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1: 18.40 Methods of Computation
A numerical approach to the recursion coefficients and quadrature abscissas and weights
These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to w ( x ) , as will be considered in the following paragraphs. … A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let N be a positive integer and define …The quadrature abscissas x n and weights w n then follow from the discussion of §3.5(vi). … Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain w ( x ) . …
2: 3.3 Interpolation
§3.3(i) Lagrange Interpolation
The nodes or abscissas z k are real or complex; function values are f k = f ( z k ) . …
3: 18.38 Mathematical Applications
The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …
4: 18.39 Applications in the Physical Sciences
Full expressions for both A x i , l and B l ( x ) are given in Yamani and Reinhardt (1975) and it is seen that | A x i , l / B l ( x i ) | 2 = w i N / w CP ( x i ) where w i N is the Gaussian-Pollaczek quadrature weight at x = x i , and w CP ( x i ) is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …