Koksma–Hlawka inequality
E824092
The Koksma–Hlawka inequality is a fundamental result in numerical analysis and discrepancy theory that bounds the error of quasi-Monte Carlo integration by the product of a function’s variation and the discrepancy of the sampling points.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Koksma inequality | 1 |
| Koksma–Hlawka inequality canonical | 1 |
| Koksma–Hlawka inequality in numerical integration | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9838927 — 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: Koksma–Hlawka inequality Context triple: [Johan Frederik Koksma, notableWork, Koksma–Hlawka inequality]
-
A.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
-
B.
Hermite constant
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
-
C.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
van der Corput method for estimating exponential sums
The van der Corput method for estimating exponential sums is a classical analytic number theory technique that provides bounds for oscillatory sums by exploiting differencing and smoothness properties of the phase function.
- 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: Koksma–Hlawka inequality Target entity description: The Koksma–Hlawka inequality is a fundamental result in numerical analysis and discrepancy theory that bounds the error of quasi-Monte Carlo integration by the product of a function’s variation and the discrepancy of the sampling points.
-
A.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
-
B.
Hermite constant
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
-
C.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
van der Corput method for estimating exponential sums
The van der Corput method for estimating exponential sums is a classical analytic number theory technique that provides bounds for oscillatory sums by exploiting differencing and smoothness properties of the phase function.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in discrepancy theory ⓘ result in numerical analysis ⓘ |
| appliesTo |
deterministic numerical integration
ⓘ
integration over the unit cube [0,1]^s ⓘ |
| assumes |
function of bounded variation in the sense of Hardy–Krause
ⓘ
point set in the s-dimensional unit cube ⓘ |
| comparedWith | Monte Carlo error bounds based on variance ⓘ |
| compares |
exact integral
ⓘ
quasi-Monte Carlo integration error ⓘ |
| errorBoundDependsOn |
discrepancy of the sampling points
ⓘ
variation of the integrand ⓘ |
| errorType | worst-case error bound ⓘ |
| field |
discrepancy theory
ⓘ
numerical analysis ⓘ quasi-Monte Carlo methods ⓘ |
| generalizedBy | various weighted discrepancy inequalities ⓘ |
| givesUpperBoundOn | absolute integration error ⓘ |
| hasVariant |
Koksma–Hlawka inequality for anchored variation
NERFINISHED
ⓘ
weighted Koksma–Hlawka inequality NERFINISHED ⓘ |
| holdsFor | any dimension s ≥ 1 ⓘ |
| implies | quasi-Monte Carlo can achieve faster convergence than Monte Carlo for smooth low-variation functions ⓘ |
| influencedDevelopmentOf | quasi-Monte Carlo theory ⓘ |
| involves | supremum over axis-aligned boxes for discrepancy ⓘ |
| isNonProbabilistic | true ⓘ |
| languageOfOriginalWork | German ⓘ |
| mathematicalDomain |
analysis
ⓘ
number theory ⓘ |
| motivates | construction of low-discrepancy sequences ⓘ |
| namedAfter |
Edmund Hlawka
NERFINISHED
ⓘ
J. F. Koksma NERFINISHED ⓘ |
| relatedTo |
Erdős–Turán–Koksma inequality
NERFINISHED
ⓘ
Faure sequence ⓘ Halton sequence NERFINISHED ⓘ Sobol sequence NERFINISHED ⓘ low-discrepancy sequences ⓘ |
| relatesConcept |
discrepancy of point sets
ⓘ
quasi-Monte Carlo integration ⓘ variation of a function ⓘ |
| statesThat | integration error is bounded by the product of variation and discrepancy ⓘ |
| timePeriod | 20th century ⓘ |
| typicalFormulation | | (1/N) Σ_{n=1}^N f(x_n) − ∫_[0,1]^s f(u) du | ≤ V_HK(f) D_N^*(x_1,…,x_N) ⓘ |
| usedIn |
analysis of quasi-Monte Carlo algorithms
ⓘ
computational finance ⓘ high-dimensional numerical integration ⓘ uncertainty quantification ⓘ |
| usesConcept |
Hardy–Krause variation
NERFINISHED
ⓘ
star discrepancy ⓘ |
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: Koksma–Hlawka inequality Description of subject: The Koksma–Hlawka inequality is a fundamental result in numerical analysis and discrepancy theory that bounds the error of quasi-Monte Carlo integration by the product of a function’s variation and the discrepancy of the sampling points.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Koksma inequality
this entity surface form:
Koksma–Hlawka inequality in numerical integration