3 Numerical Methods Areas 3.3 Interpolation 3.5 Quadrature

§3.4 Differentiation

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Referenced by:: §3.3(v)
Permalink:: http://dlmf.nist.gov/3.4
See also:: Annotations for Ch.3

§3.4(i) Equally-Spaced Nodes
§3.4(ii) Analytic Functions
§3.4(iii) Partial Derivatives

§3.4(i) Equally-Spaced Nodes

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Keywords:: Lagrange’s formula for equally-spaced nodes, differentiation, numerical
Notes:: See Hildebrand (1974, pp. 85–89). The coefficients $B_{k}^{n}$ are obtained by differentiation of the $A_{k}^{n}$ ; compare (3.4.2).
Permalink:: http://dlmf.nist.gov/3.4.i
See also:: Annotations for §3.4 and Ch.3

The Lagrange $(n+1)$ -point formula is

3.4.1		$hf^{\prime}_{t}=hf^{\prime}(x_{0}+th)=\sum_{k=n_{0}}^{n_{1}}B_{k}^{n}f_{k}+hR^% {\prime}_{n,t},$
$n_{0}<t<n_{1}$ ,
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{\prime}_{n,t}(x)$ : remainder Referenced by: §3.4(i) Permalink: http://dlmf.nist.gov/3.4.E1 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4 and Ch.3

and follows from the differentiated form of (3.3.4). The $B_{k}^{n}$ are the differentiated Lagrangian interpolation coefficients:

3.4.2		$B_{k}^{n}=\ifrac{\mathrm{d}A_{k}^{n}}{\mathrm{d}t},$
ⓘ Defines: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $A_{k}^{n}$ : Lagrangian interpolation coefficients Referenced by: §3.4(i) Permalink: http://dlmf.nist.gov/3.4.E2 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4 and Ch.3

where $A_{k}^{n}$ is as in (3.3.10).

If $f^{(n+2)}(x)$ is continuous on the interval $I$ defined in §3.3(i), then the remainder in (3.4.1) is given by

3.4.3		$hR^{\prime}_{n,t}=\frac{h^{n+1}}{(n+1)!}\left(f^{(n+1)}(\xi_{0})\frac{\mathrm{% d}}{\mathrm{d}t}\prod_{k=n_{0}}^{n_{1}}(t-k)+f^{(n+2)}(\xi_{1})\prod_{k=n_{0}}% ^{n_{1}}(t-k)\right),$
ⓘ Defines: $R^{\prime}_{n,t}(x)$ : remainder (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $!$ : factorial (as in $n!$ ) Permalink: http://dlmf.nist.gov/3.4.E3 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4 and Ch.3

where $\xi_{0}$ and $\xi_{1}\in I$ .

For the values of $n_{0}$ and $n_{1}$ used in the formulas below

3.4.4		$h\left\|R^{\prime}_{n,t}\right\|\leq h^{n+1}\left(c_{n}\left\|f^{(n+2)}(\xi_{1})% \right\|+\frac{1}{n+1}\left\|f^{(n+1)}(\xi_{0})\right\|\right),$
$n_{0}<t<n_{1}$ ,
ⓘ Symbols: $c_{n}$ : bound coefficient and $R^{\prime}_{n,t}(x)$ : remainder Permalink: http://dlmf.nist.gov/3.4.E4 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4 and Ch.3

where $c_{n}$ is defined by (3.3.12), with numerical values as in §3.3(ii).

Two-Point Formula

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See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.5		$hf^{\prime}_{t}=-f_{0}+f_{1}+hR^{\prime}_{1,t},$
$0<t<1$ .
ⓘ Symbols: $R^{\prime}_{n,t}(x)$ : remainder Permalink: http://dlmf.nist.gov/3.4.E5 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

Three-Point Formula

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See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.6		$hf^{\prime}_{t}=-\tfrac{1}{2}(1-2t)f_{-1}-2tf_{0}+\tfrac{1}{2}(1+2t)f_{1}+hR^{% \prime}_{2,t},$
$\left\|t\right\|<1$ .
ⓘ Symbols: $R^{\prime}_{n,t}(x)$ : remainder A&S Ref: 25.3.4 Permalink: http://dlmf.nist.gov/3.4.E6 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

