On the numerical condition of polynomials in Berstein form
Computer Aided Geometric Design
Calculation of pseudospectra by the Arnoldi iteration
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
On the optimal stability of the Bernstein basis
Mathematics of Computation
Matrix computations (3rd ed.)
Complexity and real computation
Complexity and real computation
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
On ray tracing parametric surfaces
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Condition-Based Complexity of Convex Optimization in Conic Linear Form via the Ellipsoid Algorithm
SIAM Journal on Optimization
Global Optimization on Funneling Landscapes
Journal of Global Optimization
A Condition Number Analysis of a Line-Surface Intersection Algorithm
SIAM Journal on Scientific Computing
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Properties of polynomial bases used in a line-surface intersection algorithm
PPAM'09 Proceedings of the 8th international conference on Parallel processing and applied mathematics: Part I
Backward error analysis in computational geometry
ICCSA'06 Proceedings of the 6th international conference on Computational Science and Its Applications - Volume Part I
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This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate cases, in which case the solution is a 1-dimensional curve. Our first main contribution is a definition of a condition number measuring reciprocal distance to degeneracy that can distinguish poor and well-conditioned instances of this problem. (Degenerate problems would be infinitely ill conditioned in our framework.) Our second contribution, which is the main novelty of our algorithm, is an analysis showing that its running time is bounded in terms of the condition number of the problem instance as well as $n$ and the polynomial degrees.