Pade approximant method

The difficulty encountered in evaluating these integrals originates primarily from the round-off errors caused by a finite number of bits available (64 bits, 128 bits, etc.) in the computer work. I call it the digit-deficiency (or d-d) problem. Although this was the suspected cause from the beginning, it was clearly identified by comparing our numerical work [57] with the result obtained by the Pade approximant method [56],... 

Numerical calculations using Kapuy s partitioning scheme have shown that for covalent systems the role of one-particle localization corrections in many-body perturbation theory is extremely important. For good quality results several orders of one-particle perturbations have to be taken into account, although the additional computational power requirement is much less in these cases than for the two-electron perturbative corrections. Another alternative for increasing the precision of the calculations is to estimate of the asymptotic behavior of the double power series expansion (24) from the first few terms by applying Canterbury approximants [31], which is a two-variable generalization of the well-known Pade approximation method. It has also been found [6, 7] that in more metallic-like systems the relative importance of the localization corrections decreases, at least in PPP approximation. 

Another way of investigating stability, that at the same time provides information on the behaviour of a given method, is what Gourlay and Morris [277] call the symbol of the algorithm, also called the symbol of the method [514] or, more logically perhaps, the stability function [286]. It is developed from Pade approximations to the general solution of the diffusion equation. Equation (14.6) can be semidiscretised to the system of odes as... 

The Lanczos algorithm and the Pade approximant have also been combined in other research fields, for example, in computational and applied mathematics [2], as well as in engineering via circuit theory [59]. The authors of Refs. [59, 60] from 1995, being apparently unaware of the earlier work [48] from 1972, rederived through a different procedure the Pad6-Lanczos approximant and called it the Pad6 via Lanczos (PVL) method. 

Pj)(z )/Q (z ) is the basis of the error analysis of proven validity in the PA. Pade approximants can be computed through many different numerical algorithms, including the most stable numerical computations via continued fractions. Moreover, unlike any other related method, for the known G(z ), both Pade polynomials Pj)(z ) and Q (z ) in the PA can be extracted by purely analytical means in their simple and concise closed forms [1]. This represents the gold standard against which all the corresponding numerical algorithms should be benchmarked for their stability and robustness. 

Later Vosko, Wilk and Nusair (VWN) [32] proposed a correlation functional that was obtained using Pade approximant interpolations of very accurate numerical calculations made by Ceperley and Alder, who used a quantum Monte Carlo method [33], The VWN correlation functional is,... 

Of course, it is also possible to rely on more sophisticated methods and, in order to calculate the exponents y and v associated with chains drawn on various lattices, Watts11 used Pade approximants. However, this method does not lead to spectacular improvement in the results. Actually, approximations of the Pade type are excellent when one tries to represent smooth analytic functions, but here this is not the case. The various functions of N which are to be interpolated or extrapolated, contain oscillating terms related to the lattice structure, and the presence of such terms, spoils the precision of the results. 

The computed complex energy eigenvalues E = F, — T/2 for different states ni, ri2, m) of a hydrogen atom are presented in Table 2, where (1/n) denotes the results of the 1/n-expansion and HPA marks the values obtained by perturbation series summation with the help of Hermite-Pade approximations [6]. The agreement between different computational methods is quite good. 

In [170] the authors discussed the numerical solution of Ordinary Differential Equations (ODEs) by using two approached the well known BDF formulae and the Piecewise-Linearized Methods. In the case of BDF method a Chord-Shamanskii iteration procedure is used for computing the nonlinear system which is produced when the BDF formula is applied. In the case of Piecewise-Linearized Methods the computation of the numerical solution at each time step is obtained using a block-oriented method based on diagonal Pade approximation. 

Malvandi A, Ganji DD (2013) A general mathematical expression of amperometric enzyme kinetics using He s variationtil iteration method with Pade approximation. J Electroaneil Chem 711 32-37... 

The Pad6 approximant theory arose more than 90 years ago. The Pade theory is a powerful method to approximate various functions, and is successfully applied in many fields of theoretical physics and chemistry. The reason for such a success resides in, first, its more rapid convergency in comparison with the corresponding Taylor series (Baker, 1965, 1975 Baker and Gammel, 1970 Graves-Morris, 1973a,b). 

It is obvious that an initial sum can be resummed in different ways. Apart from the Leroy fit parameter p mentioned above some arbitrariness arises from the different types of rational approximants one may construct. For instance, within the two-loop approximation the method of Pade-Borel resummation of a resolvent series can be done using either the [0/2] or the [1/1] approximants. The Chisholm-Borel approximation implies even more arbitrariness and demands a careful analysis of the approximants to be chosen. 

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