Four-Point Formula

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See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.7		$hf^{\prime}_{t}=\sum_{k=-1}^{2}B_{k}^{3}f_{k}+hR^{\prime}_{3,t},$
$-1<t<2$ ,
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{\prime}_{n,t}(x)$ : remainder A&S Ref: 25.3.5 (modification of) Permalink: http://dlmf.nist.gov/3.4.E7 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

3.4.8	$\displaystyle B_{-1}^{3}$	$\displaystyle=-\tfrac{1}{6}(2-6t+3t^{2}),$
	$\displaystyle B_{0}^{3}$	$\displaystyle=-\tfrac{1}{2}(1+4t-3t^{2}),$
	$\displaystyle B_{1}^{3}$	$\displaystyle=\tfrac{1}{2}(2+2t-3t^{2}),$
	$\displaystyle B_{2}^{3}$	$\displaystyle=-\tfrac{1}{6}(1-3t^{2}).$
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.4.E8 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

Five-Point Formula

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See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.9		$hf^{\prime}_{t}=\sum_{k=-2}^{2}B_{k}^{4}f_{k}+hR^{\prime}_{4,t},$
$\left\|t\right\|<2$ ,
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{\prime}_{n,t}(x)$ : remainder A&S Ref: 25.3.6 (modification of) Permalink: http://dlmf.nist.gov/3.4.E9 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

3.4.10	$\displaystyle B_{-2}^{4}$	$\displaystyle=\tfrac{1}{12}(1-t-3t^{2}+2t^{3}),$
	$\displaystyle B_{-1}^{4}$	$\displaystyle=-\tfrac{1}{6}(4-8t-3t^{2}+4t^{3}),$
	$\displaystyle B_{0}^{4}$	$\displaystyle=-\tfrac{1}{2}t(5-2t^{2}),$
	$\displaystyle B_{1}^{4}$	$\displaystyle=\tfrac{1}{6}(4+8t-3t^{2}-4t^{3}),$
	$\displaystyle B_{2}^{4}$	$\displaystyle=-\tfrac{1}{12}(1+t-3t^{2}-2t^{3}).$
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.4.E10 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

Six-Point Formula

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See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.11		$hf^{\prime}_{t}=\sum_{k=-2}^{3}B_{k}^{5}f_{k}+hR^{\prime}_{5,t},$
$-2<t<3$ ,
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{\prime}_{n,t}(x)$ : remainder Permalink: http://dlmf.nist.gov/3.4.E11 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

3.4.12	$\displaystyle B_{-2}^{5}$	$\displaystyle=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),$
	$\displaystyle B_{-1}^{5}$	$\displaystyle=-\tfrac{1}{24}(12-32t+3t^{2}+16t^{3}-5t^{4}),$
	$\displaystyle B_{0}^{5}$	$\displaystyle=-\tfrac{1}{12}(4+30t-15t^{2}-12t^{3}+5t^{4}),$
	$\displaystyle B_{1}^{5}$	$\displaystyle=\tfrac{1}{12}(12+16t-21t^{2}-8t^{3}+5t^{4}),$
	$\displaystyle B_{2}^{5}$	$\displaystyle=-\tfrac{1}{24}(6+2t-21t^{2}-4t^{3}+5t^{4}),$
	$\displaystyle B_{3}^{5}$	$\displaystyle=\tfrac{1}{120}(4-15t^{2}+5t^{4}).$
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.4.E12 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

Seven-Point Formula

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See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.13		$hf^{\prime}_{t}=\sum_{k=-3}^{3}B_{k}^{6}f_{k}+hR^{\prime}_{6,t},$
$\left\|t\right\|<3$ ,
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{\prime}_{n,t}(x)$ : remainder Permalink: http://dlmf.nist.gov/3.4.E13 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

