Large sparse least squares computations
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Orthogonal Givens factorization is a popular method for solving large sparse least squares problems. Row and column permutations of the data matrix are necessary to preserve sparsity, and reduce the computational effort during factorization. The computation of a solution is usually divided into a symbolic ordering phase, and a numerical factorization and solution phase. Some theoretical results on row ordering are obtained using a graph-theoretic representation. These results provide a basis for a symbolic Givens factorization. Column orderings are also discussed, and an efficient algorithm for the symbolic ordering phase is developed. Sometimes, due to sparsity considerations, it is advantageous to leave out some rows from the factorization, and then update only the solution with these rows. A method for updating the solution with additional rows or constraints is extended to rank-deficient problems. Finally, the application of sparse matrix methods to large unbalanced analysis of variance problems is discussed. Some of the developed algorithms are programmed and tested.