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The real two-zero algorithm: a parallel algorithm to reduce a real matrix to a real Schur form

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2 Author(s)
M. Mantharam ; Dept. of Math., City Univ. of New York, NY, USA ; P. J. Eberlein

We introduce a new method to reduce a real matrix to a real Schur form by a sequence of similarity transformations that are 3D orthogonal transformations. Two significant features of this method are that: all the transformed matrices and all the computations are done in the real field; and it can be easily parallelized. We call the algorithm that uses this method the real two-zero (RTZ) algorithm. We describe both serial and parallel implementations of the RTZ algorithm. Our tests indicate that the rate of convergence to a real Schur form is quadratic for real near-normal matrices with real distinct eigenvalues. Suppose n is the order of a real matrix A. In order to choose a sequence of 3D orthogonal transformations on A, we need to determine some ordering on triples in T={(k,l,m)|1⩽k<l<m⩽n}, where (k,l,m) defines the three coordinates under the 3D transformation. We show how the ordering of the triples used in our implementations can be generated cyclically in an algorithm

Published in:

IEEE Transactions on Parallel and Distributed Systems  (Volume:6 ,  Issue: 1 )