# §3.3 Interpolation

## §3.3(i) Lagrange Interpolation

The nodes or abscissas $z_{k}$ are real or complex; function values are $f_{k}=f(z_{k})$. Given $n+1$ distinct points $z_{k}$ and $n+1$ corresponding function values $f_{k}$, the Lagrange interpolation polynomial is the unique polynomial $P_{n}(z)$ of degree not exceeding $n$ such that $P_{n}(z_{k})=f_{k}$, $k=0,1,\dots,n$. It is given by

 3.3.1 $P_{n}(z)=\sum_{k=0}^{n}\ell_{k}(z)f_{k}=\omega_{n+1}(z)\sum_{k=0}^{n}\frac{% \widetilde{\omega}_{k}}{z-z_{k}}f_{k}=\ifrac{\sum_{k=0}^{n}\frac{\widetilde{% \omega}_{k}f_{k}}{z-z_{k}}}{\sum_{k=0}^{n}\frac{\widetilde{\omega}_{k}}{z-z_{k% }}},$ ⓘ Symbols: $f_{\NVar{t}}$: function values, $z_{\NVar{k}}$: nodes of function, $\ell_{k}(z)$: Lagrange interpolation polynomial and $\omega_{n+1}(z)$: nodal polynomial A&S Ref: 25.2.1 Referenced by: §3.3(i), Erratum (V1.1.3) for Subsection 3.3(i) Permalink: http://dlmf.nist.gov/3.3.E1 Encodings: TeX, pMML, png See also: Annotations for §3.3(i), §3.3 and Ch.3

where

 3.3.2 $\displaystyle\ell_{k}(z)$ $\displaystyle={\sideset{}{{}^{\prime}}{\prod}_{j=0}^{n}}\frac{z-z_{j}}{z_{k}-z% _{j}},$ $\displaystyle\ell_{k}(z_{j})$ $\displaystyle=\delta_{k,j}.$ ⓘ Defines: $\ell_{k}(z)$: Lagrange interpolation polynomial (locally) Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta and $z_{\NVar{k}}$: nodes of function Permalink: http://dlmf.nist.gov/3.3.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §3.3(i), §3.3 and Ch.3

Here the prime signifies that the factor for $j=k$ is to be omitted, $\delta_{k,j}$ is the Kronecker symbol, and $\omega_{n+1}(z)$ is the nodal polynomial

 3.3.3 $\omega_{n+1}(z)=\prod_{k=0}^{n}(z-z_{k}),$ ⓘ Defines: $\omega_{n+1}(z)$: nodal polynomial (locally) Symbols: $z_{\NVar{k}}$: nodes of function Referenced by: §3.3(iii), Erratum (V1.1.3) for Subsection 3.3(i) Permalink: http://dlmf.nist.gov/3.3.E3 Encodings: TeX, pMML, png See also: Annotations for §3.3(i), §3.3 and Ch.3

and the weights $\widetilde{\omega}_{k}$ are

 3.3.3_1 $\widetilde{\omega}_{k}=\frac{1}{\omega_{n+1}^{\prime}(z_{k})}={\sideset{}{{}^{% \prime}}{\prod}_{j=0}^{n}}\frac{1}{z_{k}-z_{j}}.$ ⓘ Symbols: $z_{\NVar{k}}$: nodes of function and $\omega_{n+1}(z)$: nodal polynomial Referenced by: §3.3(i), Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/3.3.E3_1 Encodings: TeX, pMML, png Addition (effective with 1.1.3): This equation was added. See also: Annotations for §3.3(i), §3.3 and Ch.3

The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. It is a direct consequence of the identity

 3.3.3_2 $1=\sum_{k=0}^{n}\ell_{k}(z)=\omega_{n+1}(z)\sum_{k=0}^{n}\frac{\widetilde{% \omega}_{k}}{z-z_{k}},$ ⓘ Symbols: $z_{\NVar{k}}$: nodes of function, $\ell_{k}(z)$: Lagrange interpolation polynomial and $\omega_{n+1}(z)$: nodal polynomial Referenced by: §3.3(i), Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/3.3.E3_2 Encodings: TeX, pMML, png Addition (effective with 1.1.3): This equation was added. See also: Annotations for §3.3(i), §3.3 and Ch.3

and according to Berrut and Trefethen (2004) it is the most efficient representation of $P_{n}(z)$.