3.4.14	$\displaystyle B_{-3}^{6}$	$\displaystyle=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),$
	$\displaystyle B_{-2}^{6}$	$\displaystyle=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),$
	$\displaystyle B_{-1}^{6}$	$\displaystyle=-\tfrac{1}{48}(36-72t-39t^{2}+52t^{3}+5t^{4}-6t^{5}),$
	$\displaystyle B_{0}^{6}$	$\displaystyle=-\tfrac{1}{18}t(49-28t^{2}+3t^{4}),$
	$\displaystyle B_{1}^{6}$	$\displaystyle=\tfrac{1}{48}(36+72t-39t^{2}-52t^{3}+5t^{4}+6t^{5}),$
	$\displaystyle B_{2}^{6}$	$\displaystyle=-\tfrac{1}{60}(9+9t-30t^{2}-20t^{3}+5t^{4}+3t^{5}),$
	$\displaystyle B_{3}^{6}$	$\displaystyle=\tfrac{1}{720}(12+8t-45t^{2}-20t^{3}+15t^{4}+6t^{5}).$
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.4.E14 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

Eight-Point Formula

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Keywords:: Lagrange’s formula for equally-spaced nodes, differentiation, numerical
See also:: Annotations for §3.4(i), §3.4 and Ch.3

3.4.15		$hf^{\prime}_{t}=\sum_{k=-3}^{4}B_{k}^{7}f_{k}+hR^{\prime}_{7,t},$
$-3<t<4$ ,
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{\prime}_{n,t}(x)$ : remainder Permalink: http://dlmf.nist.gov/3.4.E15 Encodings: TeX, pMML, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

3.4.16	$\displaystyle B_{-3}^{7}$	$\displaystyle=-\tfrac{1}{5040}(48-56t-168t^{2}+140t^{3}+35t^{4}-42t^{5}+7t^{6}),$
	$\displaystyle B_{-2}^{7}$	$\displaystyle=\tfrac{1}{720}(72-108t-213t^{2}+240t^{3}-10t^{4}-36t^{5}+7t^{6}),$
	$\displaystyle B_{-1}^{7}$	$\displaystyle=-\tfrac{1}{240}(144-360t-48t^{2}+260t^{3}-45t^{4}-30t^{5}+7t^{6}),$
	$\displaystyle B_{0}^{7}$	$\displaystyle=-\tfrac{1}{144}(36+392t-147t^{2}-224t^{3}+70t^{4}+24t^{5}-7t^{6}),$
	$\displaystyle B_{1}^{7}$	$\displaystyle=\tfrac{1}{144}(144+216t-264t^{2}-156t^{3}+85t^{4}+18t^{5}-7t^{6}),$
	$\displaystyle B_{2}^{7}$	$\displaystyle=-\tfrac{1}{240}(72+36t-267t^{2}-80t^{3}+90t^{4}+12t^{5}-7t^{6}),$
	$\displaystyle B_{3}^{7}$	$\displaystyle=\tfrac{1}{720}(48+8t-192t^{2}-20t^{3}+85t^{4}+6t^{5}-7t^{6}),$
	$\displaystyle B_{4}^{7}$	$\displaystyle=-\tfrac{1}{5040}(36-147t^{2}+70t^{4}-7t^{6}).$
ⓘ Symbols: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.4.E16 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png See also: Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

For corresponding formulas for second, third, and fourth derivatives, with $t=0$ , see Collatz (1960, Table III, pp. 538–539). For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5).

§3.4(ii) Analytic Functions

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Keywords:: analytic functions, differentiation, numerical
Permalink:: http://dlmf.nist.gov/3.4.ii
See also:: Annotations for §3.4 and Ch.3

If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))

3.4.17		$\frac{1}{k!}\,f^{(k)}(x_{0})=\frac{1}{2\pi i}\int_{C}\frac{f(\zeta)}{(\zeta-x_% {0})^{k+1}}\,\,\mathrm{d}\zeta,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral and $C$ : simple closed contour Permalink: http://dlmf.nist.gov/3.4.E17 Encodings: TeX, pMML, png See also: Annotations for §3.4(ii), §3.4 and Ch.3

where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$ , and $x_{0}$ is interior to $C$ . Taking $C$ to be a circle of radius $r$ centered at $x_{0}$ , we obtain

3.4.18		$\frac{1}{k!}\,f^{(k)}(x_{0})=\frac{1}{2\pi r^{k}}\int_{0}^{2\pi}f(x_{0}+re^{i% \theta})e^{-ik\theta}\,\mathrm{d}\theta.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral and $r$ : radius Referenced by: §3.4(ii) Permalink: http://dlmf.nist.gov/3.4.E18 Encodings: TeX, pMML, png See also: Annotations for §3.4(ii), §3.4 and Ch.3

The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).

