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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Simpson's 3/8 rule | 1 |
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 ⓘ |
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