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Lagrange polynomial

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a> (the orthogonal basis of function space).</div>

In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, the Lagrange interpolating polynomial is the unique <a href="Polynomial" title="Polynomial">polynomial</a> of lowest <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> that <a href="Polynomial_interpolation" title="Polynomial interpolation">interpolates</a> a given set of data.

Given a data set of <a href="Graph_of_a_function" title="Graph of a function">coordinate pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{j},y_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{j},y_{j})}</annotation>

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</math><img src="./dec8853489ad3dc2385adf0a049a75aed4dc0987.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.132ex; height:3.009ex;" alt="{\displaystyle (x_{j},y_{j})}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq j\leq k,}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq j\leq k,}</annotation>

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</math><img src="./29fd8346c0d58f0ac3a24b3d13902097a16c878b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.176ex; height:2.509ex;" alt="{\displaystyle 0\leq j\leq k,}" loading="lazy"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

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</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy"> are called nodes and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{j}}</annotation>

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</math><img src="./f8df4e372390588acb968986cfc388e50b930b3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.343ex;" alt="{\displaystyle y_{j}}" loading="lazy"> are called values. The Lagrange polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

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<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

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</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy"> has degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \leq k}">

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<annotation encoding="application/x-tex">{\textstyle \leq k}</annotation>

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</math><img src="./c0045683db5d3864f162fab73dcf63eb66298aae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.665ex; height:2.343ex;" alt="{\textstyle \leq k}" loading="lazy"> and assumes each value at the corresponding node, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x_{j})=y_{j}.}">

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<annotation encoding="application/x-tex">{\displaystyle L(x_{j})=y_{j}.}</annotation>

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</math><img src="./9aab9ebd62655ba529e7a99ecab6d4242621d6be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.426ex; height:3.009ex;" alt="{\displaystyle L(x_{j})=y_{j}.}" loading="lazy">

Although named after <a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>, who published it in 1795,<a href="#cite_note-1">[1]</a> the method was first discovered in 1779 by <a href="Edward_Waring" title="Edward Waring">Edward Waring</a>.<a href="#cite_note-2">[2]</a> It is also an easy consequence of a formula published in 1783 by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>.<a href="#cite_note-3">[3]</a>

Uses of Lagrange polynomials include the <a href="Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes method</a> of <a href="Numerical_integration" title="Numerical integration">numerical integration</a>, <a href="Shamir's_Secret_Sharing" class="mw-redirect" title="Shamir's Secret Sharing">Shamir's secret sharing scheme</a> in <a href="Cryptography" title="Cryptography">cryptography</a>, and <a href="Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed–Solomon error correction</a> in <a href="Coding_theory" title="Coding theory">coding theory</a>.

For equispaced nodes, Lagrange interpolation is susceptible to <a href="Runge's_phenomenon" title="Runge's phenomenon">Runge's phenomenon</a> of large oscillation.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Given a set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k+1}">

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<annotation encoding="application/x-tex">{\textstyle k+1}</annotation>

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</math><img src="./a2329c3536d3f8e618dfb1984d2bf49a0c9463cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\textstyle k+1}" loading="lazy"> nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{0},x_{1},\ldots ,x_{k}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x_{0},x_{1},\ldots ,x_{k}\}}</annotation>

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</math><img src="./d34cc561efd002ddb22980f52d61a25622b7e99e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.723ex; height:2.843ex;" alt="{\displaystyle \{x_{0},x_{1},\ldots ,x_{k}\}}" loading="lazy">, which must all be distinct, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}\neq x_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j}\neq x_{m}}</annotation>

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</math><img src="./0ba81c2f9bb5bf2b4ffbc1bf3da5e8c0e6404865.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.343ex; height:2.843ex;" alt="{\displaystyle x_{j}\neq x_{m}}" loading="lazy"> for indices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\neq m}">

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</math><img src="./479ae5bb405f30d225d22437cefba2c0aef24cc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:6.124ex; height:2.676ex;" alt="{\displaystyle j\neq m}" loading="lazy">, the Lagrange basis for polynomials of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \leq k}">

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<annotation encoding="application/x-tex">{\textstyle \leq k}</annotation>

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</math><img src="./c0045683db5d3864f162fab73dcf63eb66298aae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.665ex; height:2.343ex;" alt="{\textstyle \leq k}" loading="lazy"> for those nodes is the set of polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{\ell _{0}(x),\ell _{1}(x),\ldots ,\ell _{k}(x)\}}">

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<annotation encoding="application/x-tex">{\textstyle \{\ell _{0}(x),\ell _{1}(x),\ldots ,\ell _{k}(x)\}}</annotation>

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</math><img src="./64351193b47fbc2a09d806eb136330dab2ea41dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.06ex; height:2.843ex;" alt="{\textstyle \{\ell _{0}(x),\ell _{1}(x),\ldots ,\ell _{k}(x)\}}" loading="lazy"> each of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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<annotation encoding="application/x-tex">{\textstyle k}</annotation>

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy"> which take values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell _{j}(x_{m})=0}">

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<annotation encoding="application/x-tex">{\textstyle \ell _{j}(x_{m})=0}</annotation>

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</math><img src="./e2d230e61a31eabb57e3940240edcd783846d350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.954ex; height:3.009ex;" alt="{\textstyle \ell _{j}(x_{m})=0}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m\neq j}">

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<annotation encoding="application/x-tex">{\textstyle m\neq j}</annotation>

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</math><img src="./5468d4ebe2fafa03daf6ec106e587c6ed7d03500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.097ex; height:2.676ex;" alt="{\textstyle m\neq j}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell _{j}(x_{j})=1}">

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<annotation encoding="application/x-tex">{\textstyle \ell _{j}(x_{j})=1}</annotation>

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</math><img src="./7df6e2e80a5e795e85aef663376aba4023181918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.189ex; height:3.009ex;" alt="{\textstyle \ell _{j}(x_{j})=1}" loading="lazy">. Using the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a> this can be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell _{j}(x_{m})=\delta _{jm}.}">

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<annotation encoding="application/x-tex">{\textstyle \ell _{j}(x_{m})=\delta _{jm}.}</annotation>

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</math><img src="./2bc422d2bcb3fd08de6918a4b63c0c70d372d7c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.824ex; height:3.009ex;" alt="{\textstyle \ell _{j}(x_{m})=\delta _{jm}.}" loading="lazy"> Each basis polynomial can be explicitly described by the product:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell _{j}(x)&={\frac {(x-x_{0})}{(x_{j}-x_{0})}}\cdots {\frac {(x-x_{j-1})}{(x_{j}-x_{j-1})}}{\frac {(x-x_{j+1})}{(x_{j}-x_{j+1})}}\cdots {\frac {(x-x_{k})}{(x_{j}-x_{k})}}\\[8mu]&=\prod _{\begin{smallmatrix}0\leq m\leq k\\m\neq j\end{smallmatrix}}{\frac {x-x_{m}}{x_{j}-x_{m}}}{\vphantom {\Bigg |}}.\end{aligned}}}</annotation>

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Notice that the numerator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{m\neq j}(x-x_{m})}">

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<annotation encoding="application/x-tex">{\textstyle \prod _{m\neq j}(x-x_{m})}</annotation>

