No abstract

Given an m n matrix M with m > n, it is shown that there exists a permutation FI and an integer k such that the QR factorization MYI= Q(Ak ckBk) reveals the numerical rank of M: the k k upper-triangular matrix Ak is well conditioned, IlCkll2 is small, and Bk is linearly dependent on Ak with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take O (ran2) floating-point operations in the worst case.

An inexact Newton method is a generalization ofNewton's method for solving F(x) 0, F n _ _ in, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition F (x) + F'(xk) sk _< F(x)II for a "forcing term" r/ [0, 1). In typical applications, the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well. Promising choices of the forcing terms are given, their local convergence properties are analyzed, and their practical performance is shown on a representative set of test problems.

We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do not make use of any associated symmetric problems. Convergence results and error bounds are presented.

We i n v estigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. To perform most of the numerical computation in dense matrix kernels, we i n troduce the notion of unsymmetric supernodes. To better exploit the memory hierarchy, w e i n troduce unsymmetric supernode-panel updates and two-dimensional data partitioning. To speed up symbolic factorization, we use Gilbert and Peierls's depth-rst search with Eisenstat and Liu's symmetric structural reductions. We h a v e implemented a sparse LU code using all these ideas. We present experiments demonstrating that it is signi cantly faster than earlier partial pivoting codes. We also compare performance with Umfpack, which uses a multifrontal approach; our code is usually faster.Keywords: sparse matrix algorithms; unsymmetric linear systems; supernodes; column elimination tree; partial pivoting. AMSMOS subject classi cations: 65F05, 65F50. Computing Reviews descriptors: G.1.3 Numerical Analysis : Numerical Linear Algebra | Linear systems direct and iterative methods, Sparse and very large systems.