The Wheeler algorithm

The PD algorithm is quite efficient in a number of practical cases however, it generally becomes less stable as N increases. It is difficult to predict a priori when this will occur since it depends on the absolute values of the moments, but typically problems can be expected when N 10. Another important issue is related to the fact that for distributions with zero mean (or in other words with mi = 0) the algorithm blows up, due to a division by zero during the calculation of the coefficients of the continued fraction fa. The Wheeler algorithm, which will be reported in the next section, does not suffer from these problems. 

Sack Donovan (1971) proposed an alternative approach for the calculation of the coefficient of the recursive formula reported in Eq. (3.5) (and appearing also in the Jacobi matrix) that resulted in higher stability. This approach is based on the idea of using a different set of basis functions naif) to represent the orthogonal polynomials, rather than the usual powers of f. The improved stability results from the ability of the new polynomial basis to better sample the integration interval. The coefficients are calculated from the modified moments defined as follows  

2N - a - 1. Finally, the coefficients for the Jacobi matrix are computed as follows  

Exercise 3.2 Consider again the distribution reported in Eq. (3.16) of Exercise 3.1. Let us now calculate the quadrature approximation of order four (i.e. N = A) for p = 0 and T = 1 using the Wheeler algorithm. Eor this calculation the first eight moments of the distribution are needed  

Notice that, because the distribution has zero mean and is perfectly symmetric (zero skewness), the odd moments are all null. 

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After applying the Wheeler algorithm, the following Jacobi matrix is obtained ... 

The construction of the multivariate quadrature begins with the calculation of the univariate quadrature of order N for the first internal coordinate by using the Wheeler algorithm with the first 2N - 1 moments ... 

In summary, for each pair of A 2 = 0,..., V2 - 1 and = 1,..., 2N - 1, the quantities are calculated by solving (with the Rybicki algorithm) the linear system reported in Eq. (3.76). Subsequently these quantities are used to solve for each value of ai = 1,..., Vi the linear system reported in Eq. (3.77). Einally, the quadrature nodes for the third internal coordinate are calculated by applying the Wheeler algorithm to the first 2N3 conditional moments with respect to the third internal coordinate ... 

Examples of dependences of cr on 77 for selected values of q are shown in Figure 3.8. Note that the coefficients of the polynomial depend only on the known moments mt, and not on the weights and abscissas. Thus, can be computed first, followed by the moments ml, using Eq. (3.95), from which the weights and abscissas are found using the Wheeler algorithm. In the limit where 5 = 0 (i.e. = 3 and q = 0), all of the moments mk are... 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

The values for and with 1 < a < N are calculated from the initial condition for the moments belonging to the moment set mk(0), which in turn are calculated from the NDF. This is generally done by applying the PD or Wheeler algorithm, in which case the... 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

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