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</math><img src="./ed449492d57bf52a369b43c9ae44239a75b5438b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.809ex; height:3.343ex;" alt="{\textstyle \prod _{m\neq j}(x-x_{m})}" loading="lazy"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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<annotation encoding="application/x-tex">{\textstyle k}</annotation>

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy"> roots at the nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{x_{m}\}_{m\neq j}}">

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<annotation encoding="application/x-tex">{\textstyle \{x_{m}\}_{m\neq j}}</annotation>

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</math><img src="./e189ab93d1f72038db673233aa3d7bb5ff3b0072.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.961ex; height:3.009ex;" alt="{\textstyle \{x_{m}\}_{m\neq j}}" loading="lazy"> while the denominator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{m\neq j}(x_{j}-x_{m})}">

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<annotation encoding="application/x-tex">{\textstyle \prod _{m\neq j}(x_{j}-x_{m})}</annotation>

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</math><img src="./e5bb9bf316807a3f1905ff4feb5c5d452b3a44cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.719ex; height:3.343ex;" alt="{\textstyle \prod _{m\neq j}(x_{j}-x_{m})}" loading="lazy"> scales the resulting polynomial so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell _{j}(x_{j})=1.}">

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<annotation encoding="application/x-tex">{\textstyle \ell _{j}(x_{j})=1.}</annotation>

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</math><img src="./33467680b9da16c42de8ada5cb0083cb91010bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.836ex; height:3.009ex;" alt="{\textstyle \ell _{j}(x_{j})=1.}" loading="lazy">

The Lagrange interpolating polynomial for those nodes through the corresponding values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{0},y_{1},\ldots ,y_{k}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{y_{0},y_{1},\ldots ,y_{k}\}}</annotation>

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</math><img src="./68d409cfce7fd397c65802b1b9d62a2983d96ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.152ex; height:2.843ex;" alt="{\displaystyle \{y_{0},y_{1},\ldots ,y_{k}\}}" loading="lazy"> is the <a href="Linear_combination" title="Linear combination">linear combination</a>:

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<annotation encoding="application/x-tex">{\displaystyle L(x)=\sum _{j=0}^{k}y_{j}\ell _{j}(x).}</annotation>

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Each basis polynomial has degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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<annotation encoding="application/x-tex">{\textstyle k}</annotation>

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy">, so the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle L(x)}">

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<annotation encoding="application/x-tex">{\textstyle L(x)}</annotation>

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</math><img src="./406b3c515e637dee5fff055dbae6345d35106619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\textstyle L(x)}" loading="lazy"> has degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \leq k}">

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<annotation encoding="application/x-tex">{\textstyle \leq k}</annotation>

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</math><img src="./c0045683db5d3864f162fab73dcf63eb66298aae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.665ex; height:2.343ex;" alt="{\textstyle \leq k}" loading="lazy">, and it interpolates the data because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle L(x_{m})=\sum _{j=0}^{k}y_{j}\ell _{j}(x_{m})=\sum _{j=0}^{k}y_{j}\delta _{mj}=y_{m}.}">

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<annotation encoding="application/x-tex">{\textstyle L(x_{m})=\sum _{j=0}^{k}y_{j}\ell _{j}(x_{m})=\sum _{j=0}^{k}y_{j}\delta _{mj}=y_{m}.}</annotation>

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</math><img src="./487d9b213d62cdac2088b138c7f997bd7250cb3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.032ex; height:3.843ex;" alt="{\textstyle L(x_{m})=\sum _{j=0}^{k}y_{j}\ell _{j}(x_{m})=\sum _{j=0}^{k}y_{j}\delta _{mj}=y_{m}.}" loading="lazy">

The interpolating polynomial is unique. Proof: assume the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M(x)}">

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<annotation encoding="application/x-tex">{\textstyle M(x)}</annotation>

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</math><img src="./7145f62275c40a05527ff6ed6015baf3cd919494.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:2.843ex;" alt="{\textstyle M(x)}" loading="lazy"> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \leq k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \leq k}</annotation>

</semantics>

</math><img src="./c0045683db5d3864f162fab73dcf63eb66298aae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.665ex; height:2.343ex;" alt="{\textstyle \leq k}" loading="lazy"> interpolates the data. Then the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M(x)-L(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle M(x)-L(x)}</annotation>

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</math><img src="./3b49ecc17ea7cd5251e28c31f818aa58d9dd6683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.143ex; height:2.843ex;" alt="{\textstyle M(x)-L(x)}" loading="lazy"> is zero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k+1}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle k+1}</annotation>

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</math><img src="./a2329c3536d3f8e618dfb1984d2bf49a0c9463cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\textstyle k+1}" loading="lazy"> distinct nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{x_{0},x_{1},\ldots ,x_{k}\}.}">

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<annotation encoding="application/x-tex">{\textstyle \{x_{0},x_{1},\ldots ,x_{k}\}.}</annotation>

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</math><img src="./59fcc659012a45258162c7a6c94ae2a5f4f5181a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.37ex; height:2.843ex;" alt="{\textstyle \{x_{0},x_{1},\ldots ,x_{k}\}.}" loading="lazy"> But the only polynomial of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \leq k}">

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<annotation encoding="application/x-tex">{\textstyle \leq k}</annotation>

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</math><img src="./c0045683db5d3864f162fab73dcf63eb66298aae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.665ex; height:2.343ex;" alt="{\textstyle \leq k}" loading="lazy"> with more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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<annotation encoding="application/x-tex">{\textstyle k}</annotation>

</semantics>

</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy"> roots is the constant zero function, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M(x)-L(x)=0,}">

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<annotation encoding="application/x-tex">{\textstyle M(x)-L(x)=0,}</annotation>

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</math><img src="./c72c07b9de11193c5583c37e3dbcf40c4a5a6063.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.051ex; height:2.843ex;" alt="{\textstyle M(x)-L(x)=0,}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M(x)=L(x).}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle M(x)=L(x).}</annotation>

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</math><img src="./82376ab5eb72c2f7364b5152ac1a7ec55f6bf022.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.048ex; height:2.843ex;" alt="{\textstyle M(x)=L(x).}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Barycentric_form">Barycentric form</h2></div>

Each Lagrange basis polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell _{j}(x)}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \ell _{j}(x)}</annotation>

</semantics>

</math><img src="./f53e75a1ee792cc0c2e87806f57105813afc22a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.018ex; height:3.009ex;" alt="{\textstyle \ell _{j}(x)}" loading="lazy"> can be rewritten as the product of three parts, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell (x)=\prod _{m}(x-x_{m})}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \ell (x)=\prod _{m}(x-x_{m})}</annotation>

</semantics>

</math><img src="./e098b5ab0f866541060d93356108850063eeedfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.06ex; height:3.009ex;" alt="{\textstyle \ell (x)=\prod _{m}(x-x_{m})}" loading="lazy"> common to every basis polynomial, a node-specific constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{j}=\prod _{m\neq j}(x_{j}-x_{m})^{-1}}">

<msub>

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<annotation encoding="application/x-tex">{\textstyle w_{j}=\prod _{m\neq j}(x_{j}-x_{m})^{-1}}</annotation>

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</math><img src="./9ac635dfc9570350dc85e114bff7f690f3b696b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.724ex; height:3.509ex;" alt="{\textstyle w_{j}=\prod _{m\neq j}(x_{j}-x_{m})^{-1}}" loading="lazy"> (called the barycentric weight), and a part representing the displacement from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{j}}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\textstyle x_{j}}</annotation>

