Gaussian elimination for the solution of linear systems of equations (Q2702611)

This is a review of the main stream of research on Gaussian elimination done during the last thirty years. In five chapters, the author has done an excellent job of covering the research and development of related important algorithms. The review starts with a chapter on numerical solution of general linear systems. In this chapter Gaussian elimination for symmetric and non symmetric systems, block methods and related topics are described. This is followed by a chapter on error analysis. The third chapter is on vector and parallel algorithms for general systems. In this chapter, the well known software systems BLAS and LAPACK, as well as, triangular system solvers and LU factorization on distributed memory computers are discussed. The last two chapters are concerned with sparse matrices containing excellent descriptions of a wide variety of theoretical contributions and related computational algorithms for sparse symmetric and non symmetric systems available for serial and parallel computers. NEWLINENEWLINENEWLINEThe authors show how bipartite matching algorithms can be used to permute the rows and columns of a matrix so that the diagonal of the permuted matrix is large. The proposed algorithm computes a matching that corresponds to a permutation of a sparse matrix such that the product (or sum) of the diagonal entries is maximized. They also consider a modified version of this algorithm to compute a permutation that maximizes the smallest diagonal entry. They also investigate the influence of scaling the matrix. Results of some computational experiments are also given.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00016].