Accurate and fast computations with positive extended Schoenmakers–Coffey matrices

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

doi:10.1002/nla.2066

Cited by 8 publications

(10 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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“…Among them, first we mention here some few subclasses of nonsingular TP matrices and matrices related with them. For instance, for Generalized Vandermonde matrices [9], for Bernstein-Vandermonde matrices [10][11][12], for Said-Ball-Vandermonde matrices [13], for Cauchy-Vandermonde matrices [14], for Schoenmakers-Coffey matrices [15], for h-Bernstein-Vandermonde matrices [16] or for tridiagonal Toeplitz matrices [17], and much more subclasses of TP matrices. HRA algorithms for some algebraic problems related with other class of matrices have also been developed.…”

Section: Introductionmentioning

confidence: 99%

Green Matrices, Minors and Hadamard Products

Delgado

Peña

2023

Axioms

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Green matrices are interpreted as discrete version of Green functions and are used when working with inhomogeneous linear system of differential equations. This paper discusses accurate algebraic computations using a recent procedure to achieve an important factorization of these matrices with high relative accuracy and using alternative accurate methods. An algorithm to compute any minor of a Green matrix with high relative accuracy is also presented. The bidiagonal decomposition of the Hadamard product of Green matrices is obtained. Illustrative numerical examples are included.

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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 and fast computations with positive extended Schoenmakers–Coffey matrices

Cited by 8 publications

References 27 publications

Bjorck-Pereyra-type methods and total positivity

Bjorck-Pereyra-type methods and total positivity

Green Matrices, Minors and Hadamard Products

Accurate computations with collocation matrices of a general class of bases

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