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►A minimal cubature rule is a numerical integration rule that uses the smallest number of nodes among cubature rules of the same degree.
The nodes of these cubature rules are closely related to common zeros of OPs and they are often good points for polynomial interpolation.
Although Gaussian cubature rules rarely exist and they do not exist for centrally symmetric domains, minimal or near minimal cubature rules on the unit square are known and provide efficient numerical integration rules.
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►See Allasia and Besenghi (1987b) for the numerical computation of from (8.6.4) by means of the trapezoidal rule.
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►A numerical inversion procedure is also given for calculating the value of (with 10S accuracy), when and are specified, based on Newton’s rule (§3.8(ii)).
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►The trapezoidal rule (§3.5(i)) is then applied.
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►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations.
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►Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions.
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►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that
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►►►Figure 18.40.2: Derivative Rule inversions for carried out via Lagrange and PWCF interpolations.
…For the derivative rule Lagrange interpolation (red points) gives digits in the central region, while PWCF interpolation (blue points) gives .
Magnify►Further, exponential convergence in , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
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►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter.
…Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent.
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