Jacobi method
E157386
The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobi method canonical | 8 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382439 — 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: Jacobi method Context triple: [Gauss–Seidel method, relatedTo, Jacobi method]
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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.
-
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.
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.
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D.
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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E.
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.
- 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: Jacobi method Target entity description: The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
-
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.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm for solving linear systems
ⓘ
iterative numerical method ⓘ |
| advantage | highly parallelizable on modern hardware ⓘ |
| appliedIn |
discretized partial differential equations
ⓘ
engineering simulations ⓘ scientific computing ⓘ |
| assumes | linear system A x = b ⓘ |
| basedOn | fixed-point iteration ⓘ |
| belongsTo | classical iterative methods ⓘ |
| canBeUsedAs | preconditioner in iterative solvers ⓘ |
| category | stationary iterative method ⓘ |
| comparedTo | Gauss–Seidel method ⓘ |
| component |
D is the diagonal part of A
ⓘ
L is the strict lower triangular part of A ⓘ U is the strict upper triangular part of A ⓘ |
| convergenceDependsOn | spectral radius of the iteration matrix ⓘ |
| convergenceRateDependsOn | conditioning of the matrix A ⓘ |
| convergesIf |
spectral radius ρ(B_J) < 1
ⓘ
the coefficient matrix is strictly diagonally dominant ⓘ the coefficient matrix is symmetric positive definite ⓘ |
| differenceFrom | uses only previous-iteration values for all variables ⓘ |
| disadvantage | requires storage of two full solution vectors per iteration ⓘ |
| field | numerical linear algebra ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| implementationDetail | often implemented with two arrays for old and new iterates ⓘ |
| initialGuess | requires an initial approximation x^{(0)} ⓘ |
| iterationIndex | k denotes the iteration number ⓘ |
| iterationMatrix | B_J = D^{-1}(L + U) ⓘ |
| languageVariant | also called Jacobi iteration ⓘ |
| matrixDecomposition | A = D - L - U ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| notation | often denoted by x^{(k)} for the k-th iterate ⓘ |
| numericalProperty | error decreases approximately geometrically when convergent ⓘ |
| operatesOn | square matrices ⓘ |
| property |
all components of the new iterate can be computed in parallel
ⓘ
may converge slowly compared to other iterative methods ⓘ simple to implement ⓘ |
| relatedConcept |
Krylov subspace methods
ⓘ
Richardson iteration ⓘ successive over-relaxation ⓘ |
| requires |
access to diagonal entries of A
ⓘ
nonzero diagonal entries in the coefficient matrix ⓘ |
| stoppingCriterion |
norm of difference between successive iterates below tolerance
ⓘ
residual norm ∥Ax^{(k)} − b∥ below tolerance ⓘ |
| typicalUseCase | large sparse linear systems ⓘ |
| updateRule | x_i^{(k+1)} = (1/a_{ii}) (b_i - Σ_{j≠i} a_{ij} x_j^{(k)}) ⓘ |
| updates | each variable using values from the previous iteration ⓘ |
| usedFor |
approximating solutions of Ax = b
ⓘ
solving systems of linear equations ⓘ |
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: Jacobi method Description of subject: The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.