## Separators and Structure Prediction in Sparse Orthogonal Factorization (1993)

Citations: | 4 - 0 self |

### BibTeX

@TECHREPORT{Gilbert93separatorsand,

author = {John R. Gilbert and Esmond G. Ng and Barry W. Peyton},

title = {Separators and Structure Prediction in Sparse Orthogonal Factorization},

institution = {},

year = {1993}

}

### OpenURL

### Abstract

In the factorization A = QR of a matrix A, the orthogonal matrix Q can be represented either explicitly (as a matrix) or implicitly (as a matrix H of Householder vectors). We derive both upper and lower bounds on the number of nonzeros in H and the number of nonzeros in Q, in the case where the graph of A T A has "good" separators and A need not be square. We also derive an upper bound on the number of nonzeros in the null-basis part of Q in the case where A is the edge-vertex incidence matrix of a planar graph. The significance of these results is that they both illuminate and amplify a folk theorem of sparse QR factorization, which holds that the matrix H of Householder vectors represents the orthogonal factor of A much more compactly than Q itself. To facilitate discussion of this and related issues, we review several related results which have appeared previously. Keywords: Sparse matrix algorithms, QR factorization, separators, column intersection graph, strong Hall...