With an error term the Lagrange interpolation formula for $f$ is given by

 3.3.4 $f(z)=\sum_{k=0}^{n}\ell_{k}(z)f_{k}+R_{n}(z).$ ⓘ Symbols: $f(z)$: function, $f_{\NVar{t}}$: function values, $\ell_{k}(z)$: Lagrange interpolation polynomial and $R_{n}(x)$: remainder A&S Ref: 25.2.1 Referenced by: §3.3(ii), §3.4(i) Permalink: http://dlmf.nist.gov/3.3.E4 Encodings: TeX, pMML, png See also: Annotations for §3.3(i), §3.3 and Ch.3

If $f$, $x$ ($=z$), and the nodes $x_{k}$ are real, and $f^{(n+1)}$ is continuous on the smallest closed interval $I$ containing $x,x_{0},x_{1},\dots,x_{n}$, then the error can be expressed

 3.3.5 $R_{n}(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\,\omega_{n+1}(x),$ ⓘ Defines: $R_{n}(x)$: remainder (locally) Symbols: $!$: factorial (as in $n!$), $f(z)$: function and $\omega_{n+1}(z)$: nodal polynomial A&S Ref: 25.2.3 (modification of) Permalink: http://dlmf.nist.gov/3.3.E5 Encodings: TeX, pMML, png See also: Annotations for §3.3(i), §3.3 and Ch.3

for some $\xi\in I$. If $f$ is analytic in a simply-connected domain $D$1.13(i)), then for $z\in D$,

 3.3.6 $R_{n}(z)=\frac{\omega_{n+1}(z)}{2\pi\mathrm{i}}\int_{C}\frac{f(\zeta)}{(\zeta-% z)\omega_{n+1}(\zeta)}\,\mathrm{d}\zeta,$

where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,z_{1},z_{2},\dots,z_{n}$.

## §3.3(ii) Lagrange Interpolation with Equally-Spaced Nodes

The $(n+1)$-point formula (3.3.4) can be written in the form

 3.3.7 $f_{t}=f(x_{0}+th)=\sum_{k=n_{0}}^{n_{1}}A_{k}^{n}f_{k}+R_{n,t},$ $n_{0}, ⓘ Symbols: $f(z)$: function, $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.6 (modification of) Permalink: http://dlmf.nist.gov/3.3.E7 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3 and Ch.3

where the nodes $x_{k}=x_{0}+kh$ ($h>0$) and function $f$ are real,

 3.3.8 $\displaystyle n_{0}$ $\displaystyle=-\tfrac{1}{2}(n-\sigma),$ $\displaystyle n_{1}$ $\displaystyle=\tfrac{1}{2}(n+\sigma),$ ⓘ A&S Ref: 25.2.6 (modification of) Permalink: http://dlmf.nist.gov/3.3.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §3.3(ii), §3.3 and Ch.3
 3.3.9 $\sigma=\tfrac{1}{2}(1-(-1)^{n}),$ ⓘ Permalink: http://dlmf.nist.gov/3.3.E9 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3 and Ch.3

and $A_{k}^{n}$ are the Lagrangian interpolation coefficients defined by

 3.3.10 $A_{k}^{n}=\frac{(-1)^{n_{1}+k}}{(k-n_{0})!\,(n_{1}-k)!(t-k)}\,\prod_{m=n_{0}}^% {n_{1}}(t-m).$ ⓘ Defines: $A_{k}^{n}$: Lagrangian interpolation coefficients (locally) Symbols: $!$: factorial (as in $n!$) A&S Ref: 25.2.7 (modification of) Referenced by: §3.4(i) Permalink: http://dlmf.nist.gov/3.3.E10 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3 and Ch.3

The remainder is given by

 3.3.11 $R_{n,t}=R_{n}(x_{0}+th)=\frac{h^{n+1}}{(n+1)!}f^{(n+1)}(\xi)\prod_{k=n_{0}}^{n% _{1}}(t-k),$ ⓘ Defines: $R_{n,t}(x)$: remainder (locally) Symbols: $!$: factorial (as in $n!$) and $f(z)$: function A&S Ref: 25.2.8 (modification of) Permalink: http://dlmf.nist.gov/3.3.E11 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3 and Ch.3

where $\xi$ is as in §3.3(i).

