Lagrange interpolation polynomial
E156183
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lagrange interpolation | 1 |
| Lagrange interpolation polynomial canonical | 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.
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B.
Polynomial Root Finder
Polynomial Root Finder is a TI-84 Plus calculator application that computes the roots of polynomial equations quickly and accurately.
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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.
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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.
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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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lagrange interpolation polynomial Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lagrange interpolation