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Verified Computations Using Taylor Models and Their Applications


Numerical methods assuring confidence involve the treatment of entire sets instead of mere point evaluations. We briefly review the method of interval arithmetic that is long known for rigorous, verified computations, and all operations are conducted on intervals instead of numbers. However, interval computations suffer from overestimation, the dependency problem, the dimensionality curse, and the wrapping effect, to name a few, and those difficulties often make conventional interval based verified computational methods useless for practical challenging problems.

The method of Taylor models combines Taylor polynomials and remainder error enclosures, and operations are now conducted on Taylor models, where the bulk amount of the functional dependency is carried in the polynomial part, and the error enclosures provides a safety net to rigorously guarantee the result. Using simple and yet challenging benchmark problems, we demonstrate how the method works to bring those conventional difficulties under control. In the process, we also illustrate some ideas that lead to several Taylor model based algorithms and applications.

K. Makino, M. Berz, Lecture Notes in Computer Science 10381 (2017) 3-13


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