Polynomial, Chebyshev

Chebyshev Approximation The well known expansion of exp(— into Chebyshev polynomials T, [23] is one of the most frequently used integration technique in numerical quantum dynamics ... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

Interpolation of this type may be extremely unreliable toward the center of the region where the independent variable is widely spaced. If it is possible to select the values of x for which values of f(x) will be obtained, the maximum error can be minimized by the proper choices. In this particular case Chebyshev polynomials can be computed and interpolated [11]. 

Thus, the well-known Chebyshev polynomial defined by... 

The Chebyshev polynomials, whiGh occur in quantum chemistry and in certain numerical applications, can be obtained from the hypergeometric functions by placing a = -/ , an integer, and y — Finally, the hypergeometric... 

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. 

Other forms of vapor pressure equations, such as Cox equation (Osborn and Douslin 1974, Chao et al. 1983), Chebyshev polynomial (Ambrose 1981), Wagner s equation (Ambrose 1986), have also been widely used. Although... 

Ambrose, D., Counsell, J.F., Davenport, A.J. (1970) The use of Chebyshev polynomials for the representation of vapour pressures between the triple point and the critical point. J. Chem. Thermodyn. 2, 283-294. 

The usefulness of the Chebyshev polynomials being both efficient and accurate building blocks in numerically approximating operator functions was realized first by Tal-Ezer and Kosloff,79,123 and later by Kouri, Hoffman, and coworkers.124-129 Aspects of their pioneering work will be discussed later in this review. 

A unique and well-known property of the Chebyshev polynomials is that they can be mapped onto a cosine function ... 

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

No divergences and dependence on the contact parameters Ti 2 remain in the form for r. It shows the transmittance function (at least in the weak-coupling limit) is indeed a well-defined molecular quantity. We can rewrite equation (38), taking into account the definition of 6 (see equation (35)) and the definition of the Chebyshev polynomials of the second kind U (cos 6) — sin[(n +l)0]/sin 6 as... 

Fig. 1. Chromatogram of alkylbenzenes (reversed-phase HPLC) 100 term Chebyshev polynomial...

Fig. 6. Chebyshev approjdmation used as a filter on part of a chromatogram (0.1 pg r anthracene, 0.5 pg r benzanthracene, reversed-phase HPLQ 40 terms Chebyshev polynomial approximation.

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