On Row and Column Orderings for Sparse Least Squares Problems

Let $P_1 $ and $P_2 $ be respectively $m \times m$ and $n \times n$ permutation matrices, with $m \geqq n$. Suppose the $m \times n$ sparse matrix $\bar A = P_1 AP_2 $ is reduced to upper trapezoidal form $\left[ {\begin{array}{*{20}c} R \\ 0 \\ \end{array} } \right]$ through the application of Givens rotations sequentially to the rows of $\bar A$. It is well known that the sparsity of R depends only on the choice of $P_2 $, but the choice of $P_1 $ can drastically affect the arithmetic required to compute R. In this paper we provide a mechanism for studying the connection between good row and good column orderings, along with a modified nested dissection algorithm for finding a good $P_2 $ which automatically induces a good $P_1 $. An analysis for a model problem is given, along with some experimental results.