# abscissas

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## 4 matching pages

##### 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

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►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

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►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.
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