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</math><img src="./bed4bfb1ba3d13293c3ea916b928dfe07532195f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle x_{j}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle x}</annotation>

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}" loading="lazy">:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ell _{j}(x)=\ell (x){\dfrac {w_{j}}{x-x_{j}}}}</annotation>

</semantics>

</math>

By factoring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ell (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \ell (x)}</annotation>

</semantics>

</math><img src="./03667c75ae9fe024fa9d0d77435b622212f30688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.109ex; height:2.843ex;" alt="{\textstyle \ell (x)}" loading="lazy"> out from the sum, we can write the Lagrange polynomial in the so-called first barycentric form:

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<annotation encoding="application/x-tex">{\displaystyle L(x)=\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}.}</annotation>

</semantics>

</math><img src="./ebab4df3d50a47558e0b7b8baf10b5583dd915c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26ex; height:7.676ex;" alt="{\displaystyle L(x)=\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}.}" loading="lazy"></dd></dl>

If the weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{j}}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle w_{j}}</annotation>

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</math><img src="./326f4828cd2d8b281d5837f977d435d47450a191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.574ex; height:2.343ex;" alt="{\displaystyle w_{j}}" loading="lazy"> have been pre-computed, this requires only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(k)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(k)}</annotation>

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</math><img src="./b0dc155fff30585091c53892f3c57f96c608fe7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(k)}" loading="lazy"> operations compared to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(k^{2})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(k^{2})}</annotation>

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</math><img src="./6098494b4f8a79dbe60cdef8029e340465db8304.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.925ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(k^{2})}" loading="lazy"> for evaluating each Lagrange basis polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{j}(x)}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \ell _{j}(x)}</annotation>

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</math><img src="./69fba239d546edbcde0e858ed6f8c5bfe750823a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.018ex; height:3.009ex;" alt="{\displaystyle \ell _{j}(x)}" loading="lazy"> individually.

The barycentric interpolation formula can also easily be updated to incorporate a new node <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k+1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{k+1}}</annotation>

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</math><img src="./a54ef7b7d84549c053b24d7aa478bcde15f31056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.519ex; height:2.009ex;" alt="{\displaystyle x_{k+1}}" loading="lazy"> by dividing each of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{j}}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle w_{j}}</annotation>

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</math><img src="./326f4828cd2d8b281d5837f977d435d47450a191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.574ex; height:2.343ex;" alt="{\displaystyle w_{j}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=0\dots k}">

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<annotation encoding="application/x-tex">{\displaystyle j=0\dots k}</annotation>

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</math><img src="./179cd5c196488e69a748c4ce8a163f50a935697c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.954ex; height:2.509ex;" alt="{\displaystyle j=0\dots k}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{j}-x_{k+1})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (x_{j}-x_{k+1})}</annotation>

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</math><img src="./42d766431ac92fd60bc12ef4404c2ec094b0e7ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.408ex; height:3.009ex;" alt="{\displaystyle (x_{j}-x_{k+1})}" loading="lazy"> and constructing the new <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{k+1}}">

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<annotation encoding="application/x-tex">{\displaystyle w_{k+1}}</annotation>

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</math><img src="./23660304e2e6dc881dee23d18803026980073ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.853ex; height:2.009ex;" alt="{\displaystyle w_{k+1}}" loading="lazy"> as above.

For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x,}</annotation>

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</math><img src="./8d5bc95a23ee121a9baf899c2b222bd0675c2b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\textstyle x,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{j=0}^{k}\ell _{j}(x)=1}">

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</munderover>

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<annotation encoding="application/x-tex">{\textstyle \sum _{j=0}^{k}\ell _{j}(x)=1}</annotation>

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</math><img src="./0ac79c0fae905fafb6aa6e3755ce4a1b3147812d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.13ex; height:3.843ex;" alt="{\textstyle \sum _{j=0}^{k}\ell _{j}(x)=1}" loading="lazy"> because the constant function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g(x)=1}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle g(x)=1}</annotation>

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</math><img src="./8bd4486a6a0b1757e28aa92d1f4c14100b4a81aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.843ex;" alt="{\textstyle g(x)=1}" loading="lazy"> is the unique polynomial of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq k}">

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<annotation encoding="application/x-tex">{\displaystyle \leq k}</annotation>

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</math><img src="./17f47ca487127f924b3d2025db2eeb1d7ec1dcdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.665ex; height:2.343ex;" alt="{\displaystyle \leq k}" loading="lazy"> interpolating the data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{(x_{0},1),(x_{1},1),\ldots ,(x_{k},1)\}.}">

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<annotation encoding="application/x-tex">{\textstyle \{(x_{0},1),(x_{1},1),\ldots ,(x_{k},1)\}.}</annotation>

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</math><img src="./49deed95635702a86c4ca4b7137fe2483c578a00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.387ex; height:2.843ex;" alt="{\textstyle \{(x_{0},1),(x_{1},1),\ldots ,(x_{k},1)\}.}" loading="lazy"> We can thus further simplify the barycentric formula by dividing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)=L(x)/g(x)\colon }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x)=L(x)/g(x)\colon }</annotation>

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</math><img src="./71bf122f91856114fdca8cd5b973b83284cc4a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.606ex; height:2.843ex;" alt="{\displaystyle L(x)=L(x)/g(x)\colon }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L(x)&=\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}{\Bigg /}\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}\\[10mu]&=\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}{\Bigg /}\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./58f9eb6e3f92fb5b031bbe3a0f2fcc2834cb042e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:44.557ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}L(x)&=\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}{\Bigg /}\ell (x)\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}\\[10mu]&=\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}y_{j}{\Bigg /}\sum _{j=0}^{k}{\frac {w_{j}}{x-x_{j}}}.\end{aligned}}}" loading="lazy"></dd></dl>

This is called the second form or true form of the barycentric interpolation formula.

This second form has advantages in computation cost and accuracy: it avoids evaluation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell (x)}</annotation>

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</math><img src="./d5055bb9c668cf4608a10c9093535a8affcef7c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.109ex; height:2.843ex;" alt="{\displaystyle \ell (x)}" loading="lazy">; the work to compute each term in the denominator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{j}/(x-x_{j})}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{j}/(x-x_{j})}</annotation>

</semantics>

</math><img src="./a14ae7b3fa37b576b355a016d2217f402f54582a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.955ex; height:3.009ex;" alt="{\displaystyle w_{j}/(x-x_{j})}" loading="lazy"> has already been done in computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}w_{j}/(x-x_{j}){\bigr )}y_{j}}">

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}w_{j}/(x-x_{j}){\bigr )}y_{j}}</annotation>

</semantics>

</math><img src="./d2889fc7c6afc575a001af99acbf75c4514aaa05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.134ex; height:3.176ex;" alt="{\displaystyle {\bigl (}w_{j}/(x-x_{j}){\bigr )}y_{j}}" loading="lazy"> and so computing the sum in the denominator costs only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle k}</annotation>

</semantics>

</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy"> addition operations; for evaluation points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x}</annotation>

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}" loading="lazy"> which are close to one of the nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x_{j}}</annotation>

</semantics>

</math><img src="./bed4bfb1ba3d13293c3ea916b928dfe07532195f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle x_{j}}" loading="lazy">, <a href="Catastrophic_cancelation" class="mw-redirect" title="Catastrophic cancelation">catastrophic cancelation</a> would ordinarily be a problem for the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (x-x_{j})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle (x-x_{j})}</annotation>

</semantics>

</math><img src="./21830889bcee8af7bd5ceb750306cf8ea89faf38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.219ex; height:3.009ex;" alt="{\textstyle (x-x_{j})}" loading="lazy">, however this quantity appears in both numerator and denominator and the two cancel leaving good relative accuracy in the final result.

