Neville Elimination Research Articles

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Bidiagonal Factorizations of Filbert and Lilbert Matrices

Extensions of Filbert and Lilbert matrices are addressed in this work. They are reciprocal Hankel matrices based on Fibonacci and Lucas numbers, respectively, and both are related to Hilbert matrices. The Neville elimination is applied to provide explicit expressions for their bidiagonal factorization. As a byproduct, formulae for the determinants of these matrices are obtained. Finally, numerical experiments show that several algebraic problems involving these matrices can be solved with outstanding accuracy, in contrast with traditional approaches.

Inverse central ordering for the Newton interpolation formula

An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples.

A collection of efficient tools to work with almost strictly sign regular matrices

In this work, several algorithms have been implemented with Matlab to obtain an algorithmic characterizations of almost strictly sign regular matrices using Neville elimination.

Comparing pivoting strategies for almost strictly sign regular matrices

In this paper some properties of two-determinant pivoting for Neville elimination are presented. In particular, we consider a zero-increasing property and we show an optimal normwise growth factor. Comparisons with other pivoting strategies for Neville elimination and with Gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed. Numerical examples are included.

Backward stability with almost strictly sign regular matrices

A sign regular matrix is almost strictly sign regular if all its nontrivial minors of the same order have the same strict sign. These matrices form a subclass of sign regular matrices (matrices whose minors of the same order have the same sign). In this paper, the backward stability for almost strictly sign regular matrices is studied when Neville elimination with two-determinant pivoting strategy is applied. In addition, several numerical experiments are also presented.

Almost strictly sign regular matrices and Neville elimination with two-determinant pivoting

In 2007 Cortés and Peña introduced a pivoting strategy for the Neville elimination of nonsingular sign regular matrices and called it two-determinant pivoting. Neville elimination has been very useful for obtaining theoretical and practical properties for totally positive (negative) matrices and other related types of matrices. A real matrix is said to be almost strictly sign regular if all its nontrivial minors of the same order have the same strict sign. In this paper, some nice properties related with the application of Neville elimination with two-determinant pivoting strategy to almost strictly sign regular matrices are presented.

Computing the Bézier Control Points of the Lagrangian Interpolant in Arbitrary Dimension

The Bernstein--Bézier form of a polynomial is widely used in the fields of computer aided geometric design, spline approximation theory, and, more recently, in high order finite element methods for the solution of partial differential equations. However, if one wishes to compute the classical Lagrange interpolant relative to the Bernstein basis, then the resulting Bernstein--Vandermonde matrix is found to be highly ill-conditioned (though not as severe as in the Vandermonde case). In the univariate case of degree $n$, Marco and Martínez [Linear Algebra Appl., 422 (2007), pp. 616--628] showed, using an approach based on Neville elimination, that one can obtain an $\mathcal{O}(n^2)$ algorithm for solving the system, which also exploits the total positivity of the Bernstein basis. Remarkable as it may be, the Marco--Martínez algorithm has some drawbacks: The derivation of the algorithm is quite technical; the interplay between the ideas of total positivity and Neville elimination are not part of the standard armory of many nonspecialists; and the algorithm does not seem to extend to higher dimensional simplices. The present work addresses these issues. An alternative algorithm for solving the univariate Bernstein--Vandermonde linear system is presented that has the same complexity as the Marco--Martínez algorithm and whose stability does not seem to be in any way inferior; a simple derivation using only the basic theory of Lagrange interpolation (at least in the univariate case); and a natural generalization to the multivariate case.

Rank structure properties of rectangular matrices admitting bidiagonal-type factorizations

In this paper, we investigate the class of rectangular matrices that admit bidiagonal-type factorizations by Neville elimination without exchanges. We provide a complete characterization for a rectangular matrix to be factored as a product of bidiagonal factors and a banded factor in terms of rank structure properties. Consequently, we give a complete characterization of the class of rectangular matrices that have Neville factorizations.

