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...
