention by both the theory and the scientific computing communities. In this paper we address some “nonclassical” matrix problems that find extensive applications, notably in control theory. More precisely, we study the matrix equations AX + XAT = C and AXXB = C, the “inverse” of the eigenvalue problem (called pole assignment), and the problem of testing whether the matrix [B ABAn−1 B] has full row rank. For these problems we show two kinds of PRAM algorithms: on one side very fast, i.e. polylog time, algorithms and on the other side almost linear time and processor efficient algorithms. In the latter case, the algorithms rely on basic matrix computations that can be performed efficiently also on realistic machine models.

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This work was supported by the ESPRIT III Basic Research Programme of the EC under contract No. 9072 (Project GEPPCOM), by NSF grant under contract number DMS-9212629, and by Italian M.U.R.S.T. 40% and 60% funds.

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Part of this work was done while author was visiting the International Computer Science Institute, Berkeley, CA, 94704-1198, USA.