Using this formula to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

</semantics>

</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy"> at one of the nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

</semantics>

</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy"> will result in the <a href="Indeterminate_form" title="Indeterminate form">indeterminate</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty y_{j}/\infty }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \infty y_{j}/\infty }</annotation>

</semantics>

</math><img src="./edf81bb5c7851129642ee581905e2cbd6c0c7dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.859ex; height:3.009ex;" alt="{\displaystyle \infty y_{j}/\infty }" loading="lazy">; computer implementations must replace such results by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x_{j})=y_{j}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x_{j})=y_{j}.}</annotation>

</semantics>

Each Lagrange basis polynomial can also be written in barycentric form:

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell _{j}(x)={\frac {w_{j}}{x-x_{j}}}{\Bigg /}\sum _{m=0}^{k}{\frac {w_{m}}{x-x_{m}}}.}</annotation>

</semantics>

</math><img src="./7041ddcc610c8fa863f59c4759d192eb60eaac8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.307ex; height:7.509ex;" alt="{\displaystyle \ell _{j}(x)={\frac {w_{j}}{x-x_{j}}}{\Bigg /}\sum _{m=0}^{k}{\frac {w_{m}}{x-x_{m}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="A_perspective_from_linear_algebra">A perspective from linear algebra</h2></div>

Solving an <a href="Polynomial_interpolation#Constructing_the_interpolation_polynomial" title="Polynomial interpolation">interpolation problem</a> leads to a problem in <a href="Linear_algebra" title="Linear algebra">linear algebra</a> amounting to inversion of a matrix. Using a standard <a href="Monomial_basis" title="Monomial basis">monomial basis</a> for our interpolation polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle L(x)=\sum _{j=0}^{k}x^{j}m_{j}}">

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle L(x)=\sum _{j=0}^{k}x^{j}m_{j}}</annotation>

</semantics>

</math><img src="./4eb2e8c5d4f97e7fd7a9f6a2f2d8eb84a538257d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.861ex; height:3.843ex;" alt="{\textstyle L(x)=\sum _{j=0}^{k}x^{j}m_{j}}" loading="lazy">, we must invert the <a href="Vandermonde_matrix" title="Vandermonde matrix">Vandermonde matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})^{j}}">

<msub>

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</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{i})^{j}}</annotation>

</semantics>

</math><img src="./6d63dc9dc237d875ca66507c62abef25ea30f5bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.848ex; height:3.176ex;" alt="{\displaystyle (x_{i})^{j}}" loading="lazy"> to solve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x_{i})=y_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x_{i})=y_{i}}</annotation>

</semantics>

</math><img src="./9129c28b14481f67591dfdc23e73c053c4f23577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.559ex; height:2.843ex;" alt="{\displaystyle L(x_{i})=y_{i}}" loading="lazy"> for the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{j}}</annotation>

</semantics>

</math><img src="./14f2b12b1676ba2f42ad8d2c9ee6aa46e7667b73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.95ex; height:2.343ex;" alt="{\displaystyle m_{j}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

</semantics>

</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy">. By choosing a better basis, the Lagrange basis, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle L(x)=\sum _{j=0}^{k}l_{j}(x)y_{j}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle L(x)=\sum _{j=0}^{k}l_{j}(x)y_{j}}</annotation>

</semantics>

</math><img src="./c6f2679e9e980c7d8095211b0d56b4030f96c885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.462ex; height:3.843ex;" alt="{\textstyle L(x)=\sum _{j=0}^{k}l_{j}(x)y_{j}}" loading="lazy">, we merely get the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>, <a href="Kronecker_delta" title="Kronecker delta"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>

</semantics>

</math><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"></a>, which is its own inverse: the Lagrange basis automatically inverts the analog of the Vandermonde matrix.

This construction is analogous to the <a href="Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.

Furthermore, when the order is large, <a href="Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transformation</a> can be used to solve for the coefficients of the interpolated polynomial.

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

We wish to interpolate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}}</annotation>

</semantics>

</math><img src="./84ddac4ae10b1aa4a11741c79771a583419fb1fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}}" loading="lazy"> over the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq x\leq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq x\leq 3}</annotation>

</semantics>

</math><img src="./c3036099482087ff1830d9854178422a2d3f327a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.852ex; height:2.343ex;" alt="{\displaystyle 1\leq x\leq 3}" loading="lazy"> at the three nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\,2,\,3\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,\,2,\,3\}}</annotation>

</semantics>

</math><img src="./ba3964528ef95213642170e58a624f5953f50de7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.654ex; height:2.843ex;" alt="{\displaystyle \{1,\,2,\,3\}}" loading="lazy">:

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<mtd>

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</mrow>

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</mtd>

<mtd>

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<mtd>

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<mtr>

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<mtd>

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<msub>

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</mtd>

<mtd>

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</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=1,&&&y_{0}=f(x_{0})&=1,\\[3mu]x_{1}&=2,&&&y_{1}=f(x_{1})&=4,\\[3mu]x_{2}&=3,&&&y_{2}=f(x_{2})&=9.\end{aligned}}}</annotation>

</semantics>

</math><img src="./ea731ecebd607939d03f1af637c7dc70d167a6bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:33.005ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=1,&&&y_{0}=f(x_{0})&=1,\\[3mu]x_{1}&=2,&&&y_{1}=f(x_{1})&=4,\\[3mu]x_{2}&=3,&&&y_{2}=f(x_{2})&=9.\end{aligned}}}" loading="lazy"></dd></dl>

The node polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

</semantics>

</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> is

<msup>

</mrow>

</msup>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell (x)=(x-1)(x-2)(x-3)=x^{3}-6x^{2}+11x-6.}</annotation>

</semantics>

</math><img src="./06afb49b52761b3bf0cbddc9a4b9cb9d32c0b25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.646ex; height:3.176ex;" alt="{\displaystyle \ell (x)=(x-1)(x-2)(x-3)=x^{3}-6x^{2}+11x-6.}" loading="lazy"></dd></dl>

The barycentric weights are

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</msub>

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<mtd>

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</msup>

<msup>

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<mfrac>

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</mstyle>

</mrow>

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</mtable>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}w_{0}&=(1-2)^{-1}(1-3)^{-1}={\tfrac {1}{2}},\\[3mu]w_{1}&=(2-1)^{-1}(2-3)^{-1}=-1,\\[3mu]w_{2}&=(3-1)^{-1}(3-2)^{-1}={\tfrac {1}{2}}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./56d4a7884aee5c67d2ab29351bfd9ae86cfbfceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:31.899ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}w_{0}&=(1-2)^{-1}(1-3)^{-1}={\tfrac {1}{2}},\\[3mu]w_{1}&=(2-1)^{-1}(2-3)^{-1}=-1,\\[3mu]w_{2}&=(3-1)^{-1}(3-2)^{-1}={\tfrac {1}{2}}.\end{aligned}}}" loading="lazy"></dd></dl>

