# Newton–Cotes

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##### 1: Sidebar 9.SB1: Supernumerary Rainbows
Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. The house in the picture is Newton’s birthplace. …
Rules of closed type include the NewtonCotes formulas such as the trapezoidal rules and Simpson’s rule. …
##### 3: 3.8 Nonlinear Equations
###### §3.8(ii) Newton’s Rule
Newton’s rule is given by … Newton’s method is given by … Newton’s rule can also be used for complex zeros of $p(z)$. …
##### 4: 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. … Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …
##### 5: 8.25 Methods of Computation
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 rule (§3.8(ii)). …
##### 6: 3.3 Interpolation
###### §3.3(iv) Newton’s Interpolation Formula
Newton’s formula has the advantage of allowing easy updating: incorporation of a new point $z_{n+1}$ requires only addition of the term with $[z_{0},z_{1},\dots,z_{n+1}]f$ to (3.3.38), plus the computation of this divided difference. … For comparison, we use Newton’s interpolation formula (3.3.38) …and compute an approximation to $a_{1}$ by using Newton’s rule (§3.8(ii)) with starting value $x=-2.5$. …Then by using $x_{3}$ in Newton’s interpolation formula, evaluating $[x_{0},x_{1},x_{2},x_{3}]f=-0.26608\;28233$ and recomputing $f^{\prime}(x)$, another application of Newton’s rule with starting value $x_{3}$ gives the approximation $x=2.33810\;7373$, with 8 correct digits. …
##### 7: 9.17 Methods of Computation
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. …
##### 8: 10.74 Methods of Computation
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. …
##### 9: 6.18 Methods of Computation
Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …