Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Tight linear envelopes for splines

Abstract: A sharp bound on the distance between a spline and its B-spline control polygon is derived. The bound yields a piecewise linear envelope enclosing spline and polygon. This envelope is particularly simple for uniform splines and splines in Bernstein-Bezier form and shrinks by a factor of 4 for each uniform subdivision step. The envelope can be easily and efficiently implemented due to its explicit and constructive nature.

A Novel Piecewise Linear Recursive Convolution Approach for Dispersive Media Using the Finite-Difference Time-Domain Method

Abstract: Two novel methods for implementing recursively the convolution between the electric field and a time dependent electric susceptibility function in the finite-difference time domain (FDTD) method are presented. Both resulting algorithms are straightforward to implement and employ an inclusive susceptibility function which holds as special cases the Lorentz, Debye, and Drude media relaxations. The accuracy of the new proposed algorithms is found to be systematically improved when compared to existing standard piecewise linear recursive convolution (PLRC) approaches, it is conjectured that the reason for this improvement is that the new proposed algorithms do not make any assumptions about the time variation of the polarization density in each time interval; no finite difference or semi-implicit schemes are used for the calculation of the polarization density. The only assumption that these two new methods make is that the first time derivative of the electric field is constant within each FDTD time interval.

Dynamics of a Hénon–Lozi-type map

Abstract: We present and analyze a smooth version of the piecewise linear Lozi map. The principal motivation for this work is to develop a map, which is better amenable for an analytical treatment as compared to the Henon map and is one that still possesses the characteristics of a Henon-type dynamics. This paper is a first step. It does the comparison of the Lozi map (which is a piecewise linear version of the Henon map) with the map that we introduce. This comparison is done for fixed parameters and also through global bifurcation by changing a parameter. If e measures the degree of smoothness, we prove that, as e → 0 , the stability and the existence of the fixed points are the same for both maps. We also numerically compare the chaotic dynamics, both in the form of an attractor and of a chaotic saddle.