The Lagrange basis polynomials are

<mtr>

<mtd>

<msub>

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<mtd>

<mfrac>

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</mrow>

<mfrac>

<mrow>

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<mrow>

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</mfrac>

</mrow>

<mfrac>

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</mstyle>

</mrow>

<msup>

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</mtd>

<mtd>

<mfrac>

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</mfrac>

</mrow>

<mfrac>

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<mrow>

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</mfrac>

</mrow>

<msup>

</mrow>

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</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell _{0}(x)&={\frac {x-2}{1-2}}\cdot {\frac {x-3}{1-3}}={\tfrac {1}{2}}x^{2}-{\tfrac {5}{2}}x+3,\\[5mu]\ell _{1}(x)&={\frac {x-1}{2-1}}\cdot {\frac {x-3}{2-3}}=-x^{2}+4x-3,\\[5mu]\ell _{2}(x)&={\frac {x-1}{3-1}}\cdot {\frac {x-2}{3-2}}={\tfrac {1}{2}}x^{2}-{\tfrac {3}{2}}x+1.\end{aligned}}}</annotation>

</semantics>

</math><img src="./13c47023b84eec7fc182f39ac958e7271498cd53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.123ex; margin-bottom: -0.215ex; width:40.648ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}\ell _{0}(x)&={\frac {x-2}{1-2}}\cdot {\frac {x-3}{1-3}}={\tfrac {1}{2}}x^{2}-{\tfrac {5}{2}}x+3,\\[5mu]\ell _{1}(x)&={\frac {x-1}{2-1}}\cdot {\frac {x-3}{2-3}}=-x^{2}+4x-3,\\[5mu]\ell _{2}(x)&={\frac {x-1}{3-1}}\cdot {\frac {x-2}{3-2}}={\tfrac {1}{2}}x^{2}-{\tfrac {3}{2}}x+1.\end{aligned}}}" loading="lazy"></dd></dl>

The Lagrange interpolating polynomial is:

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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</msub>

<msub>

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</msub>

<msub>

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<msup>

</mrow>

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</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L(x)&=y_{0}\cdot \ell _{0}(x)+y_{1}\cdot \ell _{1}(x)+y_{2}\cdot \ell _{2}(x)=x^{2}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./83227e6bfef5769fa019073ac40dc02d53c5361f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.488ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}L(x)&=y_{0}\cdot \ell _{0}(x)+y_{1}\cdot \ell _{1}(x)+y_{2}\cdot \ell _{2}(x)=x^{2}.\end{aligned}}}" loading="lazy"></dd></dl>

In (second) barycentric form,

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</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

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</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

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</mrow>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

<mfrac>

<mfrac>

<mfrac>

</mfrac>

</mstyle>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mfrac>

</mfrac>

</mstyle>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

<mfrac>

<mfrac>

</mfrac>

</mstyle>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mfrac>

</mfrac>

</mstyle>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x)={\frac {\displaystyle \sum _{j=0}^{2}{\frac {w_{j}}{x-x_{j}}}y_{j}}{\displaystyle \sum _{j=0}^{2}{\frac {w_{j}}{x-x_{j}}}}}={\frac {\displaystyle {\frac {\tfrac {1}{2}}{x-1}}+{\frac {-4}{x-2}}+{\frac {\tfrac {9}{2}}{x-3}}}{\displaystyle {\frac {\tfrac {1}{2}}{x-1}}+{\frac {-1}{x-2}}+{\frac {\tfrac {1}{2}}{x-3}}}}.}</annotation>

</semantics>

</math><img src="./c24b4b791153b470b767c4e07295e9ea1f7e7ac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:50.461ex; height:15.843ex;" alt="{\displaystyle L(x)={\frac {\displaystyle \sum _{j=0}^{2}{\frac {w_{j}}{x-x_{j}}}y_{j}}{\displaystyle \sum _{j=0}^{2}{\frac {w_{j}}{x-x_{j}}}}}={\frac {\displaystyle {\frac {\tfrac {1}{2}}{x-1}}+{\frac {-4}{x-2}}+{\frac {\tfrac {9}{2}}{x-3}}}{\displaystyle {\frac {\tfrac {1}{2}}{x-1}}+{\frac {-1}{x-2}}+{\frac {\tfrac {1}{2}}{x-3}}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments. Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the <a href="Vandermonde_determinant" class="mw-redirect" title="Vandermonde determinant">Vandermonde determinant</a>.

But, as can be seen from the construction, each time a node xk changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or <a href="Newton_polynomial" title="Newton polynomial">Newton polynomials</a>.

Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as <a href="Runge's_phenomenon" title="Runge's phenomenon">Runge's phenomenon</a>; the problem may be eliminated by choosing interpolation points at <a href="Chebyshev_nodes" title="Chebyshev nodes">Chebyshev nodes</a>.<a href="#cite_note-5">[5]</a>

The Lagrange basis polynomials can be used in <a href="Numerical_integration" title="Numerical integration">numerical integration</a> to derive the <a href="Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a>.

<div class="mw-heading mw-heading2"><h2 id="Remainder_in_Lagrange_interpolation_formula">Remainder in Lagrange interpolation formula</h2></div>

When interpolating a given function f by a polynomial of degree k at the nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},...,x_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0},...,x_{k}}</annotation>

</semantics>

</math><img src="./d95e41045b132a6de6683670b3268158ce6bf4dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.972ex; height:2.009ex;" alt="{\displaystyle x_{0},...,x_{k}}" loading="lazy"> we get the remainder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)=f(x)-L(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(x)=f(x)-L(x)}</annotation>

</semantics>

</math><img src="./dfcbfc5cb40457ecc99057b2ed9454c621da9c66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.981ex; height:2.843ex;" alt="{\displaystyle R(x)=f(x)-L(x)}" loading="lazy"> which can be expressed as<a href="#cite_note-6">[6]</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(x)=f[x_{0},\ldots ,x_{k},x]\ell (x)=\ell (x){\frac {f^{(k+1)}(\xi )}{(k+1)!}},\quad \quad x_{0}<\xi <x_{k},}</annotation>

</semantics>

</math><img src="./02cf7851563959d1b06a7a07782bba6ae42c096b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.054ex; height:6.843ex;" alt="{\displaystyle R(x)=f[x_{0},\ldots ,x_{k},x]\ell (x)=\ell (x){\frac {f^{(k+1)}(\xi )}{(k+1)!}},\quad \quad x_{0}<\xi <x_{k},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[x_{0},\ldots ,x_{k},x]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f[x_{0},\ldots ,x_{k},x]}</annotation>

</semantics>

</math><img src="./64e34bb84b1de65fda77f1fa9dddac063d1f5720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.916ex; height:2.843ex;" alt="{\displaystyle f[x_{0},\ldots ,x_{k},x]}" loading="lazy"> is the notation for <a href="Divided_differences" title="Divided differences">divided differences</a>. Alternatively, the remainder can be expressed as a contour integral in complex domain as

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(x)={\frac {\ell (x)}{2\pi i}}\int _{C}{\frac {f(t)}{(t-x)(t-x_{0})\cdots (t-x_{k})}}dt={\frac {\ell (x)}{2\pi i}}\int _{C}{\frac {f(t)}{(t-x)\ell (t)}}dt.}</annotation>

