Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 52, Issue 4, Pages 1551-1572Publisher
SIAM PUBLICATIONS
DOI: 10.1137/130932387
Keywords
bivariate; polyanalytic; polynomial system
Categories
Funding
- Flanders Agency for Innovation by Science and Technology (IWT)
- Research Council KU Leuven, Optimization in Engineering (OPTEC) [OT/10/038, PF/10/002]
- Research Foundation Flanders (FWO) [G.0828.14N, G.0427.10N, G.0830.14N, G.0881.14N]
- Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) - Interuniversity Attraction Poles Programme
- Research Council KU Leuven, GOA-MaNet, Optimization in Engineering (OPTEC) [CoE EF/05/006, STRT1/08/023, CIF1]
- Belgian Network DYSCO [IUAP P7/19]
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Finding the real solutions of a bivariate polynomial system is a central problem in robotics, computer modeling and graphics, computational geometry, and numerical optimization. We propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition. In contrast to existing eigen-based solvers, the proposed algorithm does not depend on Grobner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. The method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables. Solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and Newton-Raphson refinement. Numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the same time significantly faster and more accurate than competing double precision solvers.
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