Related papers: Critical indices from self-similar root approximan…
The method of extrapolating asymptotic series, based on the Self-Similar Approximation Theory, is developed. Several important questions are answered, which makes the foundation of the method unambiguous and its application straightforward.…
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 Pad\'e summation, self-similar factor…
Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly…
A new approach to the problem of finding the asymptotical behaviour of large orders of semiclassical expansion is suggested. Asymptotics of high orders not only for eigenvalues, but also for eigenfunctions, are constructed. Thus, one can…
A method is described for the extrapolation of perturbative expansions in powers of asymptotically small coupling parameters or other variables onto the region of finite variables and even to the variables tending to infinity. The method…
We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general…
In the paper we study properties of the set of critical points for self-similar sets. We introduce simple condition that implies at most countably many critical values and we construct a self-similar set with uncountable set of critical…
A method is suggested for treating the well-known deficiency in the use of Pade approximants that are well suited for approximating rational functions, but confront problems in approximating irrational functions. We develop the approach of…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
We consider the method of self-similar renormalization for calculating critical temperatures and critical indices. A new optimized variant of the method for an effective summation of asymptotic series is suggested and illustrated by several…
The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
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 Pad\'e approximants that are known to provide the best…
An approach is suggested defining effective sums of divergent series in the form of self-similar exponential approximants. The procedure of constructing these approximants from divergent series with arbitrary noninteger powers is developed.…
Self-similar sequence transformation is an original type of nonlinear sequence transformations allowing for defining effective limits of asymptotic sequences. The method of self-similar factor transformations is shown to be regular. This…
Asymptotic expansions are derived for the tail distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances, and more generally the sum of independent copies of such random variables.…
In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…
Virial expansions are the series in powers of density assumed to be small. However, the equations of state require to consider finite densities for which virial expansions, as a rule, diverge. In order to extrapolate a virial expansion to…
Selection of descent direction at a point plays an important role in numerical optimization for minimizing a real valued function. In this article, a descent sequence is generated for the functions with bounded parameters to obtain a…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…