Pade producing approximation

It is at this point, that the various Pade approximations to the exponential functions come in. One of them is simply the sequence on the right-hand side of (14.40) cut off after the first two terms, producing the propagation equation... 

Examination of the terms to O(k ) in the SL expansion for the free energy show that the convergence is extremely slow for a RPM 2-2 electrolyte in aqueous solution at room temperature. Nevertheless, the series can be summed using a Pade approximant similar to that for dipolar fluids which gives results that are comparable in accuracy to the MS approximation as shown in figure A2.3.19(a). However, unlike the DHLL + i 2 approximation, neither of these approximations produces the negative deviations in the osmotic and activity coefficients from the DHLL observed for higher valence electrolytes at low concentrations. This can be traced to the absence of the complete renormalized second virial coefficient in these theories it is present... 

The apparent garbage produced by the perturbational series for R = 3.0 a.u. represented for the Padd approximants precise information that the absurd perturbational corrections pertain the energy of the... 2pcr state of the hydrogen atom in the electric field of the proton. Why does this happen Visibly low-order perturbational corrections, even if absolutely crazy, somehow carry information about the physics of the problem. The convergence properties of the Rayleigh-Schrddinger perturbation theory depend critically on the poles of the function approximated (see the discussion on p. 250). A pole cannot be described by any power series (as happens in perturbation theories), whereas the Pade approximants have poles built in the very essence of their construction (the denominator as a polynomial). This is why they may fit so well to the nature of the problems under study. 

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. 

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