Example

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See also:: Annotations for §3.4(ii), §3.4 and Ch.3

$f(z)=e^{z}$ , $x_{0}=0$ . The integral (3.4.18) becomes

3.4.19		$\frac{1}{k!}=\frac{1}{2\pi r^{k}}\int_{0}^{2\pi}e^{r\cos\theta}\cos\left(r\sin% \theta-k\theta\right)\,\mathrm{d}\theta.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function and $r$ : radius Permalink: http://dlmf.nist.gov/3.4.E19 Encodings: TeX, pMML, png See also: Annotations for §3.4(ii), §3.4(ii), §3.4 and Ch.3

With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals $e^{k}$ at the dominant points $\theta=0,2\pi$ , and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$ . The choice $r=k$ is motivated by saddle-point analysis; see §2.4(iv) or examples in §3.5(ix). As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.

§3.4(iii) Partial Derivatives

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Keywords:: differentiation, numerical, partial derivatives
Permalink:: http://dlmf.nist.gov/3.4.iii
See also:: Annotations for §3.4 and Ch.3

First-Order

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See also:: Annotations for §3.4(iii), §3.4 and Ch.3

For partial derivatives we use the notation $u_{t,s}=u(x_{0}+th,y_{0}+sh)$ .

3.4.20		$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{2h}\,(u_{1,0}-u_{-1,0})+O\left(h^% {2}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.21 Permalink: http://dlmf.nist.gov/3.4.E20 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.21		$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{4h}\,(u_{1,1}-u_{-1,1}+u_{1,-1}-u% _{-1,-1})+O\left(h^{2}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.22 Permalink: http://dlmf.nist.gov/3.4.E21 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

Second-Order

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See also:: Annotations for §3.4(iii), §3.4 and Ch.3

3.4.22		$\frac{{\partial}^{2}u_{0,0}}{{\partial x}^{2}}=\frac{1}{h^{2}}\,(u_{1,0}-2u_{0% ,0}+u_{-1,0})+O\left(h^{2}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.23 Permalink: http://dlmf.nist.gov/3.4.E22 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.23		$\frac{{\partial}^{2}u_{0,0}}{{\partial x}^{2}}=\frac{1}{12h^{2}}\,(-u_{2,0}+16% u_{1,0}-30u_{0,0}+16u_{-1,0}-u_{-2,0})+O\left(h^{4}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.24 Permalink: http://dlmf.nist.gov/3.4.E23 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.24		$\frac{{\partial}^{2}u_{0,0}}{{\partial x}^{2}}=\frac{1}{3h^{2}}\,(u_{1,1}-2u_{% 0,1}+u_{-1,1}+u_{1,0}-2u_{0,0}+u_{-1,0}+u_{1,-1}-2u_{0,-1}+u_{-1,-1})+O\left(h% ^{2}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.25 Permalink: http://dlmf.nist.gov/3.4.E24 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.25		$\frac{\,{\partial}^{2}u_{0,0}}{\,\partial x\,\partial y}=\frac{1}{4h^{2}}\,(u_% {1,1}-u_{1,-1}-u_{-1,1}+u_{-1,-1})+O\left(h^{2}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.26 Permalink: http://dlmf.nist.gov/3.4.E25 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.26		$\frac{\,{\partial}^{2}u_{0,0}}{\,\partial x\,\partial y}=-\frac{1}{2h^{2}}\,(u% _{1,0}+u_{-1,0}+u_{0,1}+u_{0,-1}-2u_{0,0}-u_{1,1}-u_{-1,-1})+O\left(h^{2}% \right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.27 Permalink: http://dlmf.nist.gov/3.4.E26 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

