Runge phenomenon
E620658
The Runge phenomenon is a numerical analysis effect where high-degree polynomial interpolation, especially at equally spaced points, produces large oscillations and poor approximations near the interval endpoints.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
interpolation artifact
ⓘ
numerical analysis phenomenon ⓘ |
| appliesTo |
analytic functions on closed intervals
ⓘ
equally spaced Lagrange interpolation ⓘ |
| associatedWith |
Lagrange interpolation polynomials
NERFINISHED
ⓘ
Runge function ⓘ |
| avoidedBy |
using Chebyshev-distributed nodes
ⓘ
using non-uniform interpolation nodes ⓘ using piecewise low-degree polynomials ⓘ using spline interpolation instead of high-degree polynomials ⓘ |
| cause |
equally spaced nodes leading to large Lebesgue constants
ⓘ
use of high-degree polynomials over large intervals ⓘ |
| characterizedBy |
large oscillations of interpolating polynomial
ⓘ
poor approximation near interval endpoints ⓘ |
| demonstrates |
high-degree polynomials can approximate worse than low-degree ones
ⓘ
uniform convergence of interpolation is not guaranteed ⓘ |
| describedIn | theory of polynomial interpolation error ⓘ |
| effect |
divergence of interpolating polynomials at endpoints
ⓘ
increased maximum interpolation error with higher degree ⓘ loss of uniform convergence of interpolation sequence ⓘ |
| field |
approximation theory
ⓘ
interpolation theory ⓘ numerical analysis ⓘ |
| formalProperty | interpolation error grows without bound near endpoints as degree increases for certain functions ⓘ |
| hasDomain | real-valued functions on intervals ⓘ |
| historicalPeriod | early 20th century ⓘ |
| implication |
careful choice of interpolation nodes is important
ⓘ
global high-degree polynomial interpolation can be unstable ⓘ |
| mathematicalNature | counterexample to naive expectations about interpolation convergence ⓘ |
| namedAfter | Carl Runge NERFINISHED ⓘ |
| occursIn |
global polynomial interpolation
ⓘ
interpolation on closed intervals ⓘ |
| occursWhen |
using equally spaced interpolation nodes
ⓘ
using high-degree polynomial interpolation ⓘ |
| relatedConcept |
Chebyshev interpolation
NERFINISHED
ⓘ
Chebyshev nodes NERFINISHED ⓘ Gibbs phenomenon NERFINISHED ⓘ Lebesgue constant NERFINISHED ⓘ minimax approximation ⓘ piecewise polynomial interpolation ⓘ polynomial interpolation ⓘ splines ⓘ |
| typicalExampleFunction | Runge function f(x) = 1 / (1 + 25 x^2) ⓘ |
| typicalExampleInterval | [-1, 1] ⓘ |
| typicalNodeDistribution | equally spaced points on [-1,1] ⓘ |
| usedAs | standard example in numerical analysis textbooks ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.