</semantics>

</math><img src="./61759ea0a9cb9a8c0a002e93454769d3a4955e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:71.05ex; height:6.509ex;" alt="{\displaystyle R(x)={\frac {\ell (x)}{2\pi i}}\int _{C}{\frac {f(t)}{(t-x)(t-x_{0})\cdots (t-x_{k})}}dt={\frac {\ell (x)}{2\pi i}}\int _{C}{\frac {f(t)}{(t-x)\ell (t)}}dt.}" loading="lazy"></dd></dl>

The remainder can be bound as

</mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |R(x)|\leq {\frac {(x_{k}-x_{0})^{k+1}}{(k+1)!}}\max _{x_{0}\leq \xi \leq x_{k}}|f^{(k+1)}(\xi )|.}</annotation>

</semantics>

</math><img src="./a9b94b40783455ba7df619b7add9efe1ad02616c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.973ex; height:6.676ex;" alt="{\displaystyle |R(x)|\leq {\frac {(x_{k}-x_{0})^{k+1}}{(k+1)!}}\max _{x_{0}\leq \xi \leq x_{k}}|f^{(k+1)}(\xi )|.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3></div>

Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(x)}</annotation>

</semantics>

</math><img src="./cd5e851b43895fbe06436240dc7daa4d2033f082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:2.843ex;" alt="{\displaystyle R(x)}" loading="lazy"> is zero at nodes. To find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(x)}</annotation>

</semantics>

</math><img src="./cd5e851b43895fbe06436240dc7daa4d2033f082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:2.843ex;" alt="{\displaystyle R(x)}" loading="lazy"> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{p}}</annotation>

</semantics>

</math><img src="./0bec554743fa797409a83ad8d00b4d35e110a50a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.389ex; height:2.343ex;" alt="{\displaystyle x_{p}}" loading="lazy">, define a new function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=R(x)-{\tilde {R}}(x)=f(x)-L(x)-{\tilde {R}}(x)}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)=R(x)-{\tilde {R}}(x)=f(x)-L(x)-{\tilde {R}}(x)}</annotation>

</semantics>

</math><img src="./0ce917ecac55f5b54e5d9dca2c3125ab860250e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.446ex; height:3.176ex;" alt="{\displaystyle F(x)=R(x)-{\tilde {R}}(x)=f(x)-L(x)-{\tilde {R}}(x)}" loading="lazy"> and choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tilde {R}}(x)=C\cdot \prod _{i=0}^{k}(x-x_{i})}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\tilde {R}}(x)=C\cdot \prod _{i=0}^{k}(x-x_{i})}</annotation>

</semantics>

</math><img src="./dc5ab276cda4ccddeea82141c4aa8603614c20be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.65ex; height:3.509ex;" alt="{\textstyle {\tilde {R}}(x)=C\cdot \prod _{i=0}^{k}(x-x_{i})}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is the constant we are required to determine for a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{p}}</annotation>

</semantics>

</math><img src="./0bec554743fa797409a83ad8d00b4d35e110a50a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.389ex; height:2.343ex;" alt="{\displaystyle x_{p}}" loading="lazy">. We choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>

</semantics>

</math><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+2}</annotation>

</semantics>

</math><img src="./141170bc6765f9f664fe148de50dec24f03e69f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k+2}" loading="lazy"> zeroes (at all nodes and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{p}}</annotation>

</semantics>

</math><img src="./0bec554743fa797409a83ad8d00b4d35e110a50a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.389ex; height:2.343ex;" alt="{\displaystyle x_{p}}" loading="lazy">) between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"> (including endpoints). Assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+1}</annotation>

</semantics>

</math><img src="./552a558062ed4c0486297b5b5531c5ee044dbd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k+1}" loading="lazy">-times differentiable, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(x)}</annotation>

</semantics>

</math><img src="./88b78cb2b3a60d964f3016e94f0d7bd081ebd3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.722ex; height:2.843ex;" alt="{\displaystyle L(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {R}}(x)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {R}}(x)}</annotation>

</semantics>

</math><img src="./a0bcd89b44936121ae571fe78d987f05efb3a8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:3.176ex;" alt="{\displaystyle {\tilde {R}}(x)}" loading="lazy"> are polynomials, and therefore, are infinitely differentiable, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>

</semantics>

</math><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"> will be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+1}</annotation>

</semantics>

</math><img src="./552a558062ed4c0486297b5b5531c5ee044dbd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k+1}" loading="lazy">-times differentiable. By <a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{(1)}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{(1)}(x)}</annotation>

</semantics>

</math><img src="./db33ca6fdaeae4a23ff6717d452a56478645fee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.287ex; height:3.343ex;" alt="{\displaystyle F^{(1)}(x)}" loading="lazy"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k+1}</annotation>

</semantics>

</math><img src="./552a558062ed4c0486297b5b5531c5ee044dbd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k+1}" loading="lazy"> zeroes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{(2)}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{(2)}(x)}</annotation>

</semantics>

</math><img src="./3e491d8e18261f7f63ad138a8a4b6529dc8c6aba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.287ex; height:3.343ex;" alt="{\displaystyle F^{(2)}(x)}" loading="lazy"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> zeroes... <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{(k+1)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{(k+1)}}</annotation>

</semantics>

</math><img src="./ceae6840d0dbbe0a886bb650786f0fd7a30d4f08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.283ex; height:2.843ex;" alt="{\displaystyle F^{(k+1)}}" loading="lazy"> has 1 zero, say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi ,\,x_{0}<\xi <x_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi ,\,x_{0}<\xi <x_{k}}</annotation>

</semantics>

</math><img src="./71fe66f0eee87a996fe16b2ce8bb729105b4e8a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.48ex; height:2.509ex;" alt="{\displaystyle \xi ,\,x_{0}<\xi <x_{k}}" loading="lazy">. Explicitly writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{(k+1)}(\xi )}">

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<annotation encoding="application/x-tex">{\displaystyle F^{(k+1)}(\xi )}</annotation>

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</math><img src="./7e172aacc395c9a339ad8426cddc2baf7cead395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.123ex; height:3.343ex;" alt="{\displaystyle F^{(k+1)}(\xi )}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle F^{(k+1)}(\xi )=f^{(k+1)}(\xi )-L^{(k+1)}(\xi )-{\tilde {R}}^{(k+1)}(\xi )}</annotation>

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</math><img src="./67c1dc7f35858cdd4de64db6178e5a2043fa67c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.493ex; height:3.843ex;" alt="{\displaystyle F^{(k+1)}(\xi )=f^{(k+1)}(\xi )-L^{(k+1)}(\xi )-{\tilde {R}}^{(k+1)}(\xi )}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle L^{(k+1)}=0,{\tilde {R}}^{(k+1)}=C\cdot (k+1)!}</annotation>

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</math><img src="./5f00e44d655ec9d51cd376edda54ce57eae4b3bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.793ex; height:3.843ex;" alt="{\displaystyle L^{(k+1)}=0,{\tilde {R}}^{(k+1)}=C\cdot (k+1)!}" loading="lazy"> (Because the highest power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {R}}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {R}}(x)}</annotation>

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</math><img src="./a0bcd89b44936121ae571fe78d987f05efb3a8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:3.176ex;" alt="{\displaystyle {\tilde {R}}(x)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k+1}">