Let $c_{n}$ be defined by

 3.3.12 $c_{n}=\frac{1}{(n+1)!}\max\prod_{k=n_{0}}^{n_{1}}\left|t-k\right|,$ ⓘ Defines: $c_{n}$: bound coefficient (locally) Symbols: $!$: factorial (as in $n!$) Referenced by: §3.4(i) Permalink: http://dlmf.nist.gov/3.3.E12 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3 and Ch.3

where the maximum is taken over $t$-intervals given in the formulas below. Then for these $t$-intervals,

 3.3.13 $\left|R_{n,t}\right|\leq c_{n}h^{n+1}\left|f^{(n+1)}(\xi)\right|.$ ⓘ Symbols: $f(z)$: function, $R_{n,t}(x)$: remainder and $c_{n}$: bound coefficient Permalink: http://dlmf.nist.gov/3.3.E13 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3 and Ch.3

### Linear Interpolation

 3.3.14 $\displaystyle f_{t}$ $\displaystyle=(1-t)f_{0}+tf_{1}+R_{1,t},$ $0, ⓘ Symbols: $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.9 Permalink: http://dlmf.nist.gov/3.3.E14 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3 3.3.15 $\displaystyle c_{1}$ $\displaystyle=\tfrac{1}{8},$ $0. ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.10 (modification of) Permalink: http://dlmf.nist.gov/3.3.E15 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

### Three-Point Formula

 3.3.16 $f_{t}=\sum_{k=-1}^{1}A_{k}^{2}f_{k}+R_{2,t},$ $\left|t\right|<1$, ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.11 (modification of) Permalink: http://dlmf.nist.gov/3.3.E16 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.17 $\displaystyle A_{-1}^{2}$ $\displaystyle=\tfrac{1}{2}t(t-1),$ $\displaystyle A_{0}^{2}$ $\displaystyle=1-t^{2},$ $\displaystyle A_{1}^{2}$ $\displaystyle=\tfrac{1}{2}t(t+1),$ ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.3.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.18 $c_{2}=1/(9\sqrt{3})=0.0641\ldots,$ $\left|t\right|<1$. ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.12 (modification of) Permalink: http://dlmf.nist.gov/3.3.E18 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

### Four-Point Formula

 3.3.19 $f_{t}=\sum_{k=-1}^{2}A_{k}^{3}f_{k}+R_{3,t},$ $-1, ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.13 (modification of) Permalink: http://dlmf.nist.gov/3.3.E19 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.20 $\displaystyle A_{-1}^{3}$ $\displaystyle=-\tfrac{1}{6}t(t-1)(t-2),$ $\displaystyle A_{0}^{3}$ $\displaystyle=\tfrac{1}{2}(t^{2}-1)(t-2),$ $\displaystyle A_{1}^{3}$ $\displaystyle=-\tfrac{1}{2}t(t+1)(t-2),$ $\displaystyle A_{2}^{3}$ $\displaystyle=\tfrac{1}{6}t(t^{2}-1),$ ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.3.E20 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.21 $c_{3}=\begin{cases}\tfrac{3}{128}=0.0234\ldots,&0 ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.14 (modification of) Permalink: http://dlmf.nist.gov/3.3.E21 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

