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A simple pendulum consists of a small diameter ball suspended from a string, which has negligible mass but is strong enough to not stretch. In our daily life, pendulums have many uses, such as in clocks, on a swing set, and on a sinker on a fishing line. 
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Improving series convergence: the simple pendulum and beyond.

Solomon F Duki1, T P Doerr1, Yi-Kuo Yu1

  • 1National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, Bethesda, MD 20894, United States of America.

European Journal of Physics
|September 20, 2019
PubMed
Summary
This summary is machine-generated.

Improving power series convergence is achieved by centering expansions near the average evaluation value. This straightforward method enhances computational efficiency for series approximations.

Keywords:
numerical methodsseries convergencesimple pendulum

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Area of Science:

  • Computational Physics
  • Applied Mathematics

Background:

  • Power series expansions are fundamental in physics and mathematics.
  • Choosing an expansion center is crucial for convergence and computational efficiency.
  • Traditional methods often use analytically convenient centers, not always optimal for computation.

Purpose of the Study:

  • To present a simple and effective method for improving power series convergence.
  • To demonstrate that the optimal center for series expansion is not always the most analytically convenient.
  • To enhance the computational efficiency of power series approximations.

Main Methods:

  • A novel method is proposed for selecting the center of a power series expansion.
  • The optimal center is identified as being at or near the average value of the series evaluation.
  • The method's applicability is illustrated using the simple pendulum and Mexican hat potential.

Main Results:

  • Significant improvements in series convergence rates were observed.
  • Large performance gains were demonstrated in computational efficiency.
  • The method proved effective and straightforward to implement.

Conclusions:

  • Centering power series expansions near the average evaluation value dramatically improves convergence.
  • This technique is general, effective, and can be combined with other numerical methods.
  • The presented method offers a practical approach to enhance series approximation accuracy and speed.