Predicting fill for sparse orthogonal factorization
Journal of the ACM (JACM)
Direct methods for sparse matrices
Direct methods for sparse matrices
Remarks on implementation of O(n1/2τ) assignment algorithms
ACM Transactions on Mathematical Software (TOMS)
Computing the block triangular form of a sparse matrix
ACM Transactions on Mathematical Software (TOMS)
Sparse matrices in matlab: design and implementation
SIAM Journal on Matrix Analysis and Applications
An Approximate Minimum Degree Ordering Algorithm
SIAM Journal on Matrix Analysis and Applications
Sparse Linear Least Squares Problems in Optimization
Computational Optimization and Applications
Piecewise-linear pathways to the optimal solution set in linear programming
Journal of Optimization Theory and Applications
Modifying a Sparse Cholesky Factorization
SIAM Journal on Matrix Analysis and Applications
On Algorithms for Obtaining a Maximum Transversal
ACM Transactions on Mathematical Software (TOMS)
Multiple-Rank Modifications of a Sparse Cholesky Factorization
SIAM Journal on Matrix Analysis and Applications
A New Finite Continuation Algorithm for Linear Programming
SIAM Journal on Optimization
Large-scale linear programming using the Cholesky factorization.
Large-scale linear programming using the Cholesky factorization.
Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization
ACM Transactions on Mathematical Software (TOMS)
A direct orthogonal sparse static methodology for a finite continuation hybrid LP solver
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
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Although there is good software for sparse QR factorization, there is little support for updating and downdating, something that is absolutely essential in some linear programming algorithms, for example. This article describes an implementation of sparse LQ factorization, including block triangularization, approximate minimum degree ordering, symbolic factorization, multifrontal factorization, and updating and downdating. The factor Q is not retained. The updating algorithm expands the nonzero pattern of the factor L, which is reflected in dynamic representation of L. The block triangularization is used as an "ordering for sparsity" rather than as a prerequisite for block backward substitution. In symbolic factorization, something called "element counters" is introduced to reduce the overestimation of the number of nonzeros that the commonly used methods do. Both the approximate minimum degree ordering and the symbolic factorization are done without explicitly forming the nonzero pattern of the symmetric matrix in the corresponding normal equations. Tests show that the average time used for a single update or downdate is essentially the same as the time used for a single forward or backward substitution. Other parts of the implementation show the same range of performance as existing code, but cannot be replaced because of the special character of the systems that are solved.