### Five-Point Formula

 3.3.22 $f_{t}=\sum_{k=-2}^{2}A_{k}^{4}f_{k}+R_{4,t},$ $\left|t\right|<2$, ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.15 (modification of) Permalink: http://dlmf.nist.gov/3.3.E22 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.23 $\displaystyle A_{-2}^{4}$ $\displaystyle=\tfrac{1}{24}t(t^{2}-1)(t-2),$ $\displaystyle A_{-1}^{4}$ $\displaystyle=-\tfrac{1}{6}t(t-1)(t^{2}-4),$ $\displaystyle A_{0}^{4}$ $\displaystyle=\tfrac{1}{4}(t^{2}-1)(t^{2}-4),$ $\displaystyle A_{1}^{4}$ $\displaystyle=-\tfrac{1}{6}t(t+1)(t^{2}-4),$ $\displaystyle A_{2}^{4}$ $\displaystyle=\tfrac{1}{24}t(t^{2}-1)(t+2),$ ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients A&S Ref: 25.2.15 (modification of) Permalink: http://dlmf.nist.gov/3.3.E23 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.24 $c_{4}=\begin{cases}0.0118\ldots,&\left|t\right|<1,\\ 0.0302\ldots,&1<\left|t\right|<2.\\ \end{cases}$ ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.16 (modification of) Permalink: http://dlmf.nist.gov/3.3.E24 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

### Six-Point Formula

 3.3.25 $f_{t}=\sum_{k=-2}^{3}A_{k}^{5}f_{k}+R_{5,t},$ $-2, ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.17 (modification of) Permalink: http://dlmf.nist.gov/3.3.E25 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.26 $\displaystyle A_{-2}^{5}$ $\displaystyle=-\tfrac{1}{120}t(t^{2}-1)(t-2)(t-3),$ $\displaystyle A_{-1}^{5}$ $\displaystyle=\tfrac{1}{24}t(t-1)(t^{2}-4)(t-3),$ $\displaystyle A_{0}^{5}$ $\displaystyle=-\tfrac{1}{12}(t^{2}-1)(t^{2}-4)(t-3),$ $\displaystyle A_{1}^{5}$ $\displaystyle=\tfrac{1}{12}t(t+1)(t^{2}-4)(t-3),$ $\displaystyle A_{2}^{5}$ $\displaystyle=-\tfrac{1}{24}t(t^{2}-1)(t+2)(t-3),$ $\displaystyle A_{3}^{5}$ $\displaystyle=\tfrac{1}{120}t(t^{2}-1)(t^{2}-4),$ ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients A&S Ref: 25.2.17 (modification of) Permalink: http://dlmf.nist.gov/3.3.E26 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.27 $c_{5}=\begin{cases}0.00488\ldots,&0 ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.18 (modification of) Permalink: http://dlmf.nist.gov/3.3.E27 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

### Seven-Point Formula

 3.3.28 $f_{t}=\sum_{k=-3}^{3}A_{k}^{6}f_{k}+R_{6,t},$ $\left|t\right|<3$, ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.19 (modification of) Permalink: http://dlmf.nist.gov/3.3.E28 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.29 $\displaystyle A_{-3}^{6}$ $\displaystyle=\tfrac{1}{720}t(t^{2}-1)(t-3)(t^{2}-4),$ $\displaystyle A_{-2}^{6}$ $\displaystyle=-\tfrac{1}{120}t(t^{2}-1)(t-2)(t^{2}-9),$ $\displaystyle A_{-1}^{6}$ $\displaystyle=\tfrac{1}{48}t(t-1)(t^{2}-4)(t^{2}-9),$ $\displaystyle A_{0}^{6}$ $\displaystyle=-\tfrac{1}{36}(t^{2}-1)(t^{2}-4)(t^{2}-9),$ $\displaystyle A_{1}^{6}$ $\displaystyle=\tfrac{1}{48}t(t+1)(t^{2}-4)(t^{2}-9),$ $\displaystyle A_{2}^{6}$ $\displaystyle=-\tfrac{1}{120}t(t^{2}-1)(t+2)(t^{2}-9),$ $\displaystyle A_{3}^{6}$ $\displaystyle=\tfrac{1}{720}t(t^{2}-1)(t+3)(t^{2}-4),$ ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients A&S Ref: 25.2.19 (modification of) Permalink: http://dlmf.nist.gov/3.3.E29 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 §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.30 $c_{6}=\begin{cases}0.00245\ldots,&\left|t\right|<1,\\ 0.00459\ldots,&1<\left|t\right|<2,\\ 0.0190\ldots,&2<\left|t\right|<3.\\ \end{cases}$ ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.20 (modification of) Permalink: http://dlmf.nist.gov/3.3.E30 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

