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Enhanced Neville Algorithm for Convergence Rate Estimation

Core Concepts
Efficient convergence acceleration using enhanced Neville algorithm.
The content discusses the application of the enhanced Neville algorithm for convergence acceleration in slowly convergent series. It introduces a matrix-based formulation to estimate the limit and subleading terms efficiently. Comparison with other methods like Aitken's ∆2 process and Wynn's epsilon algorithm is made, showing superior performance. Numerical examples are provided for model series and Bethe logarithms calculations. The results showcase high accuracy and efficiency in estimating convergence rates.
Bethe logarithm for hydrogen ground state: ln k0(1S) = 2.98412 85557 65497 61075 97770 90013 79796 99751 80566 17002 (100 decimal digits) Coefficients: c1 = -1, c2 = 0.81830, c3 = -0.80328, c4 = 0.98145 (verified to about 100 digits)

Deeper Inquiries

How can the enhanced Neville algorithm be adapted for different asymptotic structures

The enhanced Neville algorithm, as described in the context provided, can be adapted for different asymptotic structures by adjusting the formulation of the algorithm to suit the specific characteristics of the input series. For instance, if dealing with a series that exhibits inverse half-integer powers of the summation index in its asymptotic limits rather than inverse integer powers as shown in Eq. (2), modifications would need to be made to accommodate this new structure. This adaptation may involve redefining how partial sums are calculated or adjusting the matrix-based formulas used for convergence acceleration based on the new asymptotic behavior.

What are the implications of rational coefficients in Bethe logarithm calculations

The presence of rational coefficients in Bethe logarithm calculations has significant implications for both numerical accuracy and potential analytic representations. Rational coefficients indicate that there is a clear pattern or structure underlying these values, which could potentially lead to finding exact analytical expressions for these quantities using mathematical techniques like continued fractions or symbolic manipulation software. Additionally, rational coefficients provide insights into possible relationships between different terms within the calculation and can guide further investigations into understanding the underlying mathematics governing Bethe logarithms.

How can the PSLQ algorithm aid in finding analytic representations of numerical quantities

The PSLQ (Integer Relation Detection) algorithm can play a crucial role in finding analytic representations of numerical quantities such as those obtained from Bethe logarithm calculations. By analyzing sequences of numbers generated by these computations and searching for linear relations among them, PSLQ can identify hidden algebraic connections that may not be immediately apparent from numerical data alone. This algorithm helps mathematicians uncover patterns and relationships within datasets, leading to discoveries of exact formulas or identities that describe complex mathematical constants or functions accurately.