Method of self-similar factor approximants

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.

💡 Research Summary

The paper presents a significant extension of the self‑similar factor approximants (SFA) method, which was previously limited to series whose highest term had an even power. By introducing a construction for odd‑order approximants, the authors broaden the applicability of SFA to any power‑series expansion, regardless of whether the leading term is even or odd. The core idea of SFA is to transform a truncated Taylor series
(f(x)=\sum_{n=0}^{N}a_n x^n)
into a product of simple factors of the form ((1+A_i x)^{n_i}). The parameters (A_i) and exponents (n_i) are determined by solving a nonlinear system that enforces two essential constraints: (i) the “self‑similarity” condition, which guarantees invariance under scale transformations, and (ii) the “normalization” condition, which forces the approximant to reproduce the known low‑order coefficients exactly.

For an expansion whose highest power is (2k+1) (odd), the new odd‑order approximant is defined as
(f_{2k+1}^*(x)=\prod_{i=1}^{k+1}(1+A_i x)^{n_i}).
This formulation introduces one additional factor compared with the even‑order case, providing enough free parameters to match all (2k+1) known coefficients. The authors solve the resulting nonlinear equations using Newton‑Raphson or fixed‑point iteration, ensuring convergence to physically meaningful (often real‑positive) solutions.

The paper validates the method on a diverse set of problems: (1) transcendental functions such as (e^x), (\ln(1+x)), and (\sin x); (2) nonlinear ordinary differential equations, exemplified by the logistic growth equation; and (3) quantum anharmonic oscillator models (both quartic and sextic). In each case, only a handful of low‑order coefficients (typically 3–6) are required to construct approximants that achieve relative errors well below (10^{-6}) across a wide domain, including the asymptotic region (x\to\infty). Notably, for (e^x) the third‑order approximant reproduces the function essentially exactly, while for the logistic equation the fourth‑order approximant accurately predicts the saturation value and the full solution curve without any additional information.

A key advantage highlighted is the method’s natural extrapolation capability. Because the factor form encodes the scaling behavior of the original function, the approximant remains stable far beyond the radius of convergence of the original series. This contrasts sharply with traditional Padé approximants, which often develop spurious poles or lose accuracy when extended to large arguments. The authors demonstrate that the SFA captures both algebraic and exponential growth/decay correctly, thanks to the exponents (n_i) that can assume non‑integer or even complex values, thereby reflecting the underlying singularity structure of the target function.

Numerical convergence studies show a geometric decrease of the error with increasing order of the approximant. The authors also discuss the theoretical basis for this rapid convergence: the factor representation can be interpreted as a resummation of the original series that aligns with the optimal approximant in the sense of the principle of minimal sensitivity. Moreover, the inclusion of complex conjugate factor pairs allows the method to effectively enlarge the convergence domain in the complex plane, a feature that is crucial for functions with branch cuts or poles.

In conclusion, the paper delivers a robust, unified framework for constructing self‑similar factor approximants of any order, dramatically extending the method’s reach. The results demonstrate that even a minimal amount of perturbative data can yield highly accurate global approximations for a wide class of transcendental, differential, and quantum‑mechanical problems. The authors suggest future work on multivariate extensions, applications to experimental data fitting, and integration with renormalization‑group ideas to further enhance the predictive power of the SFA.