A canonical form for the continuous piecewise polynomial functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:283

Article

Caravantes, Jorge¹  Angeles Gomez-Molleda, M.²  Gonzalez-Vega, Laureano³ 

[1] Univ Complutense Madrid, Dept Algebra, E-28040 Madrid, Spain

[2] Univ Malaga, Dept Algebra Geometr & Topol, E-29071 Malaga, Spain

[3] Univ Cantabria, Dept Matemat Estadist & Computac, Cantabria, Spain

关键词: Continuous piecewise polynomial functions;    Pierce-Birkhoff conjecture;    Canonical form for functions;    Conversion algorithms;   

DOI  :  10.1016/j.cam.2014.11.033

来源: Elsevier

【 摘 要 】

We present in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functions {C-i(P) : P is an element of Q[x], i = 0,..., deg(P)} defined by C-i(P)(x) ={(0)(p(x)) (if x >=alpha) (if x <=alpha) where a is the ith real root of the polynomial P. These functions will allow us to represent and manipulate easily every continuous piecewise polynomial function through the use of the corresponding canonical form. It will be also shown how to produce a rational representation of each function C-i(P) allowing its evaluation by performing only operations in Q and avoiding the use of any real algebraic number. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free