Rational approximation

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)). 

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... 

Vol. 1237 Rational Approximation and its Applications in Mathematics and Physics. Proceedings, 1985. Edited by J. Gilewicz, M. Pindor and W. Siemaszko. XII, 350 pages. 1987. 

Table 2. Coefficients in the rational approximation for the screening function X(0 ofEq. (47).

P. O. Lowdin. Partitioning technique, perturbation theory, and rational approximations. Intern. J. Quantum Chem., 21 69, 1982. 

That is, each delay element becomes an allpass filter which approximates the required delay versus frequency. A diagram appears in Fig. 10.14, where Ha (z) denotes the allpass filter which provides a rational approximation toz co/c(a>)... 

In Table 11.1 /(a) is given, besides functions g(a). The right-hand side can be evaluated numerically by using a 4th order rational approximation. 

Technically it is the existence of transcendental numbers in the number continuum that makes chaos possible. We should note, however, that this does not prevent meaningful computer exploration of chaos despite the fact that computers only deal in rational approximations to algebraic and transcendental numbers (see, e.g., Hammel et al. (1987)). This fact is illustrated in Section 2.2, where we discuss some important examples of chaotic mappings. 

Although numerical integrations of equation (14) usually can be performed accurately now, some systems remain that are too complex or too stiff for the computational methods to succeed. To analyze such systems, as well as to understand better the behaviors of various systems amenable to accurate numerical integrations, rational approximations producing simplifications are wanted. Two important approximations of this type are the steady-state and partial-equilibrium approximations. 

The concept of volume filling of micropores under review is based on a radical difference of adsorption in micropores as a limiting case as compared with the opposite limiting case of adsorption on the surface of nonporous adsorbents of identical chemical nature. The concept of adsorption in micropores is of a thermodynamical nature, and its main initial principle, which has the meaning of a rational approximation, con-... 

Equation (14.131) is exact working expressions for II follow by rational approximation. 

To avoid the need for special procedures and modification of the integration algorithm, delays may be modeled by using rational approximations, e.g., Fade functions or multiple first-order lags in series. Experimentation suggests that 10 series lags is adequate for most applications, so this is used as a default. The approximation should be checked by comparison with a more detailed model where it is believed to be particularly significant. 

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