Gaussian quadrature algorithm

The Gaussian quadrature algorithm introduced in Section 3.1.1 is equivalent to approximating the univariate NDF by a sum of Dirac delta functions ... 

As has already been mentioned, Eq. (3.36) is not a Gaussian quadrature approximation its degree of accuracy is not known a priori (and strongly depends on the choice of moments on which the formula is constructed) and the algorithms for its derivation, from the moments of the NDF, are not well known (unlike for the univariate case). 

When applying this algorithm, the moment set [mo, m, ..., m2N must be known (and realizable). From the definition of the objective function and the properties of Gaussian quadrature,we have /(O) > 0. Thus, as a first step in the bounded-search algorithm, an upper bound cr+ can be determined such that /(cr+) < 0 under the condition that the moment set mj, m, 2iv-i) found from Eq. (3.93) using cr = is realizable. If no such cr+ exists, then can be chosen such that it minimizes /(cr ) and the moment set... 

Sack, R. A. Donovan, A. F. 1971 An algorithm for Gaussian quadrature given modifled moments. Numerische Mathematik 18,465 78. 

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). 

The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

In this appendix we describe a stencil algorithm which avoids many of the drawbacks of quadrature rules used in classical lattice models, while the extra computational cost is modest. The derivation consists of finding a unique and optimal set of stencil coefficients for a convolution with a Gaussian kernel, adapted to the special case of off-lattice density functional calculations. Stencil coefficients are the multipliers of the function values at corresponding grid points. 

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