Simpson's rule
E259761
Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolas through groups of data points.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Simpson's rule canonical | 5 |
| Simpson's 1/3 rule | 1 |
| composite Simpson's rule | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364433 — 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: Simpson's rule Context triple: [Riemann integral, approximatedBy, Simpson's rule]
-
A.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
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B.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
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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.
Milstein method
The Milstein method is a numerical scheme for solving stochastic differential equations that improves on the Euler–Maruyama method by including derivative terms of the diffusion coefficient for higher accuracy.
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E.
Halley’s method for solving equations
Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
- 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: Simpson's rule Target entity description: Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolas through groups of data points.
-
A.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
B.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
-
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.
Milstein method
The Milstein method is a numerical scheme for solving stochastic differential equations that improves on the Euler–Maruyama method by including derivative terms of the diffusion coefficient for higher accuracy.
-
E.
Halley’s method for solving equations
Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
numerical integration method
ⓘ
quadrature rule ⓘ |
| alternativeName |
Simpson's 1/3 rule
ⓘ
linked to:
Simpson's rule
|
| appliesTo | one-dimensional integrals ⓘ |
| approximates |
area under a curve
ⓘ
definite integral ⓘ |
| assumes | equally spaced points ⓘ |
| canBeAppliedAs |
composite Simpson's rule
ⓘ
linked to:
Simpson's rule
|
| category | numerical quadrature ⓘ |
| comparedTo |
midpoint rule
ⓘ
trapezoidal rule ⓘ |
| compositeVersionDescription | applies basic Simpson's rule on each pair of subintervals and sums results ⓘ |
| conditionOnN | n is even ⓘ |
| convergenceRate | fourth order ⓘ |
| derivationMethod | Lagrange interpolating polynomial of degree 2 ⓘ |
| errorOrder | O(h⁴) ⓘ |
| errorTerm | −(b−a)/180 · h⁴ f⁽⁴⁾(ξ) ⓘ |
| exactForPolynomialsUpToDegree | 3 ⓘ |
| field |
calculus
ⓘ
numerical analysis ⓘ |
| formula | ∫_a^b f(x) dx ≈ (h/3)[f(x₀)+4f(x₁)+2f(x₂)+4f(x₃)+…+4f(x_{n−1})+f(x_n)] ⓘ |
| generalizedTo | multiple integrals via iterated application ⓘ |
| historicalNote | formula known before Thomas Simpson but popularized by him ⓘ |
| implementationDetail | often used with adaptive step size control ⓘ |
| integrationIntervalNotation | [a,b] ⓘ |
| limitation |
less effective when function is highly oscillatory
ⓘ
requires even number of subintervals so may need to adjust partition ⓘ |
| moreAccurateThan |
midpoint rule for smooth functions
ⓘ
trapezoidal rule for smooth functions ⓘ |
| namedAfter | Thomas Simpson ⓘ |
| nodesPerPanel | 3 ⓘ |
| orderOfNewtonCotes | 2 ⓘ |
| panelsPerTwoSubintervals | 1 ⓘ |
| relatedRule |
Simpson's 3/8 rule
ⓘ
linked to:
Newton–Cotes formulas
|
| requires |
even number of subintervals
ⓘ
odd number of sample points ⓘ |
| requiresEvaluationOf | function at equally spaced nodes ⓘ |
| requiresFunctionSmoothness | f has continuous fourth derivative on [a,b] for standard error bound ⓘ |
| specialCaseOf | Newton–Cotes formulas ⓘ |
| stepSizeSymbol | h = (b − a)/n ⓘ |
| textbookTopicIn |
calculus courses
ⓘ
introductory numerical analysis courses ⓘ |
| usedIn |
engineering
ⓘ
physics ⓘ scientific computing ⓘ statistics ⓘ |
| uses | parabolic interpolation ⓘ |
| usesWeights | 1, 4, 2, 4, …, 2, 4, 1 ⓘ |
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: Simpson's rule Description of subject: Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolas through groups of data points.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Simpson's rule
linked to: Simpson's rule