Improving series convergence: the simple pendulum and beyond.

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.