Extended procedures for extrapolation to the limit

In this paper, new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed. They are based on the notions of error estimates and annihilation operators. We obtain generalizations of the discrete and confluent E-transformation, which are the most general sequence and function transformations known so far. It is shown that many transformations studied in the literature are included in our formalism. Particular cases of these procedures are discussed. Then, several extensions of the E-algorithm are given. Finally, the procedure Θ is applied to our procedures to produce new procedures for extrapolation to the limit.     

Bessel transforms and rational extrapolation

Summary A numerical method is developed which handles the Bessel transform of functions having slow rates of decrease. The method replaces the Bessel transform by a related damped transform for which the sinc quadrature rule provides an efficient and accurate approximation. It is then shown that the value of the original Bessel transform can be obtained from the damped transform by extrapolation with the Thiele algorithm.     

Addition to the Aitken method for the extrapolation of the limit of slowly convergent sequences

The class of sequences and series in which the Aitken process accelerates the convergence is considerably extended. It is proved that a proper subsequence of a slowly convergent sequence satisfies the sufficient condition for accelerating the convergence of the Aitken transformation. Two numerical examples illustrate the highly accurate limit extrapolation.     

A new approach to vector-valued rational interpolation

In this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where , and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived.     

A new formal approach to the rational interpolation problem

Summary An elegant and fast recursive algorithm is developed to solve the rational interpolation problem in a complementary way compared to existing methods. We allow confluent interpolation points, poles, and infinity as one of the interpolation points. Not only one specific solution is given but a nice parametrization of all solutions. We also give a linear algebra interpretation of the problem showing that our algorithm can also be used to handle a specific class of structured matrices.     

A new rational approximation technique based on transformational high dimensional model representation

In this work, a new rational approximation scheme, based on the recently developed Transformational High Dimensional Model Representation (THDMR) approximation method is developed. As an initial step to the construction of a rational approximation for multivariate functions via THDMR, this paper focuses on the general theoretical background of the method and gives explicit formulae for the computation of such approximants. The performance of the technique is shown by several examples both in univariate and bivariate cases.     

A new algorithm for osculatory rational interpolation

Summary A new algorithm is derived for computing continued fractions whose convergents form the elements of the osculatory rational interpolation table.     

A look-ahead strategy for the implementation of some old and new extrapolation methods

Sequence transformations, used for accelerating the convergence, are related to biorthogonal polynomials. In the particular cases of theG-transformation and the Shanks transformation (that is the -algorithm of Wynn), there is a connection with formal orthogonal polynomials. In this paper, this connection is exploited in order to propose a look-ahead strategy for the implementation of these two transformations. This strategy, which is quite similar to the strategy used for treating the same type of problems in Lanczos-based methods for solving systems of linear equations, consists in jumping over the polynomials which do not exist, thus avoiding a division by zero (breakdown) in the algorithms, and over those which could be badly computed (near-breakdown) thus leading to a better numerical stability. Numerical examples illustrate the procedure.     

On the technique for passing to the limit in nonlinear elliptic equations

We consider the problem of passing to the limit in a sequence of nonlinear elliptic problems. The “limit” equation is known in advance, but it has a nonclassical structure; namely, it contains the p-Laplacian with variable exponent p = p(x). Such equations typically exhibit a special kind of nonuniqueness, known as the Lavrent’ev effect, and this is what makes passing to the limit nontrivial. Equations involving the p(x)-Laplacian occur in many problems of mathematical physics. Some applications are included in the present paper. In particular, we suggest an approach to the solvability analysis of a well-known coupled system in non-Newtonian hydrodynamics (“stationary thermo-rheological viscous flows”) without resorting to any smallness conditions.     

Explicit extrapolation of stationary processes with generalized rational spectral density

Translated from Issledovaniya po Prikladnoi Matematike, No. 7, pp. 84–94, 1979.     

Linear extrapolation by rational functions,exponentials and logarithmic functions

In this paper linear extrapolation by rational functions with given poles is considered from an arithmetical point of view. It is shown that the classical interpolation algorithms of Lagrange, Neville-Aitken and Newton which are well known for polynomial interpolation can be extended in a natural way to this problem yielding recursive methods of nearly the same complexity. The proofs are based upon explicit representations of generalized Vandermonde-determinants which are calculated by the elimination method combined with analytical considerations. As an application a regularity criterion for certain linear sequence-transformations is given. Also, by the same method simplified recurrence relations for linear extrapolation by exponentials and logarithmic functions at special knots are derived.     

A new approach for solving repair limit problems

The usual approach to finding optimal repair limits on failure of a component is to use a finite state approximation Markov Decision Process (MDP). In this paper an alternative approach is introduced. Assuming a stochastically increasing repair cost, the optimum solution is shown to satisfy a certain two-point boundary condition, first order differential equation. An asymptotic formula for the optimal repair limit function is derived. Numerical solutions are obtained for some Weibull and Special Erlang distributed time to failure distributions. The structural form of the repair limit function results in a solution procedure which is several orders of magnitude faster than is achievable using previous methods.     

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.     

A new level-set technique for the crack-detection problem

We present a new technique for recovering key characteristics of defects such as snake-like cracks from boundary electrical measurements. We propose a shape-based reconstruction algorithm adapting the level set techniques to the situation of modelling very thin shapes. Two level set functions are employed: the first one models the location and form of the crack, and the second one models its length and connectivity. Two Hamilton-Jacobi type equations are derived to describe the evolution laws for these two level set functions in order to minimize the least squares data misfit. Numerical experiments in 2D show the efficiency of this method for reconstructing disconnected cracks in the presence of measurement noise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new technique for avoiding the Maratos effect

A well-known difficulty arising in the convergence globalization of Newton-type constrained optimization methods is the Maratos effect, which prevents these methods from achieving a superlinear convergence rate and, in many cases, reduces their general efficiency. For the sequential quadratic programming method with linesearch, a new simple and rather promising technique is proposed to avoid the Maratos effect.