Simpson's rule

E259761

numerical integration method quadrature rule

Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolas through groups of data points.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (3)

Label	Occurrences
Simpson's rule canonical	2
Simpson's 1/3 rule	1
composite Simpson's rule	1

Statements (48)

Predicate	Object
instanceOf	numerical integration method ⓘ quadrature rule ⓘ
alternativeName	Simpson's rule ⓘ surface form: Simpson's 1/3 rule
appliesTo	one-dimensional integrals ⓘ
approximates	area under a curve ⓘ definite integral ⓘ
assumes	equally spaced points ⓘ
canBeAppliedAs	Simpson's rule self-linksurface differs ⓘ surface form: composite 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	Newton–Cotes formulas ⓘ surface form: Simpson's 3/8 rule
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 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann integral → approximatedBy → Simpson's rule ⓘ

Riemann sums → contrastWith → Simpson's rule ⓘ

Simpson's rule → canBeAppliedAs → Simpson's rule self-linksurface differs ⓘ

this entity surface form: composite Simpson's rule

Simpson's rule → alternativeName → Simpson's rule ⓘ

this entity surface form: Simpson's 1/3 rule