zbMATH — the first resource for mathematics

Isolating the real roots of the piecewise algebraic variety. (English) Zbl 1165.14317
Summary: The piecewise algebraic variety, as a set of the common zeros of multivariate splines, is a kind of generalization of the classical algebraic variety. In this paper, we present an algorithm for isolating the zeros of the zero-dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials. Numerical example illustrates that the proposed algorithm is flexible.

14P10 Semialgebraic sets and related spaces
14Q99 Computational aspects in algebraic geometry
Full Text: DOI
[1] Collins, G.E., Quantifer elimination for real closed fields by cylindric algebraic decomposition, ()
[2] G.E. Collins, A.G. Akritas, Polynomial real root isolation using Descarte’s rule of signs, in: SYMSAC, 1976, pp. 272-275
[3] Rouillier, F.; Zimmermann, P., Efficient isolation of polynomial’s real roots, Journal of computational and applied mathematics, 162, 33-50, (2004) · Zbl 1040.65041
[4] Xia, B.C.; Yang, L., An algorithm for isolating the real solutions of semi-algebraic systems, Journal of symbolic computation, 34, 461-477, (2002) · Zbl 1027.68150
[5] Xia, B.C.; Zhang, T., Real solution isolation using interval arithmetic, Computers and mathematics with applications, 52, 853-860, (2006) · Zbl 1131.65041
[6] Wang, R.H., Multivariate spline functions and their applications, (2001), Science Press/Kluwer Pub. Beijing, New York
[7] Wang, R.H.; Lai, Y.S., Real piecewise algebraic variety, Journal of computational mathematics, 21, 473-480, (2003) · Zbl 1080.14547
[8] Wang, R.H.; Zhu, C.G., Piecewise algebraic varieties, Progress in natural science, 14, 568-572, (2004) · Zbl 1083.14526
[9] Y.S. Lai, R.H. Wang, J.M. Wu, Real zeros of the zero-dimensional parametric piecewise algebraic variety, Science in China Series A 52 (2009) (in press) · Zbl 1177.14096
[10] Lai, Y.S., Counting positive solutions for polynomial systems with real coefficients, Computers and mathematics with applications, 56, 6, 1587-1596, (2008) · Zbl 1155.13314
[11] Wang, R.H.; Wu, J.M., Computation of an algebraic variety on a convex polyhedron, Journal of information and computational science, 3, 903-911, (2006)
[12] Cox, D.A.; Little, J.; O’shea, D., Ideals, varieties, and algorithms, (1992), Springer Berlin
[13] Shirayanagi, K.; Sweedler, M., Remarks on automatic algorithm stabilization, J. symb. comput., 26, 761-765, (1998) · Zbl 0917.65130
[14] C. Traverso, A. Zanoni, Numerical stability and stabilization of Gröbner basis computation, in: Proc. ISSAC, 2002, pp. 262-269 · Zbl 1072.68700
[15] Moore, R.E., Interval analysis, (1966), Prentice-Hall Englewood Cliffs · Zbl 0176.13301
[16] Fan, X.C.; Deng, J.S.; Chen, F.L., Zeros of univariate interval polynomials, Journal of computational and applied mathematics, 216, 563-573, (2008) · Zbl 1146.65041
[17] Chen, F.L.; Yang, W., Applications of interval algorithm in solving algebraic equations with wu’s method, Science in China. series A, 35, 910-921, (2005)
[18] Bartlett, A.C.; Hollot, C.V.; Lin, Huang, Root location of an entire polytope of polynomials: it suffices to check the edges, Mathematics of controls, signals and systems, 1, 61-71, (1988) · Zbl 0652.93048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.