Taylor polynomial

In 2012 Cassam-Chenai and Lievin [119] presented two reduced-dimension ab initio CH4 DMSs computed at the MRCI/VQZ and MRCI/ ACV5Z levels of theory, respectively on a small grid of 100 geometries and represented them as 3-dimensional Taylor polynomial expansions in terms of normal coordinates. 

The Taylor series by itself is not numerically stable, since the individual temis can be very large even if the result is small, but other polynomials which are highly convergent can be found, e.g. Chebyshev [M, M and M] or Lancosz polynomials [, 68]. 

Mandelshtam V A and Taylor H S 1995 A simple recursion polynomial expansion of the Green s function with absorbing boundary conditions. Application to the reactive scattering J. Chem. Phys. 102... 

The simplest form of approximation to a continuous function is some polynomial. Continuous functions may be approximated in order to provide a simpler form than the original function. Truncated power series representations (such as the Taylor series) are one class of polynomial approximations. 

To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called Pade approximation, which puts the function as a ratio of two polynomials. The simplest is the first order (1/1) Pade approximation ... 

This is a form that serves many purposes. The term in the denominator introduces a negative pole in the left-hand plane, and thus probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the right-hand plane, which is needed to make a problem to become unstable. (This point will become clear when we cover Chapter 7.) Finally, the approximation is more accurate than a first order Taylor series expansion.1... 

Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

The rationale for using low degree polynomials to approximate / is based on a Taylor series expansion off around x=0. 

This is precisely the polynomial obtained by Taylor under the assumption that dCmldi is not quite constant but that the second derivative d2CJd 2 is (Taylor 19546, equation (24)). 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

Taylor s theorem permits the expansion of certain functions, often in the form of a polynomial. Only the terms which contribute in a significant way to the response are utilized, in this way facilitating the mathematical... 

The Taylor approximation model is a polynomial in the experimental factors. The rest term, R(x), becomes smaller and smaller the more polynomial terms are included in the model. R(x) accounts for the variation which is not described by the polynomial terms and will thus contain the model error. A model is considered as satisfactory if the model error is significantly less than the experimental error. 

It is well known that using an exponential or power function can also describe the portion of a polynomial curve. Indeed, these types of functions, which can represent the relationships between the process variables, accept to be developed into a Taylor expansion. This procedure can also be applied to the example of the statistical process modelling given by the general relation (5.3) [5.20]. 

When the region of expansion is properly chosen, the error e x) oscillates on both sides of the abscissa. Thus, choosing an orthogonal polynomial P x) is equivalent to demanding that the error of the approximation be zero at a finite set of points. This is in contrast to the Taylor series for which the error is zero only at one point. In this sense the orthogonal expansion is an interpolating approximation. 

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