Newton–Cotes formulas
E898476
Newton–Cotes formulas are a family of numerical integration methods that approximate definite integrals by interpolating the integrand with equally spaced polynomial points.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Newton–Cotes formulas canonical | 1 |
| Simpson's 3/8 rule | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991438 — 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: Newton–Cotes formulas Context triple: [Simpson's rule, specialCaseOf, Newton–Cotes formulas]
-
A.
Gaussian quadrature rules
Gaussian quadrature rules are numerical integration methods that approximate definite integrals by optimally choosing evaluation points and weights to achieve exactness for polynomials up to a high degree.
-
B.
Simpson's rule
Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolas through groups of data points.
-
C.
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.
-
D.
Birkhoff interpolation
Birkhoff interpolation is a generalized form of polynomial interpolation that allows prescribing function and derivative values at selected points, not necessarily in a consecutive or complete pattern.
-
E.
Runge–Kutta methods
Runge–Kutta methods are a family of iterative techniques for numerically solving ordinary differential equations with higher accuracy than simple one-step schemes.
- 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: Newton–Cotes formulas Target entity description: Newton–Cotes formulas are a family of numerical integration methods that approximate definite integrals by interpolating the integrand with equally spaced polynomial points.
-
A.
Gaussian quadrature rules
Gaussian quadrature rules are numerical integration methods that approximate definite integrals by optimally choosing evaluation points and weights to achieve exactness for polynomials up to a high degree.
-
B.
Simpson's rule
Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolas through groups of data points.
-
C.
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.
-
D.
Birkhoff interpolation
Birkhoff interpolation is a generalized form of polynomial interpolation that allows prescribing function and derivative values at selected points, not necessarily in a consecutive or complete pattern.
-
E.
Runge–Kutta methods
Runge–Kutta methods are a family of iterative techniques for numerically solving ordinary differential equations with higher accuracy than simple one-step schemes.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
numerical integration method family
ⓘ
quadrature rule family ⓘ |
| advantage | simple weights for equally spaced grids ⓘ |
| appliesTo | continuous functions on a closed interval ⓘ |
| approximate | integral of f(x) over [a,b] ⓘ |
| are | one-dimensional quadrature rules ⓘ |
| assume | equally spaced nodes ⓘ |
| assumption | function sufficiently smooth on integration interval ⓘ |
| basedOn | polynomial interpolation ⓘ |
| canBe | composite rules over subintervals ⓘ |
| characterizedBy | equally spaced interpolation points ⓘ |
| closedFormulasUse | nodes including both endpoints ⓘ |
| compositeVersion | applies basic rule on many subintervals ⓘ |
| contrastWith | Gaussian quadrature ⓘ |
| convergenceProperty | converge as step size tends to zero for smooth functions ⓘ |
| disadvantage |
Runge phenomenon for many equally spaced nodes
ⓘ
instability for high-degree polynomials ⓘ |
| errorDependsOn |
degree of interpolating polynomial
ⓘ
higher derivatives of integrand ⓘ step size ⓘ |
| hasMember |
Boole's rule
NERFINISHED
ⓘ
Simpson's 3/8 rule NERFINISHED ⓘ Simpson's rule NERFINISHED ⓘ closed Newton–Cotes formulas NERFINISHED ⓘ open Newton–Cotes formulas ⓘ trapezoidal rule NERFINISHED ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| integrationIntervalNotation | [a,b] ⓘ |
| introducedInContextOf | classical analysis ⓘ |
| mathematicalField |
computational mathematics
ⓘ
numerical analysis ⓘ |
| namedAfter |
Isaac Newton
NERFINISHED
ⓘ
Roger Cotes NERFINISHED ⓘ |
| openFormulasUse | nodes excluding endpoints ⓘ |
| relatedConcept |
numerical quadrature
ⓘ
order of accuracy ⓘ truncation error ⓘ |
| relatedTo | finite difference approximations ⓘ |
| stabilityIssue | high-degree closed formulas can be numerically unstable ⓘ |
| subclassOf | interpolatory quadrature rules ⓘ |
| typicalDomain | real-valued functions ⓘ |
| typicalUse | low-order formulas like trapezoidal and Simpson's rule ⓘ |
| usedFor | approximating definite integrals ⓘ |
| usedIn |
applied physics computations
ⓘ
engineering simulations ⓘ scientific computing ⓘ |
| useWeights | precomputed coefficients for each node ⓘ |
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: Newton–Cotes formulas Description of subject: Newton–Cotes formulas are a family of numerical integration methods that approximate definite integrals by interpolating the integrand with equally spaced polynomial points.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Simpson's 3/8 rule