Polynomial method

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

An alternative approach is to apply stronger fields and only use energies calculated for positive field strengths in generating the polynomial fit. In this case the energy is a function of both odd and even powers in the polynomial fit. We will show that the dipole moments derived from our non-BO calculations with the procedure that uses only positive fields and polynomial fits with both even and odd powers match very well the experimental results. Thus in the present work we will show results obtained using interpolations with even- and odd-power polynomials. Methods other than the finite field method exist where the noise level in the numerical derivatives is smaller (such as the Romberg method), but such methods still do not allow calculation of odd-ordered properties in the non-BO model. 

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

Selected entries from Methods in Enzymology [vol, page(s)] Graphical techniques, 210, 306 polynomial methods, 210, 307 with rational functions, 210, 311 with spline functions, 210, 312 with trigonometric functions, 210, 312... 

Y. Nesterov and A. S. Nemirovskii, Interior Point Polynomial Method in Convex Programming Theory and Applications, SIAM, Philadelphia, 1993. 

The time evolution operator exp(—/HAf/ft) acting on ( ) propagates the wave function forward in time. A number of propagation methods have been developed and we will briefly describe the following the split operator method [91,94,95], the Lanzcos method [96] and the polynomial methods such as Chebychev [93,97], Newtonian [98], Faber [99] and Hermite [100,101]. A classical comparison between the three first mentioned methods was done by Leforestier et al. [102]. 

Theoretical Analysis of Chemical Isotope Fractionation by Orthogonal Polynomial Methods... 

The isothermal compressibilities have been calculated with eq 5, using for the isothermal compressibilities of the pure substances the data from refs 53—56 (only the value for 2-butanol was taken as that for isobutanol). The data have been fitted using the Redlich—Kister equation.The values of D have been obtained from the activity coefficients or total pressure data by the sliding polynomials method. To check the accuracy of our calculations, the D values have been... 

Figure 3 Proton transfers from 5-nitro-benzisoxazole to acetate (left) and antibody 4B2 (right) in water. The changes in AG (kcal/mol) are computed using the cubic and fifth-order polynomial methods, and the exact PMF using 50 windows.

Gratitude is expressed to Auburn University and the Alabama Supercomputer Center for support of this research and to Dr. Ivan Tubert-Brohman and Professor William L. Jorgensen for their efforts on the cubic polynomial method and helpful discussions. 

In the Rys polynomial method, the first step is to use the Gaussian transform of the inverse electronic distance to separate the electron repulsion integral to, x, y and z components (Cook, 1974) using the Gaussian transform... 

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