# interpolatory rules (or formulas)

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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 …
##### 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 $\Gamma\left(a,z\right)$ 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 $P\left(a,x\right)$ 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. …
##### 9: 34.7 Basic Properties: $\mathit{9j}$ Symbol
This equation is the sum rule. It constitutes an addition theorem for the $\mathit{9j}$ symbol. …