Newton rule (or method)

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1: 3.8 Nonlinear Equations

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§3.8(ii) Newton’s Rule

… ► … ►Newton’s rule is given by … ►

Table 3.8.2: Newton’s rule for $z^4-1=0$.

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$n$	$z_n$
…

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… ►Newton’s rule can also be used for complex zeros of $p(z)$. …

2: 8.25 Methods of Computation

… ►A numerical inversion procedure is also given for calculating the value of $x$ (with 10S accuracy), when $a$ and $P(a,x)$ are specified, based on Newton’s rule (§3.8(ii)). …

3: 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. …

4: 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. …

5: 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. …

6: 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. …

7: 4.45 Methods of Computation

… ►For $x\in [-1/\mathrm{e},\mathrm{\infty })$ the principal branch $W_0\left(x\right)$ can be computed by solving the defining equation $W{\mathrm{e}}^W=x$ numerically, for example, by Newton’s rule (§3.8(ii)). …

8: 3.3 Interpolation

… ►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 $\left[x_0,x_1,x_2,x_3\right]f=-\mathrm{0.26608\hspace{0.33em}28233}$ and recomputing $f^{\prime }(x)$, another application of Newton’s rule with starting value $x_3$ gives the approximation $x=\mathrm{2.33810\hspace{0.33em}7373}$, with 8 correct digits. …

9: 3.5 Quadrature

… ► ►Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. …