Summation of the eigenvalue perturbation series by multi-valued Pade approximants: application to resonance problems and double wells

Self-similar structures and fractal transforms in approximation theory

“…There exist the so-called resummation methods ascribing finite values to divergent series [63]. The most often used among such techniques are the Borel summation [63] and the construction of Padé approximants [64], including the two-point [65] and multivalued [66,67] Padé approximants. These techniques have many known limitations.…”

Improving the accuracy of ab initio methods with summation approximants and singularity analysis

“…The higher‐degree approximants appear to have been left largely unexplored until over 80 years later, when Shafer 16 rediscovered the quadratic approximants and used them to sum Taylor series for some simple multiple valued functions. Since then, the mathematical properties of algebraic approximants have been explored in more detail 17–20 and applications have been found for them in various areas of atomic and molecular physics 21–27.…”

Asymptotic response of observables from divergent weak-coupling expansions: A fractional-calculus-assisted Padé technique

“…(v) what we achieve through (19) by using K = 10 is also much better (error reduced by about a factor of 2-3) than even the K = 50 case of the parent strategy (12). This last point is the most striking one.…”

“…This is particularly due to the strongest divergence [see (7)] of the series concerned among all those studied here, and a clear reflection of it is the extremely poor result of the parent sequence, as found in Table III. We summarize all our findings about β 0 in Table IV for the three cases of M with varying K values, viz., K = 10, 30, and 50. A few points of interest are the following: (i) There is a significant reduction in as we switch over to {S N 3 } from {S N 1 }; (ii) both the absolute percentage errors |ε| and (subscript zero now refers to the parent strategy) gradually decrease with increasing K, though the extent of reduction (12)] and the FC-assisted scheme [see (19)] for the case of quartic anharmonic oscillator (M = 2) with K = 50. The error attains a minimum at the quoted g value.…”

“…A few effective variants of the PA in estimating values of F (x) at large x have recently been put forward [9] where major references to earlier works may be found. We also tried to get reasonable estimates of β 0 [10] and α 0 [11] earlier by using specific variants of the PA. A different sort of approach is to employ multivalued algebraic approximants [12] that are constructed in the spirit of the PA, but take due care of multivaluedness of the function F (x) in the complex x plane. Needless to mention, such complexities are reflected in the observed rapid divergence of concerned series expansions.…”