Algorithms for asymptotic extrapolation
نویسنده
چکیده
Consider a power series f ∈ R[[z]], which is obtained by a precise mathematical construction. For instance, f might be the solution to some differential or functional initial value problem or the diagonal of the solution to a partial differential equation. In cases when no suitable method is beforehand for determining the asymptotics of the coefficients fn, but when many such coefficients can be computed with high accuracy, it would be useful if a plausible asymptotic expansion for fn could be guessed automatically. In this paper, we will present a general scheme for the design of such “asymptotic extrapolation algorithms”. Roughly speaking, using discrete differentiation and techniques from automatic asymptotics, we strip off the terms of the asymptotic expansion one by one. The knowledge of more terms of the asymptotic expansion will then allow us to approximate the coefficients in the expansion with high accuracy.
منابع مشابه
Mechanical Quadrature Methods and Their Extrapolations for Solving First Kind Boundary Integral Equations of Anisotropic Darcy-s Equation
The mechanical quadrature methods for solving the boundary integral equations of the anisotropic Darcy’s equations with Dirichlet conditions in smooth domains are presented. By applying the collectively compact theory, we prove the convergence and stability of approximate solutions. The asymptotic expansions for the error show that the methods converge with the order O (h), where h is the mesh ...
متن کاملNonlinear Acceleration of Stochastic Algorithms
Extrapolation methods use the last few iterates of an optimization algorithm to produce a better estimate of the optimum. They were shown to achieve optimal convergence rates in a deterministic setting using simple gradient iterates. Here, we study extrapolation methods in a stochastic setting, where the iterates are produced by either a simple or an accelerated stochastic gradient algorithm. W...
متن کاملAsymptotic analysis of extrapolation boundary conditions for LBM
We present an investigation of extrapolation boundary conditions for lattice Boltzmann method (LBM) using asymptotic analysis. Equilibrium and non-equilibrium extrapolation methods for velocity and pressure boundary conditions proposed in the literature were tested numerically in specific cases. We analyse these boundary conditions using asymptotic expansion techniques and show an improvement i...
متن کاملAsymptotic algorithm for computing the sample variance of interval data
The problem of the sample variance computation for epistemic inter-val-valued data is, in general, NP-hard. Therefore, known efficient algorithms for computing variance require strong restrictions on admissible intervals like the no-subset property or heavy limitations on the number of possible intersections between intervals. A new asymptotic algorithm for computing the upper bound of the samp...
متن کاملSelf-Similar Extrapolation of Asymptotic Series and Forecasting for Time Series
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. It is shown how the extrapolation of asymptotic series can be reformulated as forecasting for time series. The probability measure is introduced characteriz...
متن کامل