We prove that standard Gaussian random multipliers are expected to stabilize numerically both Gaussian elimination with no pivoting and block Gaussian elimination. Our tests show similar results where we applied circulant random multipliers instead of Gaussian ones.

For symmetric indeenite tridiagonal matrices, block LDL T factorization without interchanges is shown to have excellent numerical stability when a pivoting strategy of Bunch is used to choose the dimension (1 or 2) of the pivots.

We show experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the Rfactor, can lead to a substantial decrease of the number of outer parallel iteration steps in the parallel block-Jacobi SVD algorithm, whereby the details depend on the condition number and on the shape of spectrum, including the multiplicity of singular value...

A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going...

We present algorithms for the symbolic and numerical factorization phases in the direct solution of sparse unsymmetric systems of linear equations. We have modified a classical symbolic factorization algorithm for unsymmetric matrices to inexpensively compute minimal elimination structures. We give an efficient algorithm to compute a near-minimal data-dependency graph for unsymmetric multifront...

Gaussian elimination based sparse LU factorization with partial pivoting is important to many scientiic applications, but it is still an open problem to develop a high performance sparse LU code on distributed memory machines. The main diiculty is that partial pivoting operations make structures of L and U factors unpredictable beforehand. This paper presents an approach called S for paralleliz...

In this paper we present HUND, a hypergraph-based unsymmetric nested dissection ordering algorithm for reducing the fill-in incurred during Gaussian elimination. HUND has several important properties. It takes a global perspective of the entire matrix, as opposed to local heuristics. It takes into account the assymetry of the input matrix by using a hypergraph to represent its structure. It is ...

The performance of a sparse direct solver is dependent upon the pivot sequence that is chosen during the analyse phase. In the case of symmetric indefinite systems, it may be necessary to modify this sequence during the factorization to ensure numerical stability. Delaying pivots can have serious consequences in terms of time as well as the memory and flops required for the factorization and su...

The problem of surface reconstruction from a set of 3D points given by their coordinates and oriented normals is a difficult problem, which has been tackled with many different approaches. In 1999, Bernardini and colleagues introduced a very elegant and efficient reconstruction method that uses a ball pivoting around triangle edges and adds new triangles if the ball is incident to three points ...

Fast 0(n2) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermondelike matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast 0{n2) Toeplit...