Jacobi polynomial

For problems with symmetry, we construct a different set of orthogonal polynomials. We consider the problem defined over the spatial domain [0,1] and that the problem solution is symmetric about a = 0. That is to say that one of the boundary conditions is dy/dx = 0 at a = 0. We use a trial solution that is in terms of and assume a Dirichet boundary condition at x = 1. The trial function is 

We construct the polynomial P (x ) so that it satisfies the orthogonality condition 

The orthogonal polynomials described by equation (8.12.22) are Jacobi polynomials (Carnahan et ah, 1969). The value of the parameter 

There are two common values for the weighting w x ). These are u = 1 and w = —x. Roots for these Jacobi polynomials are presented in Table 8.8 for all three geometries, both values of w and various orders of N. The roots presented in Table 8.8 specify specific collocation points to be used in the method of weighted residuals. The collocation method satisfies the describing differential equation at the collocation points. In order to implement this, we need to evaluate the function and its derivatives at the collocation points. The trial function of equation (8.12.21) can also be written as 

The first derivative of the trial function at a collocation point is given by 

Because it has become so widely used, we devote the remainder of the chapter to orthogonal collocation. Before discussing additional details, we introduce a number of preliminary steps, which are needed for further development. These steps include a discussion of Jacobi polynomials (its choice has been explained above) and the Lagrangian interpolation polynomials. The Lagrangian interpolation polynomial is chosen as a convenient vehicle for interpolation between collocation points. 

The Jacobi polynomials are important in providing the optimum positions of the collocation points. Since these collocation points are not equally spaced, the Lagrangian interpolation polynomials are useful to effect an approximate solution. 

Since all finite domains can be expressed in the range [0,1] through a linear transformation, we will consider the Jacobi polynomials defined in this domain. This is critical because the orthogonality condition for the polynomials depends on the choice of the domain. The Jacobi polynomial is a solution to a class of second order differential equation defined by Eq. 3.145. 

The Jacobi polynomial of degree N has the power series representation 

are constant coefficients, and a and /8 are parameters characterizing the polynomials, as shown in Eq. 8.83. That is, is the polynomial 

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The collocation points are calculated using programs given by Villadsen and Michelsen (1978) for calculating the zeros of an arbitrary Jacobi polynomial P% P x) that satisfies the orthogonality relationship... 

The Jp in eqn (16) are Jacobi polynomials, t is time measured from the instant at which the generation of new radicals ceased, and the Bp are coefficients whose values are determined by the requirement that at t = 0 must have a particular form (5,0). ... 

This is one of the variants of the finite element methods. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their... 

The table below provides the roots of the Jacobi polynomials used as node points in orthogonal collocation, for some values of N. Values for X = 0 (i = 0) and X = 1 (i = N + 1) (the values are 0 and 1, resp.) are not included. The roots were computed using the subroutine JCOBI, modified from the original of Villadsen and Michelsen [562], discussed in Appendix C, using for a given N the call... 

There exist (4, 5, 8, 9, 27) simple direct relations, between isotope effect, structure, and force field, which do not necessarily require a complete knowledge of all molecular parameters and avoid the solution of the secular equation. These relations are, however, approximations restricted to limited ranges of temperature. [Newer approximation methods, based on expansions in Jacobi polynomials, are applicable over wide ranges of temperatures (6, i6).] In the past, before the ready availability of fast digital computers, tests of the validity of these approximations were usually fairly limited in nature, but recent extensive tests on model calculations of kinetic isotope effects have been carried out 23, 28). In addition, extensive tests of power-series approximations (not considered in the present paper) have now been performed (6,16). 

Chebyshev polynomial of the first kind, Tn""(x), which is a Jacobi polynomial defined as... 

As will be shown later, a common factor on F in the definition of Jacobi polynomials, such as ( — 1) in Equation 29, does not affect the expansion. 

The present authors have explored possibilities of using other Jacobi polynomials for expanding In — f. 

We note with interest that the use of Jacobi polynomials leads to... 

Variation of the Orthogonal Polynomial. Among the Jacobi polynomials, the Chebyshev polynomials of the first kind (y = 8 =... 

We have investigated possibilities of using Jacobi polynomials other than T (x). Our primary aim was to obtain a good approximation to In s s... 

Figure 2. Absolute error in In b(u) obtained by Jacobi polynomial expansions over the range [0,27t] as a function of u for orders n = 1 — 4...

The two parameters y and 8, which characterize a Jacobi polynomial, were varied, and RMSE values, as defined by Equations 58, 59, and 60, were numerically evaluated using an IBM-360 computer. Preliminary tests showed that satisfactory integrations were achieved by summing over 50 equally spaced points. The RMSE-surface was mapped for both the Bigeleisen-Ishida formula. Equation 44, and the modified one. Equation 52. Naturally, the best polynomial may depend on the order and... 

The values of the modulating coefficients for the Chebyshev and best Jacobi polynomials for both fixed and sub-divided ranges of the... 

RMSE as defined by Equations 58, 59, and 60 was optimized, instead of minimizing the squares of the absolute error, as is ordinarily done. The RMSE thus optimized are tabulated in Table IX. By comparison of Tables V and IX, it is obvious that within the framework of the present set of criteria. Equations 58, 59, and 60, the best possible approximation has already been achieved with the best set for L =5. For an imaginary molecule for which the assumptions leading to the present weighting, w u) oc u, are valid, the set of Jacobi polynomials given in Table IV... 

For a specific problem, one could construct a similar table of Jacobi polynomials best suited for the problem, by suitable choice of a weighting function w u), and modification of the RMSE to be optimized. In some cases it may be of interest to use the polynomial that is best at the largest range for all the calculations. Table X provides a crude idea of the degree of approximation obtainable when the polynomial of Table IV that is best at range = Stt is used for all the calculations of a given order. A comparison of Tables V and X shows that a simpler best set of polynomials used over the entire range of the expansion variable, leads... 

Jacobi Polynomials Using Subdivided Range Expansion (L = 5)... 

The bottom sections of Tables XII, XIII, XIV, and XV show results of similar calculations for the same systems, but using the best sets of Jacobi polynomials, tabulated in Tables II and IV, in place of the Chebyshev polynomials. For each best set, the calculations were carried out in the following manner using the appropriate equations. For each temperature u niax was computed from the highest frequency of the system, and the best polynomial for the range closest to u nmx was chosen—i.e, if = 4.37T the best polynomial for the range [0,47t] was selected. 

Calculated by using Equation 47, and the Best set of Jacobi polynomials tabulated in Table II. 

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