Certification Using Newton-Invariant Subspaces
- 384 Downloads
For a square system of analytic equations, a Newton-invariant subspace is a set which contains the resulting point of a Newton iteration applied to each point in the subspace. For example, if the equations have real coefficients, then the set of real points form a Newton-invariant subspace. Starting with any point for which Newton’s method quadratically converges to a solution, this article uses Smale’s \(\alpha \)-theory to certifiably determine if the corresponding solution lies in a given Newton-invariant subspace or its complement. This approach generalizes the method developed in collaboration with F. Sottile for deciding the reality of the solution in the special case that the Newton iteration defines a real map. A description of the implementation in alphaCertified is presented along with examples.
KeywordsNewton’s method Certified solutions Alpha theory Real solutions Numerical algebraic geometry
A.M.S. 2010 Mathematics Subject ClassificationPrimary 65G20 Secondary 65H10 14Q99
The author would like to thank Charles Wampler for helpful discussions related to using isotropic coordinates in kinematics.
- 1.Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for Numerical Algebraic Geometry. bertini.nd.edu
- 10.Hauenstein J.D., Sottile, F.: alphaCertified: software for certifying solutions to polynomial systems. www.math.tamu.edu/~sottile/research/stories/alphaCertified
- 17.Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R.E., Gross, K.I., Martin, C.F. (eds.) The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Laramie, Wyo 1985), pp. 185–196. Springer, New York (1986). https://doi.org/10.1007/978-1-4612-4984-9_13 CrossRefGoogle Scholar