Applied Mathematics and Computation
We introduce an interval Newton method for bounding solutions of systems of nonlinear equations. It entails three subalgorithms. The first is a Gauss-Seidel-type step. The second is a real (noninterval) Newton iteration. The third solves the linearized equations by elimination. We explain why each subalgorithm is desirable and how they fit together to provide solutions in as little as one-third or one-quarter the time required by Krawczyk's method  in our implementations.
Hansen, E R. and Greenberg, R I.. An Interval Newton Method. Applied Mathematics and Computation, 12, 2-3: 89-98, 1983. Retrieved from Loyola eCommons, Computer Science: Faculty Publications and Other Works, http://dx.doi.org/10.1016/0096-3003(83)90001-2
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Copyright © 1983 Published by Elsevier Inc.
An Interval Arithmetic Newton Method for Solving Systems of Nonlinear Equations" presented by Ronald I. Greenberg at the Illinois State Academy of Science Annual Meeting, Decatur, IL, April 1982