On the use of structural zeros in orthogonal factorization
SIAM Journal on Scientific and Statistical Computing
A direct active set algorithm for large sparse quadratic programs with simple bounds
Mathematical Programming: Series A and B
Active set algorithms for isotonic regression: a unifying framework
Mathematical Programming: Series A and B
Algorithm 686: FORTRAN subroutines for updating the QR decomposition
ACM Transactions on Mathematical Software (TOMS)
Sparse matrices in matlab: design and implementation
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Newton's Method for Large Bound-Constrained Optimization Problems
SIAM Journal on Optimization
Row Modifications of a Sparse Cholesky Factorization
SIAM Journal on Matrix Analysis and Applications
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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 Bjorck and Oreborn of the late 1980s, but without forming A^TA, 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^TA and preserved with placeholders.