Accurate computations with Lupaş matrices

Delgado, Jorge; Pea, J.M.

doi:10.1016/j.amc.2017.01.031

Cited by 11 publications

(7 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Reference ) are the rational Bernstein functions

ρ_{i}^{n} (t) : = \frac{a_{n, i} (t)}{w_{n} (t)}, i = 0, \dots, n

with

a_{n, i} (t) : = [\frac{0}{n}] q^{i (i - 1) / 2} t^{i} {(1 - t)}^{n - i}, w_{n} (t) : = \sum_{i = 0}^{n} a_{n, i} (t) = \prod_{j = 1}^{n} (1 - t + q^{j - 1} t) .

Clearly, this basis is a weighted 1/ω n ‐transformed system where the weights

w_{i}^{n} = [\frac{0}{n}] q^{i (i - 1) / 2}

can be obtained from for the particular choice a i =1 and b i = q i −1 , i =1,…, n . The bidiagonal factorization of its collocation matrices coincides with the obtained in Reference .…”

Section: Rational Weighted φ ‐Transformed Systemssupporting

confidence: 81%

“…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 and the references therein). The key tool for this purpose is provided by the algorithms of References and jointly with the use of a bidiagonal factorization of a nonsingular TP matrix, which can be obtained with HRA for some of those matrices.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Mainar

Peña

Rubio

2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

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...

show abstract

“…Reference ) are the rational Bernstein functions

ρ_{i}^{n} (t) : = \frac{a_{n, i} (t)}{w_{n} (t)}, i = 0, \dots, n

with

a_{n, i} (t) : = [\frac{0}{n}] q^{i (i - 1) / 2} t^{i} {(1 - t)}^{n - i}, w_{n} (t) : = \sum_{i = 0}^{n} a_{n, i} (t) = \prod_{j = 1}^{n} (1 - t + q^{j - 1} t) .

Clearly, this basis is a weighted 1/ω n ‐transformed system where the weights

w_{i}^{n} = [\frac{0}{n}] q^{i (i - 1) / 2}

can be obtained from for the particular choice a i =1 and b i = q i −1 , i =1,…, n . The bidiagonal factorization of its collocation matrices coincides with the obtained in Reference .…”

Section: Rational Weighted φ ‐Transformed Systemssupporting

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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

Mainar

Peña

Rubio

2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…Note that the different zero Jordan canonical forms of an ITN matrix associated with a realizable triple can be directly obtained from the values of r i and the new realizable triples following the above process taking into account equation (7). It is not necessary to calculate the full rank factorizations in echelon form.…”

Section: Jordan Canonical Forms Of Itn Matricesmentioning

confidence: 99%

“…A ∈ R n×n is called totally nonnegative if all its minors are nonnegative and it is abbreviated as TN, see for instance [1][2][3][4][5]. Due to its wide variety of applications in algebra, computer aided geometric design, differential equations, economics, quantum theory and other fields, TN matrices have been studied by several authors who have obtained properties, the Jordan structure and characterizations by using the Neville elimination [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

All Jordan canonical forms of irreducible totally non-negative matrices

Cantó

Urbano

2019

Linear and Multilinear Algebra

View full text Add to dashboard Cite

Let A ∈ R n×n be an irreducible totally nonnegative matrix with rank r and principal rank p, that is, every minor of A is nonnegative and p is the size of the largest invertible principal submatrix of A. Using Number Theory, we calculate the number of Jordan canonical forms of irreducible totally nonnegative matrices associated with a realizable triple (n, r, p). Moreover, by using full rank factorizations of A and applying the Flanders theorem we obtain all these Jordan canonical forms. Finally, some algorithms associated with these results are given.

show abstract

“…differing) signs, and otherwise only adds or subtracts input data. Some classes of totally positive matrices for which there are fast and accurate algorithms for computing their bidiagonal decompositions (A) are, for instance, Cauchy-Vandermonde, 8 generalized Vandermonde, 9 Bernstein-Vandermonde, 10 Lupaş, 11 generalized Pascal matrices, 12 collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis, 13 Wronskian matrices, 14 or Gram matrices of Bernstein bases. 15 The Lagrange basis of the space Π n (t) of the polynomials of degree less than or equal to n, widely used in polynomial interpolation, is…”

Section: Introductionmentioning

confidence: 99%

Accurate bidiagonal decomposition of Lagrange–Vandermonde matrices and applications

Marco,

Martínez,

Viaña

2023

Numerical Linear Algebra App

View full text Add to dashboard Cite

Lagrange–Vandermonde matrices are the collocation matrices corresponding to Lagrange‐type bases, obtained by removing the denominators from each element of a Lagrange basis. It is proved that, provided the nodes required to create the Lagrange‐type basis and the corresponding collocation matrix are properly ordered, such matrices are strictly totally positive. A fast algorithm to compute the bidiagonal decomposition of these matrices to high relative accuracy is presented. As an application, the problems of eigenvalue computation, linear system solving and inverse computation are solved in an efficient and accurate way for this type of matrices. Moreover, the proposed algorithms allow to solve fastly and to high relative accuracy some of the cited problems when the involved matrices are collocation matrices corresponding to the standard Lagrange basis, although such collocation matrices are not totally positive. Numerical experiments illustrating the good performance of our approach are also included.

show abstract

Accurate computations with Lupaş matrices

Cited by 11 publications

References 18 publications

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

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

All Jordan canonical forms of irreducible totally non-negative matrices

Accurate bidiagonal decomposition of Lagrange–Vandermonde matrices and applications

Contact Info

Product

Resources

About