An analytical method for approximating high-order Galerkin solutions

The high-order Galerkin procedures for computing forced oscillations of nonlinear systems are generally impractical to apply analytically. In this paper, an analytical method for approximating high-order Galerkin solutions is presented, which is applicable to ordinary differential equations with polynomial nonlinearities. The procedure is restricted to oscillations whose predominant Fourier component is the fundamental; hence subharmonic oscillations may be allowed, whereas superharmonic oscillations are excluded. The approach taken is to consider only the first-order effects of the higher harmonics. An example is given which demonstrates that the accuracy of the method can be quite impressive.     

Optimal a priori error bounds for the Rayleigh-Ritz method

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.

     

The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation

In this paper, the homotopy analysis method for solving the nonlinear modified Korteweg-de Vries equation is implemented with approximate initial conditions. We discuss the case when the problem has solitons or breathers. Some numerical examples are presented.     

Convergence conditions for a restarted GMRES method augmented with eigenspaces

We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A‐invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

An a posteriori error estimation of the Rayleigh-Ritz eigenvalues

Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL ₂(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=Bu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.     

A regularization method for approximating the inverse Laplace transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum     

An elementary algebraic method for approximating average radii of first and second ring electrons

In this paper, experimental ionization energies are used to determine algebraic equations whose solutions approximate relativistic quantum mechanical estimates of average atomic radii. These algebraic equations are nonlinear and are solved on a digital computer by Newton's method. Pairing is an essential element in the formulation.     

A finite element method for approximating the time-harmonic Maxwell equations

Summary In this paper we study the use of Nédélec's curl conforming finite elements to approximate the time-harmonic Maxwell equations on a bounded domain. The analysis is complicated by the fact that the bilinear form is not coercive, and the principle part has a very large null-space. This difficulty is circumvented by using a discrete Helmholtz decomposition of the error vector. Numerical results are presented that compare two different linear elements.Research supported in part by grants from AFOSR and NSF     

An algorithm for approximating the singular triplets of complex symmetric matrices

We present an algorithm for the approximation of the dominant singular values and corresponding right and left singular vectors of a complex symmetric matrix. The method is based on two short-term recurrences first proposed by Saunders, Simon and Yip [24] for a non-Hermitian linear system solver. With symmetric matrices, the recurrence can be modified so as to generate a tridiagonal symmetric matrix from which the original triplets can be approximated. The recurrence formally resembles the Lanczos method, in spite of substantial differences which make usual convergence results inapplicable. Implementation aspects are discussed, such as re-orthogonalization and the use of alternative representation matrices. The method is very efficient over existing approaches which do not exploit the symmetry of the problem. Numerical experiments on application problems validate the analysis, while showing satisfactory results, especially on dense matrices. © 1997 by John Wiley & Sons, Ltd.     

The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

We extend the Rayleigh-Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh-Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh-Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.     

A domain decomposition method for approximating the conformal modules of long quadrilaterals

Summary This paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals. The method has been studied already by us and also by D. Gaier and W.K. Hayman, but only in connection with a special class of quadrilaterals, viz. quadrilaterals where: (a) the defining domain is bounded by two parallel straight lines and two Jordan arcs, and (b) the four specified boundary points are the four corners where the arcs meet the straight lines.Our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above.     

An asymptotically optimal algorithm for approximating bounded analytic functions

Summary. Let be the unit disk in the complex plane and let be a compact, simply connected subset of , whose boundary is assumed to belong to the class . Let be the unit ball of the Hardy space . A linear algorithm is constructed for approximating functions in . The algorithm is based on sampling functions in the Fejer points of and it produces the error Here denotes the space of continuous functions on and is the Green capacity of with respect to . Moreover it is shown that the algorithm is asymptotically optimal in the sense of -widths. Received July 7, 1994     

An approximating polynomial algorithm for a sequence partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. We assume that the cardinalities of the clusters are fixed. The center of one cluster has to be optimized and is defined as the average value over all vectors in this cluster. The center of the other cluster lies at the origin. The partition satisfies the condition: the difference of the indices of the next and previous vectors in the first cluster is bounded above and below by two given constants. We propose a 2-approximation polynomial algorithm to solve this problem.     

A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix

A fast method for computing all the eigenvalues of a Hamiltonian matrix M is given. The method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties. The algorithm requires about one-fourth the number of floating-point operations and one-half the space of the standard QR algorithm. The computed eigenvalues are shown to be the exact eigenvalues of a matrix M + E where ∥E∥ depends on the square root of the machine precision. The accuracy of a computed eigenvalue depends on both its condition and its magnitude, larger eigenvalues typically being more accurate.