Rational function approximation

Often the enhancement factor is written as a rational function this is motivated by the observation that the gradient expansion diverges and by the knowledge that Pade approximants are often very effective approaches for resummation of divergent series. A representative functional of this form is the Depristo-Kress functional  

Modern Theoretical Chemistry Electronic Structure and Reactivity 

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The rational function approximations to the Lorentzian and Gaussian lineshapes were written as... 

Rational Function Approximations to Implicit and Transcendental Functions... 

Figure 7.30. Rational function approximation to electron mobility data.

The Rational Function Optimization (RFO) expands the function in terms of a rational approximation instead of a straight second-order Taylor series (eq. (14.3)). 

Functions with localized strong inflections or poles may be approximated by rational functions of the general form... 

Since the machine performs only arithmetic operations (and these only approximately), iff is anything but a rational function it must be approximated by a rational function, e.g., by a finite number of terms in a Taylor expansion. If this rational approximation is denoted by fat this gives rise to an error fix ) — fa(x ), generally called the truncation error. Finally, since even the arithmetic operations are carried out only approximately in the machine, not even fjx ) can usually be found exactly, and still a third type of error results, fa(x ) — / ( ) called generated error, where / ( ) is the number actually produced by the machine. Thus, the total error is the sum of these... 

As an alternative procedure to predict coefficients of a radial function p(x) for electric dipolar moment, one might attempt to convert the latter function from polynomial form, as in formula 91, which has unreliable properties beyond its range of validity from experimental data, into a rational function [13] that conforms to properties of electric dipolar moment as a function of intemuclear distance R towards limits of united and separate atoms. When such a rational function is constrained to yield the values of its derivatives the same as coefficients pj in a polynomial representation, that rational function becomes a Fade approximant. For CO an appropriate formula that conforms to properties described above would be... 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

Global strategies for minimization are needed whenever the current estimate of the minimizer is so far from x that the local model is not a good approximation to fix) in the neighborhood of x. Three methods are considered in this section the quadratic model with line search, trust region (restricted second-order) minimization and rational function (augmented Hessian) minimization. 

Fig. 13. Normalized Gaussian (G) lineshape (l/cr /n) exp(-5/

An approximate solution of Equation 22 is obtained from Equation 23 by suitably approximating the matrix exponential e K This is accomplished by the Fade approximants of the exponential function. These Fade approximants are rational functions of the form... 

Although in recent years specific cases of enzyme-enzyme interaction have been approximated in various ways and treated by various modifications of the Michaelis-Menten Formalism, no general method for dealing with this class of mechanisms has developed from this approach. In any case, the rate laws that result from such interactions can be quite different in mathematical form from the rational functions characteristic of the Michaelis-Menten Formalism. 

The simplest and the most powerful rational function, is the Pad6 approximant, which is introduced by a ratio of two polynomials P[(z )... 

The EF algorithm [57] is based on the work of Ceijan and Miller [69] and, in particular, Simons and coworkers [70, 71]. It is closely related to the augmented Hessian (rational function) approach[25]. We have seen in section B3.5.2.5 that this is equivalent to adding a constant level shift (damping factor) to the diagonal elements of the approximate Hessian H. An appropriate level shift effectively makes the Hessian positive definite, suitable for minimization. 

When these composite rational functions are approximated by power-law functions, the description of the entire system can be written (5,7,3,1) ... 

Exact Representation. Althou the power-law formalism was originally derived as an approximation, one can often obtain an exact representation of rational-function nonllnearltles by Introducing additional variables (8). This obviously extends the generality and utility of the power-law formalism. Any disadvantage caused by the Increase In number of variables Is more than offset by the decrease In complexity of the nonllnearltles. The number of parameter values plus Initial conditions necessary to characterize a system remains the same In either representation. 

Once the value of is calculated for different Hn, an extrapolation for A = 0 is performed. For the extrapolation, we can use a polynomial approximation with the Neville algorithm, which is equivalent to the Richardson extrapolation for this specific case, or better still, a rational function using the Bulirsch-Stoer algorithm. The convergence of the method can be assessed by comparing the two different values of the extrapolation. 

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