Theory of linear and integer programming
Theory of linear and integer programming
Curves and surfaces for computer aided geometric design: a practical guide
Curves and surfaces for computer aided geometric design: a practical guide
Comparison of interval methods for plotting algebraic curves
Computer Aided Geometric Design
How good are interior point methods? Klee–Minty cubes tighten iteration-complexity bounds
Mathematical Programming: Series A and B
Nonlinear systems solver in floating-point arithmetic using LP reduction
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
| Hi-index | 0.00 |
Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate several polytopes based on the tensorial Bernstein basis, and we formulate a polytope for the quadratic patch Qn:= (x1, ..., xn, x21, ..., x2n, x1x2, ..., xn-1xn) by projections. This Bernstein polytope has Θ(n2) hyperplanes. We give the number of vertices, the number of hyperplanes, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.