Sparsity analysis of the QR factorization




Hare, D.
Johnson, C. R.
Olesky, D. D.
Van den Driessche, P.

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Given only the zero-nonzero pattern of an m-by-n matrix A of full column rank, which entries of Q and which entries of R in its QR factorization must be zero and which entries may be nonzero? We give a complete answer to this question, which involves an interesting interplay between combinatorial structure and the algebra implicit in orthogonality. To this end some new sparse structural concepts are introduced, and an algorithm to determine the structure of Q is given. The structure of R then follows immediately from that of Q and A. The computable zero/nonzero structures for the matrices Q and R are proven to be tight, and the conditions on the pattern for A are the weakest possible (namely that it allows matrices A with full column rank). This complements existing work that focussed upon R and then only under an additional combinatorial assumption (the strong Hall property).


Originally published March 1990. Revised November 1991.


QR factorization, bipartite graph, Hall property, sparse matrix, combinatorial matrix theory, orthogonality