Editing Lagrange polynomial (section)

{{Short description|Polynomials used for interpolation}}
{{Distinguish|text = [[Legendre polynomials]] (the orthogonal basis of function space)}}
[[Image:Lagrange polynomial.svg|thumb|upright=1.5|This image shows, for four points (<span style="color:#5e81B5;">(&minus;9,&nbsp;5)</span>, <span style="color:#e19c24;">(&minus;4,&nbsp;2)</span>, <span style="color:#8FB131;">(&minus;1,&nbsp;&minus;2)</span>, <span style="color:#EC6235;">(7,&nbsp;9)</span>), the (cubic) interpolation polynomial <span style="color:black;">''L''(''x'')</span> (dashed, black), which is the sum of the ''scaled'' basis polynomials <span style="color:#5e81B5;">y<sub>0</sub>''ℓ''<sub>0</sub>(''x'')</span>, <span style="color:#e19c24;">y<sub>1</sub>''ℓ''<sub>1</sub>(''x'')</span>, <span style="color:#8FB131;">y<sub>2</sub>''ℓ''<sub>2</sub>(''x'')</span> and <span style="color:#EC6235;">y<sub>3</sub>''ℓ''<sub>3</sub>(''x'')</span>. The interpolation polynomial passes through all four control points, and each ''scaled'' basis polynomial passes through its respective control point and is 0 where ''x'' corresponds to the other three control points.]]

In [[numerical analysis]], the '''Lagrange interpolating polynomial''' is the unique [[polynomial]] of lowest [[degree of a polynomial|degree]] that [[polynomial interpolation|interpolates]] a given set of data.

Given a data set of [[graph of a function|coordinate pairs]] <math>(x_j, y_j)</math> with <math>0 \leq j \leq k,</math> the <math>x_j</math> are called ''nodes'' and the <math>y_j</math> are called ''values''. The Lagrange polynomial <math>L(x)</math> has degree <math display=inline>\leq k</math> and assumes each value at the corresponding node, <math>L(x_j) = y_j.</math>

Although named after [[Joseph-Louis Lagrange]], who published it in 1795,<ref>
{{cite book |last=Lagrange |first=Joseph-Louis |author-link= Joseph-Louis Lagrange |title=Leçons Elémentaires sur les Mathématiques |language=fr |year=1795 |chapter=Leçon Cinquième. Sur l'usage des courbes dans la solution des problèmes |place=Paris}} Republished in {{cite book |last=Lagrange |first=Joseph-Louis |editor-last=Serret |editor-first=Joseph-Alfred |editor-link=Joseph-Alfred Serret |display-authors=0 |title=Oeuvres de Lagrange |year=1877 |volume=7 |publisher=Gauthier-Villars |pages=[https://archive.org/details/oeuvresdelagrang07lagr/page/271 271–287] }} Translated as {{cite book |last=Lagrange |first=Joseph-Louis |display-authors=0 |translator-last=McCormack |translator-first=Thomas J. |title=Lectures on Elementary Mathematics |edition=2nd |publisher=Open Court |year=1901 |chapter=Lecture V. On the Employment of Curves in the Solution of Problems |chapter-url=https://archive.org/details/lecturesonelemen00lagriala/page/127 |pages=127–149}}</ref> the method was first discovered in 1779 by [[Edward Waring]].<ref>{{cite journal
 |title=Problems concerning interpolations
 |first=Edward |last=Waring |author-link=Edward Waring
 |journal=[[Philosophical Transactions of the Royal Society]]
 |year=1779 |volume=69 |pages=59–67
 |url=https://archive.org/details/philosophicaltra6917roya/page/59
 |doi=10.1098/rstl.1779.0008
 |doi-access=
 }}</ref> It is also an easy consequence of a formula published in 1783 by [[Leonhard Euler]].<ref>{{Cite journal
 | last1=Meijering | first1=Erik
 | title=A chronology of interpolation: from ancient astronomy to modern signal and image processing
 | doi=10.1109/5.993400 | year=2002
 | journal=Proceedings of the IEEE | volume=90 | issue=3 | pages=319–342
 | url = http://bigwww.epfl.ch/publications/meijering0201.pdf}}</ref>

Uses of Lagrange polynomials include the [[Newton–Cotes formulas|Newton–Cotes method]] of [[numerical integration]], [[Shamir's Secret Sharing|Shamir's secret sharing scheme]] in [[cryptography]], and [[Reed–Solomon error correction]] in [[coding theory]].

For equispaced nodes, Lagrange interpolation is susceptible to [[Runge's phenomenon]] of large oscillation.