Richardson iteration
E157388
Richardson iteration is an early iterative method for solving linear systems and other operator equations, based on repeated relaxation steps to progressively improve an approximate solution.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Richardson iteration canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382441 — 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: Richardson iteration Context triple: [Gauss–Seidel method, relatedTo, Richardson iteration]
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A.
Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
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B.
Picard iteration
Picard iteration is a successive approximation method used to construct solutions to ordinary differential equations and establish their existence and uniqueness.
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C.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
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D.
Crank–Nicolson scheme
The Crank–Nicolson scheme is a finite difference method for numerically solving time-dependent partial differential equations, especially parabolic ones like the heat equation, known for its second-order accuracy and unconditional stability.
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E.
Gaussian elimination
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
- 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: Richardson iteration Target entity description: Richardson iteration is an early iterative method for solving linear systems and other operator equations, based on repeated relaxation steps to progressively improve an approximate solution.
-
A.
Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
-
B.
Picard iteration
Picard iteration is a successive approximation method used to construct solutions to ordinary differential equations and establish their existence and uniqueness.
-
C.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
-
D.
Crank–Nicolson scheme
The Crank–Nicolson scheme is a finite difference method for numerically solving time-dependent partial differential equations, especially parabolic ones like the heat equation, known for its second-order accuracy and unconditional stability.
-
E.
Gaussian elimination
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
iterative method
ⓘ
method for solving linear systems ⓘ numerical algorithm ⓘ |
| advantage |
low memory requirements
ⓘ
simple implementation ⓘ |
| appliesTo |
linear systems of equations
ⓘ
operator equations ⓘ |
| canBePreconditionedBy |
left preconditioner
ⓘ
right preconditioner ⓘ |
| canDivergeIf | relaxation parameter is chosen too large ⓘ |
| category | linear solver ⓘ |
| convergenceDependsOn |
magnitude of relaxation parameter
ⓘ
spectrum of the matrix A ⓘ |
| convergesIf | spectral radius of (I - \omega A) is less than 1 ⓘ |
| field |
numerical analysis
ⓘ
numerical linear algebra ⓘ |
| generalization | preconditioned Richardson method ⓘ |
| goal | progressive improvement of an approximate solution ⓘ |
| hasParameter |
relaxation parameter
ⓘ
step size ⓘ |
| hasVariant |
accelerated Richardson iteration
ⓘ
damped Richardson iteration ⓘ |
| historicalSignificance | one of the earliest iterative methods for linear systems ⓘ |
| isSpecialCaseOf |
fixed-point iteration
ⓘ
stationary iterative method ⓘ |
| limitation |
sensitivity to relaxation parameter
ⓘ
slow convergence without good parameter choice ⓘ |
| namedAfter | Lewis Fry Richardson ⓘ |
| relatedTo |
Gauss–Seidel method
ⓘ
Jacobi method ⓘ gradient descent ⓘ preconditioned Richardson iteration ⓘ successive over-relaxation ⓘ |
| requires | choice of relaxation parameter ⓘ |
| typicalUse |
discretized partial differential equations
ⓘ
large sparse linear systems ⓘ |
| updateFormula | x_{k+1} = x_k + \,\omega\,(b - A x_k) ⓘ |
| usedIn |
iterative refinement schemes
ⓘ
time-marching interpretations of elliptic problems ⓘ |
| uses | relaxation steps ⓘ |
| yearProposed | 1910s ⓘ |
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: Richardson iteration Description of subject: Richardson iteration is an early iterative method for solving linear systems and other operator equations, based on repeated relaxation steps to progressively improve an approximate solution.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.