Extremal and optimal properties of B-bases collocation matrices
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Numerische Mathematik
Extremal and optimal properties of B-bases collocation matrices Jorge Delgado1 · J. M. Peña2 Received: 4 February 2019 / Revised: 18 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Totally positive matrices are related with the shape preserving representations of a space of functions. The normalized B-basis of the space has optimal shape preserving properties. Bernstein polynomials, B-splines and rational Bernstein bases are examples of normalized B-bases. It is proven that the minimal eigenvalue and singular value of a collocation matrix of a normalized B-basis is bounded below by the minimal eigenvalue and singular value of the corresponding collocation matrix of any normalized totally positive basis of the same space. The optimal conditioning for the ∞-norm of a collocation matrix of a normalized B-basis among all the normalized totally positive bases of a space of functions is also shown. Numerical examples confirm the theoretical results and answer related questions. Mathematics Subject Classification 15A18 · 15A12 · 65F35 · 15B48 · 65D17
1 Introduction The minimal eigenvalue and singular value of a nonsingular matrix play an important role in many problems of Numerical Analysis. In this paper, we prove that the minimal eigenvalue and singular value of the collocation matrices of some systems of functions (such as Bernstein polynomials and B-splines) are not less than the minimal eigenvalue and singular value of the collocation matrices of other bases of their generated spaces. The bases considered in this paper are the bases associated with shape preserving rep-
This research was partially funded by the Spanish research Grant PGC2018-096321-B-I00 (MCIU/AEI), by Gobierno de Aragón (E41-17R) and Feder 2014-2020 “Construyendo Europa desde Aragón”.
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Jorge Delgado [email protected]
1
Departamento de Matemática Aplicada, Escuela Universitaria Politécnica de Teruel, Universidad de Zaragoza, 44003 Teruel, Spain
2
Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain
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J. Delgado , J. M. Peña
resentations (see [2,3]), whose corresponding collocation matrices are totally positive. Totally positive matrices, which are also called totally nonnegative in the literature, play an important role in many fields, such as approximation theory, computer aided geometric design (CAGD), mechanics, differential or integral equations, statistics, combinatorics, economics and biology (see [1,11,13,16] or [24]). A matrix is totally positive (TP) if all its minors are nonnegative. Relevant properties of TP matrices about algebraic computations with high relative accuracy have been found recently (cf. [10,17,18]). In fact, for some classes of TP matrices adequately parameterized, one can compute their eigenvalues, singular values, inverses or the solutions of some linear systems with high relative accuracy independently of their conditioning (see [9,10,18,20]). This holds for many popular matrices, such as positive Vandermonde matrices o
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