Expansion, orthogonal polynomial

Orthogonal Polynomial Expansion of the Spectral Density Operator and the Calculation of Bound State Energies and Eigenfunctions. 

A computationally efficient method of function fitting using an orthogonal polynomial expansion is presented for approximating continuous wall temperature profiles. 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

Figure 4. Exact and finite orthogonal polynomial expansion calculation of the equilibrium (42) F ONOj...

Bather than using the Chapman-Enskog procedure directly, we shall employ the technique of Burnett,12 which involves an expansion of the distribution function in a set of orthogonal polynomials in particle-velocity space. 

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

The random phase error of a wavefront which has passed through turbulence may be expressed as a weighted sum of orthogonal polynomials. The usual set of polynomials for this expansion is the Zernike polynomials, which... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

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. 

The use of a finite-basis expansion to represent the continuum is reminiscent of the use of quadratures to represent an integration. Heller, Reinhardt and Yamani (1973) showed that use of the Laguerre basis (5.56) is equivalent to a Gaussian-type quadrature rule. The underlying orthogonal polynomials were shown by Yamani and Reinhardt (1975) to be of the Pollaczek (1950) class. 

D. De Fazio, S. Cavalli, and V. Aquilanti, Orthogonal polynomials of a discrete variable as expansion basis sets in quantum mechanics. The hyperquantization algorithm. Int. J. Quant. Chem., 93 91-111,2003. 

In packed-bed, flow-through electrodes, concentration and potential variation within the bed can also give more than one steady state. The convective transport equation with axial dispersion, coupled with Ohm s law for the electrode potential, was solved recently (418) by polynomial expansion and orthogonal collocation within the bed, to determine multiplicity regions. 

The integral terms in parentheses are known for the family of orthogonal polynomials. With finite N and known moments, this system of linear equations has the form M = AC and can be solved to find fhe expansion coefficients Ca(t, x). Thus, the presumed NDF n (t, X, Vp) is a unique function of a finite set of moments, and the latter are found by solving the moment-transport equations using n to close the unclosed terms. The fact that... 

The functions G (x) appearing above are the Meixner-Sheffer orthogonal polynomials the prime sign in (42) denotes their x derivative, and ct = Int(.v/2) is the integer part of s/2. If we omit the terms proportional to e and higher, we get the expansion for the FD squeezed vacuum as... 

Based on the Christoffel-Darboux formula (20) it can be shown that this procedure leads to a functional expansion which becomes an interpolation formula on the integration points, vK<7 ) = (<7 )- As an example consider an expansion by the Chebychev orthogonal polynomials g,(q) = T (q) with the constant weights W(qi) = 2/tt. The quadrature points q, are the zeros of the Chebychev polynomial of degree N + 1. On inserting Eq. (16) into the functional expansion Eq. (2) becomes... 

Choosing the generating function UNg as the polynomial, UNf (q) = (q - qx)(q - 2) " (q qi) (q q ) leads to the well-known Lagrange interpolation formula. Figure 4 shows the expansion function g (q) which is based on the zeros of the Cheby-chev orthogonal polynomial of order Ng. Another choice appropriate for evenly distributed sampling points is based on the global function NNg(q) = sin(2 Tr /A ). It is closely related to the Fourier method described in the next section. 

Orthogonal polynomials are a very useful set of expansion functions on grids. The simplest case is to define gn(q) as w(q)Pn(q), where P (q) is a member of the set of orthogonal polynomials, and w(q) is a weight function. These functions obey the continuous orthogonal relation defined in the domain D ... 

If the expansion functions are derived from orthogonal polynomials, the matrix d can be obtained from the recursion relation for the orthogonal polynomials (32). If there is a fast transform for G (which is true for the Chebychev polynomial expansion), then applying Eq. (38) will scale as 0(Ng log Ng). 

The terms in parenthesis are Legendre polynomials. Their advantage over the ordinary polynomial expansion is that the coefficients py are orthogonal. This means that the values of py found by regressing experimental data are independent of each other, and in consequence the accuracy can be adapted to the number of data points. 

Obviously any basis set method is heavily reliant on the choice of appropriate expansion functions. Conventional vibrational basis set have usually been constructed from products of one-dimensional expansions of orthogonal polynomials. In particular Hermite or associated Laguerre... 

A, B and are expanded in a series of orthogonal basis tensors according to the procedure outlined in Section 4.2.1.1 for pure gases. As noted there, this procedure leads to an infinite set of linear equations for the expansion coefficients, and each of the transport coefficients is itself related to just one of the expansion coefficients. The result is that the transport properties of a multicomponent gas mixture can be expressed formally as the ratio of two infinite determinants. Various orders of approximations to the transport coefficients can then be generated by retaining only a limited number of terms in the polynomial expansion. There are various subtleties associated with the nomenclature of orders of approximation which need to be considered carefully. Here almost exclusively the lowest order of approximation is considered, which is again remarkably accurate. Details of higher-order approximations may be found elsewhere (McCourt et al. 1990 Ross et al. 1992). 

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