Topic
Rate of convergence
About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.
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TL;DR: In this paper, the convergence of Pade approximants is studied under two types of assumptions: in the first case the function f to be approximated has to have all its singularities in a compact set E ⊆ C of capacity zero (the function may be multi-valued in C \ E ), and in the second case f has to be analytic in a domain possessing a certain symmetry property (this notion is defined and discussed below).
212 citations
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TL;DR: An algebraic multigrid method is presented which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum and is analytically shown to hold for the model Poisson problem.
Abstract: We consider the iterative solution of large sparse symmetric positive definite linear systems. We present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum. The coarsening is based on the aggregation of the unknowns. A key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two-grid convergence rate is bounded by a user-defined parameter. For a sensible choice of this parameter, it is shown that the recursive use of the two-grid procedure yields a convergence independent of the number of levels, provided that one uses a proper AMLI-cycle. On the other hand, the computational cost per iteration step is of optimal order if the mean aggregate size is large enough. This cannot be guaranteed in all cases but is analytically shown to hold for the model Poisson problem. For more general problems, a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method. The experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations (PDEs). The set includes two- and three-dimensional problems, with both structured and unstructured grids, some of them with local refinement and/or reentering corner, and possible jumps or anisotropies in the PDE coefficients.
212 citations
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TL;DR: In this paper, a real-space ab initio method for electronic structure calculations in terms of nonorthogonal orbitals defined on a grid is proposed, which substantially reduces the computational cost for very large systems.
Abstract: We have formulated and implemented a real-space ab initio method for electronic structure calculations in terms of nonorthogonal orbitals defined on a grid. A multigrid preconditioner is used to improve the steepest descent directions used in the iterative minimization of the energy functional. Unoccupied or partially occupied states are included using a density matrix formalism in the subspace spanned by the nonorthogonal orbitals. The freedom introduced by the nonorthogonal real-space description of the orbitals allows for localization constraints that linearize the cost of the most expensive parts of the calculations, while keeping a fast convergence rate for the iterative minimization with multigrid acceleration. Numerical tests for carbon nanotubes show that very accurate results can be obtained for localization regions with radii of 8 bohr. This approach, which substantially reduces the computational cost for very large systems, has been implemented on the massively parallel Cray T3E computer and tested on carbon nanotubes containing more than 1000 atoms.
211 citations
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TL;DR: In this paper, a sequence of iterative methods improving Newton's method for solving nonlinear equations is presented, and the order of convergence is derived analytically, and then rederived by applying symbolic computation of Maple.
Abstract: In this paper, we present a sequence of iterative methods improving Newton's method for solving nonlinear equations. The Adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses. The orders of convergence are derived analytically, and then rederived by applying symbolic computation of Maple. Some numerical illustrations are given.
210 citations
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TL;DR: The implicit finite difference scheme with the shifted Grunwald formula is employed to discretize fractional diffusion equations and the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant; hence the convergence rate of the proposed iterative algorithm is superlinear.
210 citations