Sign regular matrices and Neville elimination

Applied Numerical Mathematics, 1993

Scaled partial pivoting with respect to the [,-norm and Euclidean norm are studied for Gauss and Neville elimination applied to totally positive linear systems. It is proved that in exact arithmetic row exchanges are not necessary. The same result holds, for sufficiently high precision arithmetic, in Gauss elimination and also, for a class of totally positive ma&ices (which included B-spline collocation matrices), in Neville elimination. A geometrical interpretation of these results is given.

Fuel and Energy Abstracts

Neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations. The growth factor is an indicator of the numerical stability of an algorithm. In the literature, bounds for the growth factor of Neville elimination with some pivoting strategies have appeared. In this work, we determine all the matrices such that the minimal upper bound of the growth factor of Neville elimination with those pivoting strategies is reached.

Linear Algebra and its Applications, 2008

In this paper it is shown that Neville elimination is suited to exploit the rank structure of an order-r quasiseparable matrix A ∈ C n×n by providing a condensed decomposition of A as product of unit bidiagonal matrices, all together specified by O(nr) parameters, at the cost of O(nr 3) flops. An application of this result for eigenvalue computation of totally positive rank-structured matrices is also presented.

Applied Mathematics and Computation, 2010

Neville elimination is an elimination procedure alternative to Gaussian elimination. It is very useful when dealing with totally positive matrices, for which nice stability results are known. Here we include examples, most of them test matrices used in MATLAB which are not totally positive matrices, where Neville elimination outperforms Gaussian elimination.

International Journal of Computer Mathematics, 2009

Lecture Notes in Computer Science, 2002

This paper presents a parallel algorithm to solve linear equation systems. This method, known as Neville elimination, is appropriate especially for the case of totally positive matrices (all its minors are non-negative). We discuss one common way to partition coefficient matrix among processors. In our mapping, called columwise block-cyclic-striped mapping, the matrix is divided into blocks of complete columns and

2010

An n × n real matrix is called sign regular if, for each k (1 k n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.

Linear Algebra and its Applications, 2002

In this paper we prove that Neville elimination can be matricially described by elementary matrices. A PLU-factorization is obtained for any n × m matrix, where P is a permutation matrix, L is a lower triangular matrix (product of bidiagonal factors) and U is an upper triangular matrix. This result generalizes the Neville factorization usually applied to characterize the totally positive matrices. We prove that this elimination procedure is an alternative to Gaussian elimination and sometimes provides a lower computational cost.

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999

Neville elimination is an elimination method which creates zeros in a column of a matrix by adding to each row an appropriate multiple of the previous one. This elimination procedure, alternative to Gaussian elimination, has proved to be useful when dealing with some special classes of matrices such as totally positive matrices. In this paper we analyze block and partitioned Neville elimination, necessary steps in order to adapt Neville elimination to many high performance computer architectures. 0 1999 Academic des sciences&ditions scientifiques et medicales Elsevier SAS Dhveloppement de l'&ilimination de Neville partition&e et par blocs RCSUmC* Note prLsentCe par Philippe G. CIARLET. 0764~4442/99/03291091 0 1999 AcadCmie des science&ditions scientifiques et mtdicales Elsevier SAS. Tous droits r&ervts.

Linear Algebra and Its Applications, 2001

In this paper, a method to compute the solution of a system of linear equations by means of Neville elimination is described using two kinds of partitioning techniques: Block and Block-striped. This type of approach is especially suited to the case of totally positive linear systems, which is present in different fields of application. Although Neville elimination carried out more floating point operations than Gaussian elimination in some cases, in this study we confirm that these advantages disappear when we use multiprocessor systems. On the other hand, the overall parallel run time of Neville elimination is better than Gauss time as Neville elimination uses a lower cost communication model.