Certification Using Newton-Invariant Subspaces
نویسنده
چکیده
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 Newtoninvariant subspace. Starting with any point for which Newton’s method quadratically converges to a solution, this article uses Smale’s α-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 of 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.
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