abscissas

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1: 18.40 Methods of Computation

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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

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§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 ${|A_{x_i,l}/B_l(x_i)|}^2$ = $w_i^N/w^{\mathrm{CP}}(x_i)$ where $w_i^N$ is the Gaussian-Pollaczek quadrature weight at $x=x_i$, and $w^{\mathrm{CP}}(x_i)$ is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …