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interpolatory rules (or formulas)


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1: 3.5 Quadrature
§3.5(iv) Interpolatory Quadrature Rules
An interpolatory quadrature ruleIf we add - 1 and 1 to this set of x k , then the resulting closed formula is the frequently-used Clenshaw–Curtis formula, whose weights are positive and given by …
§3.5(v) Gauss Quadrature
2: 3.8 Nonlinear Equations
§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. …
3: 8.25 Methods of Computation
See Allasia and Besenghi (1987b) for the numerical computation of Γ ( a , z ) from (8.6.4) by means of the trapezoidal rule. … A numerical inversion procedure is also given for calculating the value of x (with 10S accuracy), when a and P ( a , x ) are specified, based on Newton’s rule3.8(ii)). …
4: 9.17 Methods of Computation
The trapezoidal rule3.5(i)) is then applied. … Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule3.8(ii)) or Halley’s rule3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …
5: 7.22 Methods of Computation
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. …
6: About MathML
As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. …
7: 10.74 Methods of Computation
Newton’s rule3.8(i)) or Halley’s rule3.8(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. …
8: 29.20 Methods of Computation
A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …
9: 34.7 Basic Properties: 9 j Symbol
This equation is the sum rule. It constitutes an addition theorem for the 9 j symbol. …
10: 1.4 Calculus of One Variable
Chain Rule
L’Hôpital’s Rule