Laplacian

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Keywords:: Laplacian, numerical approximations
See also:: Annotations for §3.4(iii), §3.4 and Ch.3

3.4.27	$\displaystyle\nabla^{2}u$	$\displaystyle=\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{% {\partial y}^{2}}\,.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.30 Permalink: http://dlmf.nist.gov/3.4.E27 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3
3.4.28	$\displaystyle\nabla^{2}u_{0,0}$	$\displaystyle=\frac{1}{h^{2}}\,(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1}-4u_{0,0})+O% \left(h^{2}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding and $u$ : function A&S Ref: 25.3.30 Permalink: http://dlmf.nist.gov/3.4.E28 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.29		$\nabla^{2}u_{0,0}=\frac{1}{12h^{2}}\left(-60u_{0,0}+16(u_{1,0}+u_{0,1}+u_{-1,0% }+u_{0,-1})-(u_{2,0}+u_{0,2}+u_{-2,0}+u_{0,-2})\right)+O\left(h^{4}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding and $u$ : function A&S Ref: 25.3.31 Permalink: http://dlmf.nist.gov/3.4.E29 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

Fourth-Order

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See also:: Annotations for §3.4(iii), §3.4 and Ch.3

3.4.30	$\displaystyle\frac{{\partial}^{4}u_{0,0}}{{\partial x}^{4}}$	$\displaystyle=\frac{1}{h^{4}}\,(u_{2,0}-4u_{1,0}+6u_{0,0}-4u_{-1,0}+u_{-2,0})+% O\left(h^{2}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.28 Permalink: http://dlmf.nist.gov/3.4.E30 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3
3.4.31	$\displaystyle\frac{\,{\partial}^{4}u_{0,0}}{\,{\partial}^{2}x\,{\partial}^{2}y}$	$\displaystyle=\frac{1}{h^{4}}\,(u_{1,1}+u_{-1,1}+u_{1,-1}+u_{-1,-1}-2u_{1,0}-2% u_{-1,0}-2u_{0,1}-2u_{0,-1}+4u_{0,0})+O\left(h^{2}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.29 Permalink: http://dlmf.nist.gov/3.4.E31 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

Biharmonic Operator

ⓘ

Keywords:: biharmonic operator, differentiation, numerical, numerical approximation, partial derivatives
See also:: Annotations for §3.4(iii), §3.4 and Ch.3

3.4.32		$\nabla^{4}u=\frac{{\partial}^{4}u}{{\partial x}^{4}}+2\frac{\,{\partial}^{4}u}% {\,{\partial}^{2}x\,{\partial}^{2}y}+\frac{{\partial}^{4}u}{{\partial y}^{4}}\,.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $u$ : function A&S Ref: 25.3.32 Permalink: http://dlmf.nist.gov/3.4.E32 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.33		$\nabla^{4}u_{0,0}=\frac{1}{h^{4}}\,(20u_{0,0}-8(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,% -1})+2(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})+(u_{0,2}+u_{2,0}+u_{-2,0}+u_{0,-2}% ))+O\left(h^{2}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding and $u$ : function A&S Ref: 25.3.32 Permalink: http://dlmf.nist.gov/3.4.E33 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

3.4.34		$\nabla^{4}u_{0,0}=\frac{1}{6h^{4}}\,(184u_{0,0}-(u_{0,3}+u_{0,-3}+u_{3,0}+u_{-% 3,0})+14(u_{0,2}+u_{0,-2}+u_{2,0}+u_{-2,0})-77(u_{0,1}+u_{0,-1}+u_{1,0}+u_{-1,% 0})+20(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})-(u_{1,2}+u_{2,1}+u_{1,-2}+u_{2,-1}% +u_{-1,2}+u_{-2,1}+u_{-1,-2}+u_{-2,-1}))+O\left(h^{4}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding and $u$ : function A&S Ref: 25.3.33 Permalink: http://dlmf.nist.gov/3.4.E34 Encodings: TeX, pMML, png See also: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives.

For additional formulas involving values of $\nabla^{2}u$ and $\nabla^{4}u$ on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546).

3.3 Interpolation 3.5 Quadrature

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