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<annotation encoding="application/x-tex">{\displaystyle k+1}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 0=f^{(k+1)}(\xi )-C\cdot (k+1)!}</annotation>

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</math><img src="./5f9163103accfd252f3efe065e0e2a8124bcd7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.845ex; height:3.343ex;" alt="{\displaystyle 0=f^{(k+1)}(\xi )-C\cdot (k+1)!}" loading="lazy"></dd></dl>

The equation can be rearranged as<a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle C={\frac {f^{(k+1)}(\xi )}{(k+1)!}}}</annotation>

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</math><img src="./4274932aa9b93a42285fbb7a352d6bdd300f7d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.329ex; height:6.843ex;" alt="{\displaystyle C={\frac {f^{(k+1)}(\xi )}{(k+1)!}}}" loading="lazy"></dd></dl>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{p})=0}">

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<annotation encoding="application/x-tex">{\displaystyle F(x_{p})=0}</annotation>

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</math><img src="./2a6ccd8938f44fab3e7c0be08647724091d202fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.2ex; height:3.009ex;" alt="{\displaystyle F(x_{p})=0}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x_{p})={\tilde {R}}(x_{p})={\frac {f^{k+1}(\xi )}{(k+1)!}}\prod _{i=0}^{k}(x_{p}-x_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle R(x_{p})={\tilde {R}}(x_{p})={\frac {f^{k+1}(\xi )}{(k+1)!}}\prod _{i=0}^{k}(x_{p}-x_{i})}</annotation>

</semantics>

</math><img src="./6fce96e92f0e33a534876eacd830be0b77193a73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.152ex; height:7.343ex;" alt="{\displaystyle R(x_{p})={\tilde {R}}(x_{p})={\frac {f^{k+1}(\xi )}{(k+1)!}}\prod _{i=0}^{k}(x_{p}-x_{i})}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Derivatives">Derivatives</h2></div>

The dth <a href="Derivative" title="Derivative">derivative</a> of a Lagrange interpolating polynomial can be written in terms of the derivatives of the basis polynomials,

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<annotation encoding="application/x-tex">{\displaystyle L^{(d)}(x):=\sum _{j=0}^{k}y_{j}\ell _{j}^{(d)}(x).}</annotation>

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</math><img src="./18bc133e45c7a1203b37c72b118632058cc5b8fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.756ex; height:7.676ex;" alt="{\displaystyle L^{(d)}(x):=\sum _{j=0}^{k}y_{j}\ell _{j}^{(d)}(x).}" loading="lazy"></dd></dl>

Recall (see <a href="#Definition">§ Definition</a> above) that each Lagrange basis polynomial is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell _{j}(x)&=\prod _{\begin{smallmatrix}m=0\\m\neq j\end{smallmatrix}}^{k}{\frac {x-x_{m}}{x_{j}-x_{m}}}.\end{aligned}}}</annotation>

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</math>

The first derivative can be found using the <a href="Product_rule#Product_of_more_than_two_factors" title="Product rule">product rule</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell _{j}'(x)&=\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\Biggl [}{\frac {1}{x_{j}-x_{i}}}\prod _{\begin{smallmatrix}m=0\\m\not =(i,j)\end{smallmatrix}}^{k}{\frac {x-x_{m}}{x_{j}-x_{m}}}{\Biggr ]}\\[5mu]&=\ell _{j}(x)\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\frac {1}{x-x_{i}}}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./9f7f5339a0a09570403bcdcf4176fbdc6871d07b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:39.58ex; height:19.676ex;" alt="{\displaystyle {\begin{aligned}\ell _{j}'(x)&=\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\Biggl [}{\frac {1}{x_{j}-x_{i}}}\prod _{\begin{smallmatrix}m=0\\m\not =(i,j)\end{smallmatrix}}^{k}{\frac {x-x_{m}}{x_{j}-x_{m}}}{\Biggr ]}\\[5mu]&=\ell _{j}(x)\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\frac {1}{x-x_{i}}}.\end{aligned}}}" loading="lazy"></dd></dl>

The second derivative is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell _{j}''(x)&=\sum _{\begin{smallmatrix}i=0\\i\neq j\end{smallmatrix}}^{k}{\frac {1}{x_{j}-x_{i}}}{\Biggl [}\sum _{\begin{smallmatrix}m=0\\m\neq (i,j)\end{smallmatrix}}^{k}{\Biggl (}{\frac {1}{x_{j}-x_{m}}}\prod _{\begin{smallmatrix}n=0\\n\neq (i,j,m)\end{smallmatrix}}^{k}{\frac {x-x_{n}}{x_{j}-x_{n}}}{\Biggr )}{\Biggr ]}\\[10mu]&=\ell _{j}(x)\sum _{0\leq i<m\leq k}{\frac {2}{(x-x_{i})(x-x_{m})}}\\[10mu]&=\ell _{j}(x){\Biggl [}{\Biggl (}\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\frac {1}{x-x_{i}}}{\Biggr )}^{2}-\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\frac {1}{(x-x_{i})^{2}}}{\Biggr ]}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./9d9f334e0ce26b8fd713e808986bf11888a7899e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.838ex; width:60.625ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}\ell _{j}''(x)&=\sum _{\begin{smallmatrix}i=0\\i\neq j\end{smallmatrix}}^{k}{\frac {1}{x_{j}-x_{i}}}{\Biggl [}\sum _{\begin{smallmatrix}m=0\\m\neq (i,j)\end{smallmatrix}}^{k}{\Biggl (}{\frac {1}{x_{j}-x_{m}}}\prod _{\begin{smallmatrix}n=0\\n\neq (i,j,m)\end{smallmatrix}}^{k}{\frac {x-x_{n}}{x_{j}-x_{n}}}{\Biggr )}{\Biggr ]}\\[10mu]&=\ell _{j}(x)\sum _{0\leq i<m\leq k}{\frac {2}{(x-x_{i})(x-x_{m})}}\\[10mu]&=\ell _{j}(x){\Biggl [}{\Biggl (}\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\frac {1}{x-x_{i}}}{\Biggr )}^{2}-\sum _{\begin{smallmatrix}i=0\\i\not =j\end{smallmatrix}}^{k}{\frac {1}{(x-x_{i})^{2}}}{\Biggr ]}.\end{aligned}}}" loading="lazy"></dd></dl>

The third derivative is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ell _{j}'''(x)&=\ell _{j}(x)\sum _{0\leq i<m<n\leq k}{\frac {3!}{(x-x_{i})(x-x_{m})(x-x_{n})}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./ac4f188dcf492e0453194c6e2db922593a19de33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:51.586ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\ell _{j}'''(x)&=\ell _{j}(x)\sum _{0\leq i<m<n\leq k}{\frac {3!}{(x-x_{i})(x-x_{m})(x-x_{n})}}\end{aligned}}}" loading="lazy"></dd></dl>

and likewise for higher derivatives.

Note that all of these formulas for derivatives are invalid at or near a node. A method of evaluating all orders of derivatives of a Lagrange polynomial efficiently at all points of the domain, including the nodes, is converting the Lagrange polynomial to power basis form and then evaluating the derivatives.

