A sparse counterpart of Reichel and Gragg’s package QRUP

Abstract

We describe how to maintain the triangular factor of a sparse QR factorization when columns are added and deleted and Q cannot be stored for sparsity reasons. The updating procedures could be thought of as a sparse counterpart of Reichel and Gragg’s package QRUP. They allow us to solve a sequence of sparse linear least squares subproblems in which each matrix B k is an independent subset of the columns of a fixed matrix A , and B k + 1 is obtained by adding or deleting one column. Like Coleman and Hulbert [T. Coleman, L. Hulbert, A direct active set algorithm for large sparse quadratic programs with simple bounds, Math. Program. 45 (1989) 373–406], we adapt the sparse direct methodology of Björck and Oreborn of the late 1980s, but without forming A T A , which may be only positive semidefinite. Our Matlab 5 implementation works with a suitable row and column numbering within a static triangular sparsity pattern that is computed in advance by symbolic factorization of A T A and preserved with placeholders.