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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Runge phenomenon canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6800912 — 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: Runge phenomenon Context triple: [Lagrange interpolation polynomial, relatedTo, Runge phenomenon]
-
A.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
B.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
-
C.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
D.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
-
E.
Hermite interpolation
Hermite interpolation is a numerical analysis method for constructing a polynomial that matches both function values and specified derivatives at given data points.
- 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: Runge phenomenon Target entity description: 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.
-
A.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
B.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
-
C.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
D.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
-
E.
Hermite interpolation
Hermite interpolation is a numerical analysis method for constructing a polynomial that matches both function values and specified derivatives at given data points.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
interpolation artifact
ⓘ
numerical analysis phenomenon ⓘ |
| appliesTo |
analytic functions on closed intervals
ⓘ
equally spaced Lagrange interpolation ⓘ |
| associatedWith |
Lagrange interpolation polynomials
ⓘ
linked to:
Lagrange interpolation polynomial
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 ⓘ |
| occursIn |
global polynomial interpolation
ⓘ
interpolation on closed intervals ⓘ |
| occursWhen |
using equally spaced interpolation nodes
ⓘ
using high-degree polynomial interpolation ⓘ |
| relatedConcept |
Chebyshev interpolation
ⓘ
Chebyshev nodes ⓘ Gibbs phenomenon ⓘ Lebesgue constant ⓘ 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 ⓘ |
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: Runge phenomenon Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.