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Title: Computer-Assisted Proofs and Symbolic Computations
Authors: Krämer, Walter
Keywords: Computer-Assisted Proofs
Symbolic Computations
Self-Validating Numerical Methods
Dynamical System
Verified Periodic Orbit
Issue Date: 2010
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Journal of Computing, Vol. 4, No 1, (2010), 73p-84p
Abstract: We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying theorems fast finite precision arithmetic is used. The results are absolutely rigorous. To demonstrate the power of reliable symbolic-numeric computations we investigate in some details the verification of very long periodic orbits of chaotic dynamical systems. The verification is done directly in Maple, e.g. using the Maple Power Tool intpakX or, more efficiently, using the C++ class library C-XSC.
ISSN: 1312-6555
Appears in Collections:Volume 4 Number 1

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