Accurate computations with totally positive Bernstein-Vandermonde matrices

2018

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.

“…For a given sequence of nodes, a < t 1 <⋯< t n +1 < b , we call fg‐Vandermonde matrix (fg‐V matrix) of order n +1 and nodes t 1 ,…, t n +1 the collocation matrix

M_{n + 1, t_{1}, ⋯, t_{n + 1}} : = {((), ((), \binom{n}{j - 1}) f^{j - 1} false(t_{i} false) g^{n - j + 1} false(t_{i} false))}_{1 \leq i, j \leq n + 1} .

Let us observe that the matrix class includes the Bernstein–Vandermonde matrix

B_{n + 1, t_{1}, ⋯, t_{n + 1}} = {((), ((), \binom{n}{j - 1}) t_{i}^{j - 1} {false(1 - t_{i} false)}^{n - j + 1})}_{1 \leq i, j \leq n + 1},

obtained by considering the collocation matrix of the Bernstein basis of polynomials of degree n on the interval [0,1], at nodes 0< t 1 <⋯< t n +1 <1. Marco et al proved that

B_{n + 1, t_{1}, ⋯, t_{n + 1}}

is an STP matrix. In the following result, we are going to prove that fg‐V matrices are also STP.…”

Section: Bidiagonal Factorization For the Collocation Matrices Of A Gmentioning

confidence: 99%

“…This has been achieved in some important subclasses of TP matrices with applications to computer-aided geometric design (cf. other works 2,3,9,10 ), finance (cf. the work of Delgado et al 11 ), or combinatorics (cf.…”

Section: Introductionmentioning

confidence: 99%

Accurate computations with collocation matrices of a general class of bases

Mainar

2018

“…for i = 1, … ,n − 1, and p 11 = 1. Moreover, if the parameters (u i ) 1 ≤ i ≤ n are known to HRA , then the computation of (13) and (14) (and so, of the (G # ) given by (12)) can be performed to HRA and with 2n − 1 quotients and n − 1 additions, subtractions, and products.…”

Section: Theoremmentioning

confidence: 99%

“…Demmel). Among the classes of matrices for which algorithms to HRA have been constructed, we can mention some subclasses of nonsingular totally positive (TP) matrices . As shown in Koev, for a nonsingular TP matrix A , the adequate parameterization to obtain computations to HRA is its bidiagonal factorization

scriptB scriptD (A)

.…”

Section: Introductionmentioning

confidence: 99%

“…Among the classes of matrices for which algorithms to HRA have been constructed, we can mention some subclasses of nonsingular totally positive (TP) matrices. [13][14][15][16] As shown in Koev, 17,18 for a nonsingular TP matrix A, the adequate parameterization to obtain computations to HRA is its bidiagonal factorization (A). Given a matrix A = (a ij ) 1 ≤ i, j ≤ n , the conversion matrix of A is the matrix A # : = (a n + 1 − i, n + 1 − j ) 1 ≤ i, j ≤ n .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Accurate and fast computations with positive extended Schoenmakers–Coffey matrices

Delgado

2016

Summary Schoenmakers–Coffey matrices are correlation matrices with important financial applications. Several characterizations of positive extended Schoenmakers–Coffey matrices are presented. This paper provides an accurate and fast method to obtain the bidiagonal decomposition of the conversion of these matrices, which in turn can be used to compute with high relative accuracy the eigenvalues and inverses of positive extended Schoenmakers–Coffey matrices. Numerical examples are included.

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

Mainar

Rubio

2020

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...