### Eight-Point Formula

 3.3.31 $f_{t}=\sum_{k=-3}^{4}A_{k}^{7}f_{k}+R_{7,t},$ $-3, ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients, $R_{n,t}(x)$: remainder and $f_{\NVar{t}}$: function values A&S Ref: 25.2.21 (modification of) Permalink: http://dlmf.nist.gov/3.3.E31 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.32 $\displaystyle A_{-3}^{7}$ $\displaystyle=-\tfrac{1}{5040}t(t^{2}-1)(t-3)(t-4)(t^{2}-4),$ $\displaystyle A_{-2}^{7}$ $\displaystyle=\tfrac{1}{720}t(t^{2}-1)(t-2)(t-4)(t^{2}-9),$ $\displaystyle A_{-1}^{7}$ $\displaystyle=-\tfrac{1}{240}t(t-1)(t-4)(t^{2}-4)(t^{2}-9),$ $\displaystyle A_{0}^{7}$ $\displaystyle=\tfrac{1}{144}(t^{2}-1)(t-4)(t^{2}-4)(t^{2}-9),$ $\displaystyle A_{1}^{7}$ $\displaystyle=-\tfrac{1}{144}t(t+1)(t-4)(t^{2}-4)(t^{2}-9),$ $\displaystyle A_{2}^{7}$ $\displaystyle=\tfrac{1}{240}t(t^{2}-1)(t+2)(t-4)(t^{2}-9),$ $\displaystyle A_{3}^{7}$ $\displaystyle=-\tfrac{1}{720}t(t^{2}-1)(t+3)(t-4)(t^{2}-4),$ $\displaystyle A_{4}^{7}$ $\displaystyle=\tfrac{1}{5040}t(t^{2}-1)(t^{2}-4)(t^{2}-9),$ ⓘ Symbols: $A_{k}^{n}$: Lagrangian interpolation coefficients Permalink: http://dlmf.nist.gov/3.3.E32 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 §3.3(ii), §3.3(ii), §3.3 and Ch.3
 3.3.33 $c_{7}=\begin{cases}0.00106\ldots,&0 ⓘ Symbols: $c_{n}$: bound coefficient A&S Ref: 25.2.22 (modification of) Permalink: http://dlmf.nist.gov/3.3.E33 Encodings: TeX, pMML, png See also: Annotations for §3.3(ii), §3.3(ii), §3.3 and Ch.3

## §3.3(iii) Divided Differences

The divided differences of $f$ relative to a sequence of distinct points $z_{0},z_{1},z_{2},\dots$ are defined by

 3.3.34 $\displaystyle\left[z_{0}\right]f$ $\displaystyle=f_{0},$ $\displaystyle\left[z_{0},z_{1}\right]f$ $\displaystyle=(\left[z_{1}\right]f-\left[z_{0}\right]f)/(z_{1}-z_{0}),$ $\displaystyle\left[z_{0},z_{1},z_{2}\right]f$ $\displaystyle=(\left[z_{1},z_{2}\right]f-\left[z_{0},z_{1}\right]f)/(z_{2}-z_{% 0}),$ ⓘ Defines: $\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$: divided difference Symbols: $f(z)$: function, $f_{\NVar{t}}$: function values and $z_{\NVar{k}}$: nodes of function A&S Ref: 25.1.4 Referenced by: Erratum (V1.1.2) for Equation (3.3.34) Permalink: http://dlmf.nist.gov/3.3.E34 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png Errata (effective with 1.1.2): The leading divided difference operators were previously omitted from these formulas, due to programming error. Reported 2021-06-01 by Nico Temme See also: Annotations for §3.3(iii), §3.3 and Ch.3

