Generating Very-High-Precision Frobenius Series with Apriori Estimates of Coefficients

The Frobenius method can be used to compute solutions of ordinary linear differential equations by generalized power series. Each series converges in a circle which at least extends to the nearest singular point; hence exponentially fast inside the circle. This makes this method well suited for very-high-precision solutions of such equations. It is useful for this purpose to have prior knowledge of the behaviour of the series. We show that the magnitude of its coefficients can be apriori predicted to surprisingly high accuracy, employing a Legendre transformation of the WKB approximated solutions of the equation.

💡 Research Summary

The paper revisits the classical Frobenius method for solving linear ordinary differential equations (ODEs) and demonstrates how it can be turned into a highly efficient tool for computing solutions with thousands of decimal digits of precision. A Frobenius series expands a solution around a regular singular point as

  y(x)=∑ₙ aₙ (x−x₀)^{n+ρ},

where ρ is the indicial exponent and a₀≠0. Because the radius of convergence extends at least to the nearest other singularity, the series converges exponentially fast inside that disc. This rapid convergence makes the method attractive for high‑precision work, but practical implementation is hampered by the fact that the coefficients aₙ often grow or decay extremely fast, leading to overflow, underflow, or loss of significant digits when naïvely computed with floating‑point arithmetic.

The authors’ central contribution is a systematic way to predict the magnitude of aₙ before any numerical evaluation. They start from a WKB (Wentzel–Kramers–Brillouin) approximation of the original ODE, which yields an asymptotic solution of the form

  ψ(x)≈exp