Sparse Linear Least Squares Problems in Optimization
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This paper studies the solution of the linear least squares problem for a large and sparse $m$ by $n$ matrix $A$ with $m \ge n$ by $QR$ factorization of $A$ and transformation of the right-hand side vector $b$ to $Q^Tb$. A multifrontal-based method for computing $Q^Tb$ using Householder factorization is presented. A theoretical operation count for the $K$ by $K$ unbordered grid model problem and problems defined on graphs with $\sqrt{n}$-separators shows that the proposed method requires $O(N_R)$ storage and multiplications to compute $Q^Tb$, where $N_R=O(n \log n)$ is the number of nonzeros of the upper triangular factor $R$ of $A$. In order to introduce BLAS-2 operations, Schreiber and Van Loan's storage-efficient WY representation [SIAM J. Sci. Stat. Comput., 10 (1989), pp. 53--57] is applied for the orthogonal factor $Q_i$ of each frontal matrix $F_i$. If this technique is used, the bound on storage increases to $O(n(\log n)^2)$. Some numerical results for the grid model problems as well as Harwell--Boeing problems are provided.