Hyperbolic Householder Algorithms for Factoring Structured Matrices

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Abstract
Efficient algorithms for computing triangular decompositions of Hermitian matrices with small displacement rank using hyperbolic Householder matrices are derived. These algorithms can be both vectorized and parallelized. Implementations along with performance results on an Alliant FX/80, Cray X-MP/48, and Cray-2 are discussed. The use of Householder-type transformations is shown to improve performance for problems with nontrivial displacement ranks. In special cases, the general algorithm reduces to the well-known Schur algorithm for factoring Toeplitz matrices and Elden’s algorithm for solvig structured regularization problems. It gives a Householder formulation to the class of algorithms based on hyperbolic rotations studied by Kailath, Lev-Ari, Chun, and their colleagues for Hermitian matrices with small displacement structure. In addition, an extension to the efficient factorization of indefinite systems is described.

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