rhombus rule
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1: 3.10 Continued Fractions
2: 3.5 Quadrature
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§3.5(i) Trapezoidal Rules
… ►The composite trapezoidal rule is … ►§3.5(ii) Simpson’s Rule
… ►§3.5(iv) Interpolatory Quadrature Rules
… ► …3: 3.8 Nonlinear Equations
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§3.8(ii) Newton’s Rule
… ► … ►Newton’s rule is given by … ►Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …4: 8.25 Methods of Computation
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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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5: 9.17 Methods of Computation
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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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6: About MathML
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►As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful.
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7: 7.22 Methods of Computation
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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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8: 18.40 Methods of Computation
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Derivative Rule Approach
►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that … ► ►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). …9: 23.5 Special Lattices
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►The parallelogram , , , , is a rhombus: see Figure 23.5.1.
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►The rhombus
, , , can be regarded as the union of two equilateral triangles: see Figure 23.5.2.
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