The product-difference PD algorithm

The product-difference (PD) algorithm was developed by Gordon (1968) and is based on the theory of continued fractions of Stieltjes. The first step is to construct a matrix P with 

The coefficients of the continued fraction a necessary to determine the coefficients of the three-term recursive relation are generated by setting the first element equal to zero ( 1 = 0), and then computing the others according to the following recursive relationship  

Exercise 3.1 Consider the following normal (or Gaussian) distribution  

The moments of this distribution can be easily calculated through the following equations (reported only for the first nine moments)  

Use the PD algorithm to calculate the quadrature approximation of order four (i.e. N = A) for p = 5 and cr = 1. For this calculation the first eight moments of the distribution are needed  

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This system of 2 M nonlinear equations is ill-conditioned for large M, but can be efficiently solved using the product-difference (PD) algorithm introduced by McGraw (1997). Thus, given the set of 2 M moments on the left-hand side of Eq. (107), the PD algorithm returns wm and lm for m — 1., M. The closed microscopic transport equation for the moments can then be written for k — 0,..., 2 M— 1 as... 

Compute the weights and abscissas from the given moments by use of the product-difference (PD)-algorithm. 

The procedure can be iterated, considering now the Volterra integro-differential equation (3.44) for x(/), then for 2(0. and so on. The implications of Eq. (3.54) have been discussed in more detail in Chapter III, with comments on product-difference (PD) algorithms. 

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). 

Matrix in Golub-Welsch algorithm for which the eigensys-tem are the Gauss-Lobatto quadrature points and weights Product difference matrix of order (2Nq- -l)(2Nq- -l) in PD algorithm... 

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