Accurate and fast computations with positive extended Schoenmakers–Coffey matrices

Delgado, Jorge; Peña, Guillermo; Peña, J. M.

doi:10.1002/nla.2066

Cited by 7 publications

(4 citation statements)

References 27 publications

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“…Finally, we compute the solution of the linear system by using the algorithm TNSolve of P. Koev, by previously computing B = BD(A) by means of the algorithm TNCauchyBD of P. Koev, with the instruction B = TNCauchyBD(x,y), by defining x = [0,1,2,3,4,5,6] and y = [1,2,3,4,5,6,7]. In this case the relative error is 1.4e − 16, which again confirms the high relative accuracy to be expected of this algorithm when f has an alternating sign pattern.…”

Section: Bidiagonal Factorization and Neville Eliminationmentioning

confidence: 99%

“…We can compute B = BD(A) by means of the algorithm TNCauchyBD of P. Koev, with the instruction B = TNCauchyBD(x,y), by defining x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], y = [1,2,3,4,5,6,7,8,9,10] (P. Koev is using as entries of the Cauchy matrix c ij = 1/(x i + y j )). Then the eigenvalues are computed by means of the instruction TNEigenvalues(B).…”

Section: Eigenvalue and Singular Value Problemsmentioning

confidence: 99%

“…The availability of the algorithms of Koev in [26] has encouraged the search for new algorithms for the bidiagonal decomposition of various classes of totally positive structured matrices, including matrices from new application fields such as [6] (see [39] for a recent account).…”

Section: Eigenvalue and Singular Value Problemsmentioning

confidence: 99%

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Bjorck-Pereyra-type methods and total positivity

Martinez

2018

Preprint

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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, 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 showed the crucial role of total positivity for 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 behaviour of these algorithms when compared with algorithms which ignore the matrix structure are also included.

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Section: Bidiagonal Factorization and Neville Eliminationmentioning

confidence: 99%

Section: Eigenvalue and Singular Value Problemsmentioning

confidence: 99%

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Bjorck-Pereyra-type methods and total positivity

Martinez

2018

Preprint

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“…other works), finance (cf. the work of Delgado et al), or combinatorics (cf. the work of Delgado et al).…”

Section: Introductionmentioning

confidence: 99%

Accurate computations with collocation matrices of a general class of bases

Mainar

Peña

2018

Numerical Linear Algebra App

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Summary In this paper, we provide algorithms for computing the bidiagonal decomposition of the collocation matrices of a very general class of bases of interest in computer‐aided geometric design and approximation theory. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these matrices, such as the calculation of their inverses, their eigenvalues, or their singular values. Numerical experiments illustrate the results.

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Accurate bidiagonal decomposition of collocation matrices of weighted ϕ‐transformed systems

Mainar

Peña

Rubio

2020

Numerical Linear Algebra App

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Given a system of functions, we introduce the concept of weighted -transformed system, which will include a very large class of useful representations in Statistics and Computer Aided Geometric Design. An accurate bidiagonal decomposition of the collocation matrices of these systems is obtained. This decomposition is used to present computational methods with high relative accuracy for solving algebraic problems with collocation matrices of weighted -transformed systems such as the computation of eigenvalues, singular values, and the solution of some linear systems. Numerical examples illustrate the accuracy of the performed computations. K E Y W O R D S accurate computations, bidiagonal decompositions, rational basis INTRODUCTIONIn this article, we present a very general procedure for generating, from an initial system and a positive function , new systems of functions useful for many applications. These systems, which we call weighted -transformed systems, arise with relevant probability distributions. They also include important rational bases 1,2 as well as systems belonging to spaces mixing algebraic, trigonometric, and hyperbolic polynomials, which are useful in many applications of Approximation Theory and Computer Aided Geometric Design (CAGD). The weighted -transformed systems inherit from the initial system its accuracy when computing with its collocation matrices. The accurate computation with structured classes of matrices is an important issue in Numerical Linear Algebra and it is receiving increasing attention in the recent years (cf. References 3-5). For this purpose, a parametrization adapted to the structure of the considered matrices is needed. Let us recall that an algorithm can be performed with high relative accuracy (HRA) if it does not include subtractions (except of the initial data), that is, if it only includes products, divisions, sums of numbers of the same sign, and subtractions of the initial data (cf. Reference 6). Performing an algorithm with HRA is a very desirable goal because it implies that the relative errors of the computations are of the order of the machine precision, independently of the size of the condition number of the considered problem. Let us recall that a totally positive (TP) matrix has all its minors nonnegative. TP matrices arise in many applications (cf. Reference 7). It is known that, for some subclasses of TP matrices, many algebraic computations can be performed with HRA. For instance, the computation of their eigenvalues, singular values, or the solutions of linear systems Ax = b such that the components of b have alternating signs (see Reference 8 and the references therein). The key tool for this purpose is provided by the algorithms of References 6 and 9 jointly with the use of a bidiagonal factorization of a nonsingular TP matrix, which can be obtained with HRA for some of those matrices. Up to now, this has been achieved with some relevant subclasses of TP matrices with applications Numer Linear Algebra Appl. 2020;27:e2295.wileyonlinelibrary.com/journa...

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Accurate and fast computations with positive extended Schoenmakers–Coffey matrices

Cited by 7 publications

References 27 publications

Bjorck-Pereyra-type methods and total positivity

Bjorck-Pereyra-type methods and total positivity

Accurate computations with collocation matrices of a general class of bases

Accurate bidiagonal decomposition of collocation matrices of weighted ϕ‐transformed systems

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