Numerical methods polynomial approximation

Numerical methods are often used to find the roots of polynomials. A detailed discussion of these techniques is given under Numerical Analysis and Approximate Methods. ... 

It can be proved that this numerical method is of order 1 in h = max diam(7j). As mentioned above, higher order methods can be obtained by first using curved tesserae instead of planar triangles and then increasing the degree of the polynomial approximation on each tessera (/ , or P2 BEM [2]). 

Wajge et al. (1997) attempted to develop rigorous PDAE model for packed batch distillation with and without chemical reaction and used finite difference and orthogonal collocation techniques to solve such model. The main purpose of the study was to investigate the efficiencies of the numerical methods employed. The authors observed that the collocation techniques are computationally more efficient compared to the finite difference method, however the order of approximating polynomial needs to be carefully chosen to achieve a right balance between accuracy and efficiency. See the original reference for further details. 

NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, Richard Hamming. Classic (ext stresses frequency approach in coverage of algorithms, polynomial approximation, Fourier approximation, exponential approximation, other topics. Revised and enlarged 2nd edition. 721pp. 55 854. 

Most numerical methods for solving polynomial equations are based on an initial guess for the answer followed by some iterative procedure for obtaining successively better approximations. A preliminary guess may be obtained by neglecting one or more items in equation viia. A convenient way to obtain a root for this equation is to plot various values of/([H ]) versus [H ] and to locate where /([H" ]) crosses the zero axis. Programmable scientific... 

All steps of these numerical methods use the given advantages of raw-files Equal step widths (very often 0.02° in 26) for the complete pattern and about equal half-widths of all peaks (for routine measurements 0.1-0.2°). A very powerful method is the application of sliding polynomial fitting procedures, which rely on a constant step-width and an approximately constant peak form, and can by used for steps 1, 3, and 5. 

Any approximation of the derivative of a function in terms of values of that function at a discrete set of points is called a finite difference approximation. There are several ways of constructing such approximations, the Taylor series approach illustrated above is frequently used in numerical analysis because it supplies the added benefit that information about the error is obtained. Another method uses interpolation to provide estimates of derivatives. In particular, we use interpolation to fit a smooth curve through the data points and differentiate the resulting curve to get the desired result. A collection of low order approximations (i.e., first to fourth order polynomial approximations) of first and second order derivative terms can be found in textbooks like [49, 50, 167]. 

Equation (478) is the exact analytical geometry expression of capillary rise in a cylindrical tube having a circular cross section, which considers the deviation of the meniscus from sphericity, so that the curvature corresponds to (AP = A pgy) at each point on the meniscus, where y is the elevation of that point above the flat liquid level (y = z+h). Unfortunately, this relation cannot be solved analytically. Numerous approximate solutions have been offered, such as application of the Bashforth and Adams tables in 1883 (see Equation (476)) derivation of Equation (332) by Lord Rayleigh in 1915 a polynomial fit by Lane in 1973 (see Equation (482)) and other numerical methods using computers in modern times. 

This DAE system can be solved by means of numerical methods used for stiff ODE systems [15]. BDF methods replace i by a polynomial approximation. The DASSL solver [16], which employs this technique, is our used numerical code. 

The conventional Ewald summation method works well for simulations of small periodic systems, but the computation can become prohibitively expensive when large systems are involved, in which the particle number exceeds lO. Several numerical techniques have been used to enhance the performance of the traditional Ewald method with mixed results. For example, look-up tables and polynomial approximations have been suggested. The algorithmic performance can also be optimized through the parameter 1 113,114 determines both the extension of the short-range interaction... 

For nonlinear reaction kinetics, a numerical solution of the balance Equation 4.121 is carried out. For example, for second-order kinetics, R = kcACB, with an arbitrary stoichiometry, the generation rate expressions, ta = —va CaCb and tb = —vb caCb, are inserted into the mass balance expression, which is solved numerically using, for example, a polynomial approximation (orthogonal collocation method). The performances of the normal dispersion model and its segregated or maximum-mixed variants are compared in Figure 4.34. The symbols are explained in the figure. The comparison reveals that the differences between the segregated, maximum-mixed, and normal axial dispersion models are notable at moderate Damkohler numbers R = Damkohler number). 

In order to obtain a system of algebraic equations, the PBE (12.399) and the boundary conditions must be transformed into a discrete form. In spectral methods, the solution function is approximated in terms of a polynomial solution function expansion (12.408). The differentiation of the discrete solution approximation were presented as (12.410) and (12.411). The PBE (12.399) is an integro-differential equation. Thus, appropriate quadrature rules are required for the numerical solution. Integral approximations can be presented on the form ... 

The numerical methods commonly used to carry out either TDW or TIW calculations are the symmetric split operator (SSO) approximation to the evolution operator, exp(-i//r//i), and the Chebychev polynomial expansions of either exp(-i//r/fi) or the causal Green s function, E — H 4- ie)". These methods are described in great detail in the literature, but are too technically involved to go into in this article. Of course, there are other propagation methods as well which are being used or are under development. One which is very interesting and which has the attractive feature of being readily applied even in the case of an explicit time-dependence in the Hamiltonian is the modified Cayley method . ... 

The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-77) can be replaced by an integration formula, and Eq. (3-77) may be written... 

The function 7(f) can be chosen for the whole reaction time interval, or two or three subsequent temperature-time data points 7(fi-i), 7(fi), and 7(fi+i) can be approximated by polynomials of second or third order 7,(f), respectively. These polynomials will then be used in a procedure for numerical integration in each integration step i. This method has been successfully applied in a kinetic study of the partial oxidation of hydrocarbons (Skrzypek et al., 1975, Krajewski etai, 1975, 1976, 1977). 

The approximations inherent in Brouwer diagrams can be bypassed by writing the appropriate electroneutrality equation as a polynomial equation and then solving this numerically using a computer. (This is not always a computationally trivial task.) To illustrate this method, the examples given in Sections 7.5 and 7.6, the MX system, will be rewritten in this form. 

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

Starting with the method described above, extensive tables of the numerical values of mean energies, integrals of electrostatic and constant of spin-orbit interactions are presented in [137] for the ground and a large number of excited configurations, for atoms of boron up to nobelium and their positive ions. They are obtained by approximation of the corresponding Hartree-Fock values by polynomials (21.20) and (21.22). Such data can be directly utilized for the calculation of spectral characteristics of the above-mentioned elements or they can serve as the initial parameters for semi-empirical calculations [138]. 

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