L?H�pital rule for derivatives

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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 … ►Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …

2: 1.5 Calculus of Two or More Variables

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§1.5(i) Partial Derivatives

… ►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, and ${\partial }^{2}f/\partial y\partial x$ are continuous. … ►

Chain Rule

… ►Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty }$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. … ►

§1.5(vi) Jacobians and Change of Variables

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3: 18.40 Methods of Computation

… ►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). ►

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 … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. … ►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). …

4: 3.5 Quadrature

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§3.5(i) Trapezoidal Rules

… ►The composite trapezoidal rule is … ►

§3.5(ii) Simpson’s Rule

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§3.5(iv) Interpolatory Quadrature Rules

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5: 9.17 Methods of Computation

… ►The trapezoidal rule (§3.5(i)) is then applied. … ►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to $\mathrm{Ai}\left(z\right)$, $\mathrm{Bi}\left(z\right)$, and their derivatives. … ►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. … ►For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).

6: 1.6 Vectors and Vector-Valued Functions

… ►where $𝐧$ is the unit vector normal to $𝐚$ and $𝐛$ whose direction is determined by the right-hand rule; see Figure 1.6.1. ►

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Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify

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1.6.19 $$\nabla =𝐢\frac{\partial }{\partial x}+𝐣\frac{\partial }{\partial y}+𝐤\frac{\partial }{\partial z}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E19

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See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►where $\partial g/\partial n=\nabla g\cdot 𝐧$ is the derivative of $g$ normal to the surface outwards from $V$ and $𝐧$ is the unit outer normal vector. …

7: 1.4 Calculus of One Variable

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§1.4(iii) Derivatives

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Higher Derivatives

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Chain Rule

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Leibniz’s Formula

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L’Hôpital’s Rule

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8: 10.74 Methods of Computation

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§10.74(vi) Zeros and Associated Values

►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. Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example. Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. … ► …

9: 37.20 Mathematical Applications

… ►A minimal cubature rule is a numerical integration rule that uses the smallest number of nodes among cubature rules of the same degree. The nodes of these cubature rules are closely related to common zeros of OPs and they are often good points for polynomial interpolation. Although Gaussian cubature rules rarely exist and they do not exist for centrally symmetric domains, minimal or near minimal cubature rules on the unit square are known and provide efficient numerical integration rules. …

10: Bibliography R

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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Referenced by:

3rd item.

Encodings:

BibTeX

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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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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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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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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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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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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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

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