and so on. Explicitly, the divided difference of order $n$ is given by

 3.3.35 $\left[z_{0},z_{1},\dots,z_{n}\right]f=\sum_{k=0}^{n}\left(\ifrac{f(z_{k})}{% \prod_{\begin{subarray}{c}0\leq j\leq n\\ j\neq k\end{subarray}}(z_{k}-z_{j})}\right).$ ⓘ Symbols: $\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$: divided difference, $f(z)$: function and $z_{\NVar{k}}$: nodes of function A&S Ref: 25.1.5 (in slightly different form) Permalink: http://dlmf.nist.gov/3.3.E35 Encodings: TeX, pMML, png See also: Annotations for §3.3(iii), §3.3 and Ch.3

If $f$ and the $z_{k}$ ($=x_{k}$) are real, and $f$ is $n$ times continuously differentiable on a closed interval containing the $x_{k}$, then

 3.3.36 $\left[x_{0},x_{1},\dots,x_{n}\right]f=\frac{f^{(n)}(\xi)}{n!}$ ⓘ Symbols: $\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$: divided difference, $!$: factorial (as in $n!$) and $f(z)$: function A&S Ref: 25.1.10 Permalink: http://dlmf.nist.gov/3.3.E36 Encodings: TeX, pMML, png See also: Annotations for §3.3(iii), §3.3 and Ch.3

and again $\xi$ is as in §3.3(i). If $f$ is analytic in a simply-connected domain $D$, then for $z\in{D}$,

 3.3.37 $\left[z_{0},z_{1},\dots,z_{n}\right]f=\frac{1}{2\pi\mathrm{i}}\int_{C}\frac{f(% \zeta)}{\omega_{n+1}(\zeta)}\,\mathrm{d}\zeta,$

where $\omega_{n+1}(\zeta)$ is given by (3.3.3), and $C$ is a simple closed contour in ${D}$ described in the positive rotational sense and enclosing $z_{0},z_{1},\dots,z_{n}$.

## §3.3(iv) Newton’s Interpolation Formula

This represents the Lagrange interpolation polynomial in terms of divided differences:

 3.3.38 $f(z)=\left[z_{0}\right]f+(z-z_{0})\left[z_{0},z_{1}\right]f+(z-z_{0})(z-z_{1})% \left[z_{0},z_{1},z_{2}\right]f+\cdots+(z-z_{0})(z-z_{1})\cdots(z-z_{n-1})% \left[z_{0},z_{1},\dots,z_{n}\right]f+R_{n}(z).$ ⓘ Symbols: $\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$: divided difference, $f(z)$: function, $z_{\NVar{k}}$: nodes of function and $R_{n}(x)$: remainder A&S Ref: 25.2.26 Referenced by: §3.3(v), §3.3(v), §3.3(iv) Permalink: http://dlmf.nist.gov/3.3.E38 Encodings: TeX, pMML, png See also: Annotations for §3.3(iv), §3.3 and Ch.3

The interpolation error $R_{n}(z)$ is as in §3.3(i). 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 $\left[z_{0},z_{1},\dots,z_{n+1}\right]f$ to (3.3.38), plus the computation of this divided difference. Another advantage is its robustness with respect to confluence of the set of points $z_{0},z_{1},\dots,z_{n}$. For example, for $k+1$ coincident points the limiting form is given by $\left[z_{0},z_{0},\dots,z_{0}\right]f=f^{(k)}(z_{0})/k!$.

## §3.3(v) Inverse Interpolation

In this method we interchange the roles of the points $z_{k}$ and the function values $f_{k}$. It can be used for solving a nonlinear scalar equation $f(z)=0$ approximately. Another approach is to combine the methods of §3.8 with direct interpolation and §3.4.

