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Self-similar factor approximants.

S Gluzman1, V I Yukalov, D Sornette

  • 1Institute of Geophysics and Planetary Physics, University of California-Los Angeles, Los Angeles, CA 90095, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 15, 2003
PubMed
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New self-similar factor approximants accurately reconstruct functions from asymptotic expansions. These novel methods outperform traditional techniques like Padé approximants for complex function approximation.

Area of Science:

  • Numerical Analysis
  • Applied Mathematics
  • Theoretical Physics

Background:

  • Reconstructing functions from asymptotic expansions is a challenging problem in applied mathematics.
  • Existing methods, such as Padé approximants, have limitations in accuracy and scope.
  • Self-similar approximation theory offers a novel framework for developing improved approximants.

Purpose of the Study:

  • To derive a new class of approximants, termed self-similar factor approximants.
  • To enhance the accuracy and applicability of function reconstruction from asymptotic expansions.
  • To demonstrate the superiority of these new approximants over existing methods.

Main Methods:

  • The derivation is based on self-similar approximation theory, treating approximant generation as a dynamical system.

Related Experiment Videos

  • Control functions for convergence are determined by accuracy-through-order conditions.
  • The new approximants generalize Padé approximants and relate to previously developed self-similar exponential and root approximants.

    • Main Results:

    • Self-similar factor approximants can exactly reproduce a wide range of functions, including non-algebraic types.
    • For other functions, they provide highly accurate approximations, surpassing the accuracy of Padé approximants.
    • The method demonstrates generality and accuracy, even for problems where conventional techniques struggle.

    Conclusions:

    • Self-similar factor approximants represent a significant advancement in function reconstruction from asymptotic expansions.
    • These approximants offer superior accuracy and broader applicability compared to existing methods.
    • The self-similar approximation framework provides a powerful tool for developing advanced numerical techniques.