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

Abstract

In this paper, based on the homotopy continuation method and the interval Newton method, an efficient algorithm is introduced to isolate the real roots of semi-algebraic system. Tests on some random examples and a variety of problems including transcendental functions arising in many applications show that the new algorithm reduces the cost substantially compared with the traditional symbolic approaches.

Eccentric pie charts and an unusual pie cutting

“…This step may bring false solutions, and we will see that it does so indeed. The square root in equations (15) and (16) can be eliminated likewise, with another possibility to have false solutions. Equations (8)- (11), (15), and (16) are replaced by polynomial equations…”

“…The equations of the eccentric pie chart include non-polynomial ones; however, an approximation by the Taylor series, written up to the first few terms, is polynomial. This idea of polynomial approximation has also been applied by Ji et al 16 [Example 4.4], Lim, 17 and Yalc xinbas x. 18 The roots of the approximating polynomial system can be found by the homotopy method, 13 then they are used as starting points of a Newton iteration for the approximated system of non-polynomial equations.…”