<div class="mw-heading mw-heading2"><h2 id="Finite_fields">Finite fields</h2></div>

The Lagrange polynomial can also be computed in <a href="Finite_field" title="Finite field">finite fields</a>. This has applications in <a href="Cryptography" title="Cryptography">cryptography</a>, such as in <a href="Shamir's_Secret_Sharing" class="mw-redirect" title="Shamir's Secret Sharing">Shamir's Secret Sharing</a> scheme.

<ul><li><a href="Neville's_algorithm" title="Neville's algorithm">Neville's algorithm</a></li>

<li><a href="Newton_polynomial" title="Newton polynomial">Newton form</a> of the interpolation polynomial</li>

<li><a href="Bernstein_polynomial" title="Bernstein polynomial">Bernstein polynomial</a></li>

<li><a href="Carlson's_theorem" title="Carlson's theorem">Carlson's theorem</a></li>

<li><a href="Lebesgue_constant" title="Lebesgue constant">Lebesgue constant</a></li>

<li><a href="Chebfun" title="Chebfun">The Chebfun system</a></li>

<li><a href="Table_of_Newtonian_series" title="Table of Newtonian series">Table of Newtonian series</a></li>

<li><a href="Frobenius_covariant" title="Frobenius covariant">Frobenius covariant</a></li>

<li><a href="Sylvester's_formula" title="Sylvester's formula">Sylvester's formula</a></li>

<li><a href="Finite_difference_coefficient" title="Finite difference coefficient">Finite difference coefficient</a></li>

<li><a href="Hermite_interpolation" title="Hermite interpolation">Hermite interpolation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLagrange1795" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange, Joseph-Louis</a> (1795). "Leçon Cinquième. Sur l'usage des courbes dans la solution des problèmes". Leçons Elémentaires sur les Mathématiques (in French). Paris.</cite> Republished in <cite id="CITEREFLagrange1877" class="citation book cs1"><a href="Joseph-Alfred_Serret" title="Joseph-Alfred Serret">Serret, Joseph-Alfred</a>, ed. (1877). Oeuvres de Lagrange. Vol. 7. Gauthier-Villars. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/oeuvresdelagrang07lagr/page/271">271–287</a>.</cite> Translated as <cite id="CITEREFLagrange1901" class="citation book cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/lecturesonelemen00lagriala/page/127">"Lecture V. On the Employment of Curves in the Solution of Problems"</a>. Lectures on Elementary Mathematics. Translated by McCormack, Thomas J. (2nd ed.). Open Court. 1901. pp. 127–149.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWaring1779" class="citation journal cs1"><a href="Edward_Waring" title="Edward Waring">Waring, Edward</a> (1779). <a rel="nofollow" class="external text" href="https://archive.org/details/philosophicaltra6917roya/page/59">"Problems concerning interpolations"</a>. <a href="Philosophical_Transactions_of_the_Royal_Society" title="Philosophical Transactions of the Royal Society">Philosophical Transactions of the Royal Society</a>. 69: 59–67. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1779.0008">10.1098/rstl.1779.0008</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMeijering2002" class="citation journal cs1">Meijering, Erik (2002). <a rel="nofollow" class="external text" href="http://bigwww.epfl.ch/publications/meijering0201.pdf">"A chronology of interpolation: from ancient astronomy to modern signal and image processing"</a> (PDF). Proceedings of the IEEE. 90 (3): 319–342. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F5.993400">10.1109/5.993400</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFBerrutTrefethen2004" class="citation journal cs1"><a href="Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Berrut, Jean-Paul</a>; Trefethen, Lloyd N. (2004). <a rel="nofollow" class="external text" href="https://people.maths.ox.ac.uk/trefethen/barycentric.pdf">"Barycentric Lagrange Interpolation"</a> (PDF). SIAM Review. 46 (3): 501–517. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004SIAMR..46..501B">2004SIAMR..46..501B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0036144502417715">10.1137/S0036144502417715</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFQuarteroniSaleri2003" class="citation book cs1"><a href="Alfio_Quarteroni" title="Alfio Quarteroni">Quarteroni, Alfio</a>; Saleri, Fausto (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fE1W5jsU4zoC&pg=PA66">Scientific Computing with MATLAB</a>. Texts in computational science and engineering. Vol. 2. Springer. p. 66. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-44363-6</bdi>.</cite>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_878.htm">"Chapter 25, eqn 25.2.3"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 878. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170215114507/http://www.sam.nitk.ac.in/sites/default/Numerical_Methods/Interpolation/interpolation.pdf">"Interpolation"</a> (PDF). pp. 12–15. Archived from <a rel="nofollow" class="external text" href="https://sam.nitk.ac.in/sites/default/Numerical_Methods/Interpolation/interpolation.pdf">the original</a> (PDF) on 2017-02-15.</cite>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Algorithm_Implementation" class="extiw external" title="wikibooks:Algorithm Implementation">Algorithm Implementation</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Algorithm_Implementation/Mathematics/Polynomial_interpolation" class="extiw external" title="wikibooks:Algorithm Implementation/Mathematics/Polynomial interpolation">Polynomial interpolation</a></div></div>

</div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Lagrange_interpolation_formula">"Lagrange interpolation formula"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="https://www.alglib.net/interpolation/polynomial.php">ALGLIB</a> has an implementations in C++ / C# / VBA / Pascal.</li>

<li><a rel="nofollow" class="external text" href="https://www.gnu.org/software/gsl/">GSL</a> has a polynomial interpolation code in C</li>

<li><a rel="nofollow" class="external text" href="https://stackoverflow.com/questions/11029615/lagrange-interpolation-method/11552763">SO</a> has a MATLAB example that demonstrates the algorithm and recreates the first image in this article</li>

<li><a rel="nofollow" class="external text" href="https://nm.mathforcollege.com/chapter-05-04-lagrange-method/">Lagrange Method of Interpolation — Notes, PPT, Mathcad, Mathematica, MATLAB, Maple</a></li>

<li><a rel="nofollow" class="external text" href="http://www.math-linux.com/spip.php?article71">Lagrange interpolation polynomial</a> on www.math-linux.com</li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html">"Lagrange Interpolating Polynomial"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://mathformeremortals.wordpress.com/2013/01/15/bicubic-interpolation-excel-worksheet-function/">Excel Worksheet Function for Bicubic Lagrange Interpolation</a></li>

<li><a rel="nofollow" class="external text" href="https://learn.64bitdragon.com/articles/computer-science/numerical-analysis/lagrange-interpolation">Lagrange polynomials in Python</a></li></ul>

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</style><div id="Joseph-Louis_Lagrange25" style="font-size:114%;margin:0 4em"><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Lagrange's_four-square_theorem" title="Lagrange's four-square theorem">Lagrange's four-square theorem</a></li>

<li><a href="Lagrange's_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem (group theory)</a></li>

<li><a href="Lagrange's_identity" title="Lagrange's identity">Lagrange's identity</a></li>

<li><a href="Lagrange's_identity_(boundary_value_problem)" title="Lagrange's identity (boundary value problem)">Lagrange's identity (boundary value problem)</a></li>

<li><a href="Lagrange's_trigonometric_identities" class="mw-redirect" title="Lagrange's trigonometric identities">Lagrange's trigonometric identities</a></li>

<li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Lagrange's mean value theorem</a></li>

<li><a href="Lagrange_stability" title="Lagrange stability">Lagrange stability</a></li>

<li><a href="Lagrange_point" title="Lagrange point">Lagrange point</a></li></ul>

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