# L?H�pital rule

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## 1—10 of 34 matching pages

##### 1: 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

… ► …##### 2: 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. …##### 3: 8.25 Methods of Computation

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►See Allasia and Besenghi (1987b) for the numerical computation of $\mathrm{\Gamma}(a,z)$ 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 $x$ (with 10S accuracy), when $a$ and $P(a,x)$ are specified, based on Newton’s rule (§3.8(ii)).
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##### 4: 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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##### 5: 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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##### 6: 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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##### 7: 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 $N$, 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 $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty},\mathrm{\infty})$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …##### 8: 10.74 Methods of Computation

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►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.
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##### 9: 29.20 Methods of Computation

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►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.
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##### 10: Bibliography R

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (4), pp. 91.
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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (3), pp. 56–64.
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Finite-sum rules for Macdonald’s functions and Hankel’s symbols.
Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
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