# Self-similarly corrected Padé approximants for the indeterminate problem

@article{Gluzman2015SelfsimilarlyCP, title={Self-similarly corrected Pad{\'e} approximants for the indeterminate problem}, author={Simon Gluzman and Vyacheslav I. Yukalov}, journal={The European Physical Journal Plus}, year={2015}, volume={131}, pages={1-21} }

Abstract.A method is suggested for treating the well-known deficiency in the use of Padé approximants that are well suited for approximating rational functions, but confront problems in approximating irrational functions. We develop the approach of self-similarly corrected Padé approximants, making it possible to essentially increase the class of functions treated by these approximants. The method works well even in those cases where the standard Padé approximants are not applicable, resulting…

## 22 Citations

### Critical indices from self-similar root approximants

- Mathematics
- 2017

Abstract.The method of self-similar root approximants has earlier been shown to provide accurate interpolating formulas for functions for which small-variable expansions are given and the behaviour…

### Additive self-similar approximants

- MathematicsJournal of Mathematical Chemistry
- 2016

A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory, allowing for accurate extrapolation and interpolation of asymptotic series. The method is…

### Padé and Post-Padé Approximations for Critical Phenomena

- MathematicsSymmetry
- 2020

We discuss and apply various direct extrapolation methods for calculation of the critical points and indices from the perturbative expansions my means of Padé-techniques and their various post-Padé…

### Modified Padé–Borel Summation

- MathematicsAxioms
- 2023

We revisit the problem of calculating amplitude at infinity for the class of functions with power-law behavior at infinity by means of a resummation procedure based on the truncated series for small…

### Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

- PhysicsInternational Journal of Modern Physics B
- 2020

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even…

### Optimal Mittag-Leffler Summation

- MathematicsAxioms
- 2022

A novel method of an optimal summation is developed that allows for calculating from small-variable asymptotic expansions the characteristic amplitudes for variables tending to infinity. The method…

### Optimized Factor Approximants and Critical Index

- Computer ScienceSymmetry
- 2021

The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations, and based on expansions with only two coefficients and known critical points, a minimal model of critical phenomena is considered.

### Self-similarly corrected Padé approximants for nonlinear equations

- Mathematics, Physics
- 2019

We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Pade approximants t...

### Methods of Retrieving Large-Variable Exponents

- MathematicsSymmetry
- 2022

Methods of determining, from small-variable asymptotic expansions, the characteristic exponents for variables tending to infinity are analyzed. The following methods are considered: diff-log Padé…

## References

SHOWING 1-10 OF 90 REFERENCES

### Convergence Rates of Padé and Padé-Type Approximants

- Mathematics
- 1996

A comparison is made between Pade and Pade-type approximants. LetQnbe thenth orthonormal polynomial with respect to a positive measure?with compact support inC. We show that for functions of the…

### Self-similar factor approximants.

- Mathematics, Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions, which include a variety of nonalgebraic functions, and provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Padé approximant.

### Padé approximants and efficient analytic continuation of a power series

- Mathematics
- 2002

This survey reflects the current state of the theory of Pade approximants, that is, best rational approximations of power series. The main focus is on the so-called inverse problems of this theory,…

### Extrapolation of perturbation-theory expansions by self-similar approximants

- MathematicsEuropean Journal of Applied Mathematics
- 2014

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on…

### EXTRAPOLATION OF POWER SERIES BY SELF-SIMILAR FACTOR AND ROOT APPROXIMANTS

- Physics
- 2004

The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of…

### Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: Application to anharmonic oscillators

- Mathematics
- 1998

The divergent Rayleigh-Schrodinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are…

### Method of self‐similar approximations

- Physics
- 1991

A new method is suggested to approximately find out the limit for a sequence of functions when solely several first terms of a sequence are known. The method seems to be very useful for those…

### Pade and Hermite-Pade approximation and orthogonality

- Mathematics
- 2006

We give a short introduction to Pade approximation (rational approximation to a function with close contact at one point) and to Hermite-Pade approximation (simultaneous rational approximation to…