Showing papers on "Auxiliary function published in 1997"
TL;DR: In this article, two methods for computing the coefficients of the asymptotic series near the transition point are discussed, and auxiliary functions that can be computed more efficiently than the coefficients in the first method, and do not need the tabulation of many coefficients.
Abstract: Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.
32 citations
TL;DR: In this paper, it is shown that the auxiliary functions A are closely related to the Gaussian hypergeometric function F and that A n 21 few simple finite-series expansions are also presented, which seem to be very convenient for numerical computations.
Abstract: The so-called few-body auxiliary functions play a very important role in bound-state calculations for nonrelativistic few-body systems. In particular, the functions of the first, second, and third orders are used in four-body calculations. By means of an original approach developed for the first time in the present study, we have found exact . finite-series representations for the auxiliary functions An F 3 .It is shown that the n auxiliary functions A are closely related to the Gaussian hypergeometric function F. A n 21 few simple finite-series expansions are also presented, which seem to be very convenient for numerical computations. This work opens a new avenue in the study of auxiliary functions and their properties. Q 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 269)278, 1997
19 citations
TL;DR: In this article, the authors studied the initial value problem for the third order Benjamin-Ono equation in the weighted Sobolev spaces and proved its well-posedness in H s = H s T L 2,w here s> 3 ; 0.
Abstract: Here we continue the study of the initial value problem for the third order Benjamin-Ono equation in the weighted Sobolev spaces H s = H s T L 2 ,w here s> 3 ; 0. The result is the proof of well-posedness of the afore mentioned problem in H s ;s >3 ;2 [0; 1]. The proof involves the use of parabolic regularization, the Riesz-Thorin interpolation theorem and the construction technique of auxiliary functions.
7 citations
Journal Article•
TL;DR: In this paper, two methods for computing the coefficients of the asymptotic series near the transition point are discussed, one based on expanding the coefficients in Maclaurin series and the other based on auxiliary functions.
Abstract: Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we don't need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.
2 citations
TL;DR: In this article, a generating function Γ(e) for closed Euler diagrams with constant step is introduced for Hamiltonian circuits in a rectangular lattice of dimensionN×M. The two-dimensional Jordan-Wigner type transformations that were introduced in [7] are also used.
Abstract: In the present study a generating function
$$\bar \Gamma ^{(h)} $$
is considered for Hamiltonian circuits in a rectangular lattice of dimensionN×M. A generating function Γ(e) for closed Euler diagrams (with valence number for the vertices δ=0, 2, 4) with constant step is introduced for this lattice. It is proved that these two generating functions coincide as in the case of the corresponding generating functions relative to a single node (in limit as N, M → ∞). To construct the proof, an auxiliary function that is, in fact, the statistical sum for the two-dimensional Ising model is introduced. The two-dimensional Jordan-Wigner type transformations that were introduced in [7] are also used.
TL;DR: A new algorithm is introduced based on a fast one-dimensional newtonian procedure applied to the objective value of an auxiliary function for solving Formula math, which is an effective tool for solving initial vector coding problems.
Abstract: We consider the problem Formula math. where ∫ 1 ..., ∫ m : R n - R n are (generally nonlinear) differentiable functions, Ω ⊂ R n and n, m can be large. We introduce a new algorithm for solving this problem that can be implemented in rather modest computer environments. The new method is based on a fast one-dimensional newtonian procedure applied to the objective value of an auxiliary function. We report numerical experiments, which suggest that the new algorithm, combined with a powerful strategy for minimization on spheres, can be an effective tool for solving initial vector coding problems.