Lagrange interpolation polynomial

E156183

interpolation method mathematical formula numerical analysis concept polynomial

The Lagrange interpolation polynomial is a classical formula in numerical analysis that constructs a unique polynomial passing through a given set of data points, widely used for interpolation and approximation.

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All labels observed (3)

Label	Occurrences
Lagrange interpolation	4
Lagrange interpolation polynomial canonical	3
Lagrange interpolation polynomials	1

How this entity was disambiguated

This entity first appeared as the object of triple T1358584 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Lagrange interpolation polynomial
Context triple: [Joseph-Louis Lagrange, knownFor, Lagrange interpolation polynomial]

A. Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
B. Polynomial Root Finder
Polynomial Root Finder is a TI-84 Plus calculator application that computes the roots of polynomial equations quickly and accurately.
C. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
D. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
E. method of least squares
The method of least squares is a fundamental mathematical technique for estimating unknown parameters by minimizing the sum of squared differences between observed and predicted values, widely used in statistics, data fitting, and regression analysis.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Lagrange interpolation polynomial
Target entity description: The Lagrange interpolation polynomial is a classical formula in numerical analysis that constructs a unique polynomial passing through a given set of data points, widely used for interpolation and approximation.

A. Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
B. Polynomial Root Finder
Polynomial Root Finder is a TI-84 Plus calculator application that computes the roots of polynomial equations quickly and accurately.
C. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
D. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
E. method of least squares
The method of least squares is a fundamental mathematical technique for estimating unknown parameters by minimizing the sum of squared differences between observed and predicted values, widely used in statistics, data fitting, and regression analysis.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	interpolation method ⓘ mathematical formula ⓘ numerical analysis concept ⓘ polynomial ⓘ
advantage	does not require solving linear systems for interpolation ⓘ explicit closed-form expression ⓘ
appliesTo	complex-valued functions ⓘ functions defined on intervals or discrete sets ⓘ real-valued functions ⓘ
assumes	interpolation nodes are pairwise distinct ⓘ
basisDefinition	L_j(x) = Π_{m≠j} (x - x_m)/(x_j - x_m) ⓘ
component	Lagrange basis polynomial ⓘ
degree	at most n if there are n+1 data points ⓘ
disadvantage	computationally expensive to update when adding new nodes ⓘ numerically unstable for high-degree interpolation on equidistant nodes ⓘ
errorTerm	E(x) = f^{(n+1)}(ξ)/(n+1)! Π_{j=0}^n (x - x_j) ⓘ involves (n+1)th derivative of the interpolated function ⓘ
field	approximation theory ⓘ numerical analysis ⓘ polynomial interpolation ⓘ
formula	P(x) = Σ_{j=0}^n y_j L_j(x) ⓘ
generalizationOf	linear interpolation ⓘ quadratic interpolation ⓘ
input	function values at interpolation nodes ⓘ set of interpolation nodes ⓘ
namedAfter	Joseph-Louis Lagrange ⓘ
notation	often denoted by P_n(x) or L_n(x) ⓘ
origin	18th century ⓘ
output	polynomial function ⓘ
property	uniqueness of interpolating polynomial for distinct nodes ⓘ
purpose	construct a polynomial passing through given points ⓘ interpolate a function from discrete data points ⓘ
relatedTo	Newton interpolation polynomial ⓘ Runge phenomenon ⓘ Vandermonde matrix ⓘ barycentric interpolation formula ⓘ
representation	linear combination of Lagrange basis polynomials ⓘ
requires	arithmetic operations on field elements ⓘ
satisfies	P(x_j) = y_j for all interpolation nodes x_j ⓘ
stability	improved by barycentric form ⓘ
usedFor	curve fitting through exact data points ⓘ function approximation ⓘ numerical differentiation ⓘ numerical integration (via interpolatory quadrature) ⓘ
usedIn	computer graphics ⓘ data fitting ⓘ finite element method shape function construction ⓘ signal processing ⓘ

How these facts were elicited