Eigenvalue localization and Neville elimination

Neville elimination is an elimination procedure alternative to Gaussian elimination and very adequate when dealing with some special classes of matrices. In this paper, we present pivoting strategies such that the radii of the Geršgorin circles of the Schur complements through Neville elimination with these pivoting strategies reduce their length and we consider classes of matrices important in many applications. We include illustrative examples comparing the results with those obtained with Gaussian elimination and showing that our hypotheses are necessary.

Componentwise backward error analysis of Neville elimination

In this paper, we perform a backward error analysis of Neville elimination whenever there need row exchanges. Componentwise backward error bounds are presented for this elimination procedure applied to any nonsingular matrices. Consequently, it is shown that Neville elimination with two-determinant pivoting proposed by Cortes and Peña (2007) [5] is an excellent method for the triangularization of nonsingular sign regular matrices including totally nonnegative and totally nonpositive matrices, which has a pleasantly small componentwise relative backward error. In particular, a small componentwise relative error bound is also provided for the bidiagonal factorization of totally nonnegative matrices.

On the characterization of almost strictly sign regular matrices

An algorithmic characterization of the nonsingular almost strictly sign regular matrices by Neville elimination is presented.

Increasing data locality and introducing Level-3 BLAS in the Neville elimination

In this paper we present two new algorithmic variants to compute the Neville elimination, with and without pivoting, which improve data locality and cast most of the computations in terms of high-performance Level 3 BLAS. The experimental evaluation on a state-of-the-art multi-core processor demonstrates that the new blocked algorithms exhibit a much higher degree of concurrency and better cache usage, yielding higher performance while offering numerical accuracy akin to that of the traditional columnwise variant in most cases.

A note on matrices with maximal growth factor for Neville elimination

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.

An efficient and scalable block parallel algorithm of Neville elimination as a tool for the CMB maps problem

This paper analyses the performance of several versions of a block parallel algorithm in order to apply Neville elimination in a distributed memory parallel computer. Neville elimination is a procedure to transform a square matrix A into an upper triangular one. This analysis must take into account the algorithm behaviour as far as execution time, efficiency and scalability are concerned. Special attention has been paid to the study of the scalability of the algorithms trying to establish the relationship existing between the size of the block and the performance obtained in this metric. It is important to emphasize the high efficiency achieved in the studied cases and that the experimental results confirm the theoretical approximation. Therefore, we have obtained a high predicting ability tool of analysis. Finally, we will present the elimination of Neville as an efficient tool in detecting point sources in cosmic microwave background maps.

A collection of examples where Neville elimination outperforms Gaussian elimination

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.

Neville elimination: an efficient algorithm with application to chemistry

Solving linear systems is often required in chemical problems. Besides, birth and death processes occur in many chemical phenomena and the matrices associated to these processes are totally positive, that is, all their minors are nonnegative. Neville elimination is an elimination procedure very useful when dealing with these matrices. Convergence and stability of iterative refinement using Neville elimination are analyzed, in particular when the coefficient matrix is totally positive. Other applications to chemistry are commented and numerical experiments are shown.

Neville elimination on multi- and many-core systems: OpenMP, MPI and CUDA

This paper describes several parallel algorithmic variations of the Neville elimination. This elimination solves a system of linear equations making zeros in a matrix column by adding to each row an adequate multiple of the preceding one. The parallel algorithms are run and compared on different multi- and many-core platforms using parallel programming techniques as MPI, OpenMP and CUDA.

Growth factors of pivoting strategies associated with Neville elimination

Neville elimination is a direct method for solving linear systems. Several pivoting strategies for Neville elimination, including pairwise pivoting, are analyzed. Bounds for two different kinds of growth factors are provided. Finally, an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed using random matrices.

Iterative refinement for Neville elimination

Neville elimination is an elimination procedure that is very useful when dealing with totally positive matrices. We provide a sufficient condition for the convergence of the iterative refinement using Neville elimination.