### Example

To compute the first negative zero $a_{1}=-2.33810\;7410\ldots$ of the Airy function $f(x)=\operatorname{Ai}\left(x\right)$9.2). The inverse interpolation polynomial is given by

 3.3.39 $x(f)=\left[f_{0}\right]x+(f-f_{0})\left[f_{0},f_{1}\right]x+(f-f_{0})(f-f_{1})% \left[f_{0},f_{1},f_{2}\right]x;$ ⓘ Symbols: $\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$: divided difference, $f(z)$: function and $f_{\NVar{t}}$: function values Permalink: http://dlmf.nist.gov/3.3.E39 Encodings: TeX, pMML, png See also: Annotations for §3.3(v), §3.3(v), §3.3 and Ch.3

compare (3.3.38). With $x_{0}=-2.2$, $x_{1}=-2.3$, $x_{2}=-2.4$, we obtain

 3.3.40 $x=-2.2+1.44011\;1973(f-0.09614\;53780)+0.08865\;85832\*(f-0.09614\;53780)(f-0.% 02670\;63331),$ ⓘ Symbols: $f(z)$: function Permalink: http://dlmf.nist.gov/3.3.E40 Encodings: TeX, pMML, png See also: Annotations for §3.3(v), §3.3(v), §3.3 and Ch.3

and with $f=0$ we find that $x=-2.33823\;2462$, with 4 correct digits. By using this approximation to $x$ as a new point, $x_{3}=x$, and evaluating $\left[f_{0},f_{1},f_{2},f_{3}\right]x=1.12388\;6190$, we find that $x=-2.33810\;7409$, with 9 correct digits.

For comparison, we use Newton’s interpolation formula (3.3.38)

 3.3.41 $f(x)=0.09614\;53780+0.69439\;04495(x+2.1)-0.03007\;14275(x+2.2)(x+2.3),$ ⓘ Symbols: $f(z)$: function Permalink: http://dlmf.nist.gov/3.3.E41 Encodings: TeX, pMML, png See also: Annotations for §3.3(v), §3.3(v), §3.3 and Ch.3

with the derivative

 3.3.42 $f^{\prime}(x)=0.55906\;90257-0.06014\;28550x,$ ⓘ Symbols: $f(z)$: function Permalink: http://dlmf.nist.gov/3.3.E42 Encodings: TeX, pMML, png See also: Annotations for §3.3(v), §3.3(v), §3.3 and Ch.3

and compute an approximation to $a_{1}$ by using Newton’s rule (§3.8(ii)) with starting value $x=-2.5$. This gives the new point $x_{3}=-2.33934\;0514$. Then by using $x_{3}$ in Newton’s interpolation formula, evaluating $\left[x_{0},x_{1},x_{2},x_{3}\right]f=-0.26608\;28233$ 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.

## §3.3(vi) Other Interpolation Methods

For Hermite interpolation, trigonometric interpolation, spline interpolation, rational interpolation (by using continued fractions), interpolation based on Chebyshev points, and bivariate interpolation, see Bulirsch and Rutishauser (1968), Davis (1975, pp. 27–31), and Mason and Handscomb (2003, Chapter 6). These references also describe convergence properties of the interpolation formulas.

For interpolation of a bounded function $f$ on $\mathbb{R}$ the cardinal function of $f$ is defined by

 3.3.43 $C(f,h)(x)=\sum_{k=-\infty}^{\infty}f(kh)S(k,h)(x),$ ⓘ Defines: $C(f,h)$: cardinal function of $f$ (locally) Symbols: $f(z)$: function and $S(k,h)(x)$: Sinc function Permalink: http://dlmf.nist.gov/3.3.E43 Encodings: TeX, pMML, png See also: Annotations for §3.3(vi), §3.3 and Ch.3

where

 3.3.44 $S(k,h)(x)=\frac{\sin\left(\pi(x-kh)/h\right)}{\pi(x-kh)/h},$ ⓘ Defines: $S(k,h)(x)$: Sinc function (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $\sin\NVar{z}$: sine function Permalink: http://dlmf.nist.gov/3.3.E44 Encodings: TeX, pMML, png See also: Annotations for §3.3(vi), §3.3 and Ch.3

is called the Sinc function. For theory and applications see Stenger (1993, Chapter 3).