Section-wise piecewise-linear functions: Canonical representation, properties, and applications

Abstract

This paper presents a new closed form analytical formula for representing n-dimensional surfaces and scalar functions of n variables which are piecewise-linear over each cross section obtained by freezing any combination of n - 1 of the n coordinates. This new section-wise piecewise-linear representation can be easily programmed with efficient computer storage. It is a global representation in the sense that a single formula is used to compute for f(x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ,x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ,...,x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> ) for all values of (x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ,...., x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> ). Since this representation is expressed in closed analytic form, it allows standard mathematical operations and manipulations to be carried out in theorectical studies, In particular, it led to the possibility of deriving explicit closed form expressions for system parameters and design formulas. Examples are given which illustrate the potential applications of this representation in the modeling and analysis of nonlinear devices, circuits and systems.