Intersection points algorithm for piecewise algebraic curves based on Groebner bases

Abstract

Piecewise algebraic curve is defined as the zero set of a bivariate spline. In this paper, we mainly study the intersection points algorithm for two given piecewise algebraic curves based on Groebner bases. Given a domain D and a partition , we present a flow and introduce the truncated signs, and then represent the two piecewise algebraic curves in the global form. We get their Groebner bases with respect to a lexicographic order and adopt the interval arithmetic in the back-substitution process, which makes the algorithm numerically precise. An example is also presented to show the algorithm's feasibility and effectiveness.

Numerical Method for Real Root Isolation of Semi-Algebraic System and Its Applications

“…However, the number of bisections, the order of Taylor expansion and the error control are deserved to further study. 4 2 + 39x 2 2 = 0. −7 − 7x 2 + 4x 1 x 3 − 2x 1 x 4 + 5x 2 x 3 + 5x 2 x 4 = 0.…”

“…Now suppose we have obtained the isolated real root intervals of polynomial system f defined in (4). Let X = ([a 1 , b 1 ], · · · , [a n , b n ]) be an isolated real root interval of f , andx be the accurate zero of f such thatx ∈ X.…”

“…In the following, we will present a method to deal with the non-negative systems n in semi-algebraic system defined in (4). • Substituting real − roots{t} into n, and suppose n(real − roots{t}) = ([a 1 , b 1 ], · · · , [a s , b s ]);…”

“…Hence, f and g have 5 real points. Methods given by Wang, et al, based on interval method, Groebner basis in [1,2,3,4,5] can also find the commom points of piecewise algebraic curve, but they did not consider these points which lie in the boundary.…”

“…The problem of counting and isolating real solutions of nonlinear system is an important topic in computing geometry and many other applications in various fields, e.g., the real intersection points for piecewise algebraic curve [1,2,3,4,5], the stability of a large class of biological networks [6,7], discovering non-linear ranking functions of loop programs in [8,9], automated proving inequality type theorem [10] and so on.…”