An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

Abstract

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, an algorithm is presented to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

Real root classification of parametric spline functions

“…Thus, it is of theoretic and practical significance to study the counting and isolating the real roots of spline functions and its related problems. There exists several work on this issue [6][7][8][9][10][11][12][13]. For univariate case, Goodman [6] and de Boor [7] studied the relationship between the number of real roots of a univariate spline and the sequence of its B-spline coefficients, which provides new bounds on the number of real roots of the spline function.…”

“…For multivariate case, Lai et al [11,12] gave the method to compute the supremum and its distribution of the distinct torsion-free real zeros of a given parametric piecewise polynomial system. In 2011, Wu and Zhang [13] presented an algo-S n ½x 0 ; x 1 ; . .…”