Jacobi eigenvalue algorithm
E697940
The Jacobi eigenvalue algorithm is an iterative numerical method for computing all eigenvalues and eigenvectors of a real symmetric matrix by applying a sequence of orthogonal similarity transformations.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Jacobi eigenvalue algorithm canonical | 1 |
| cyclic Jacobi method | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7921598 — 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 eigenvalue algorithm Context triple: [Jacobi matrix, usedIn, Jacobi eigenvalue algorithm]
-
A.
Jacobi method
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.
-
B.
Richardson iteration
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.
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C.
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.
-
D.
arpack
arpack is a numerical software library for efficiently computing a few eigenvalues and eigenvectors of large sparse matrices, commonly used in scientific computing and machine learning.
-
E.
Jacobi
Jacobi is a German surname most famously associated with the 19th-century mathematician Carl Gustav Jacob Jacobi, known for his foundational work in elliptic functions and number theory.
- 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 eigenvalue algorithm Target entity description: The Jacobi eigenvalue algorithm is an iterative numerical method for computing all eigenvalues and eigenvectors of a real symmetric matrix by applying a sequence of orthogonal similarity transformations.
-
A.
Jacobi method
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.
-
B.
Richardson iteration
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.
-
C.
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.
-
D.
arpack
arpack is a numerical software library for efficiently computing a few eigenvalues and eigenvectors of large sparse matrices, commonly used in scientific computing and machine learning.
-
E.
Jacobi
Jacobi is a German surname most famously associated with the 19th-century mathematician Carl Gustav Jacob Jacobi, known for his foundational work in elliptic functions and number theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
eigenvalue algorithm
ⓘ
iterative method ⓘ matrix diagonalization method ⓘ numerical algorithm ⓘ |
| accuracy | high relative accuracy for well-separated eigenvalues ⓘ |
| advantage |
conceptually simple
ⓘ
high accuracy for eigenvectors ⓘ produces orthogonal eigenvectors explicitly ⓘ |
| application |
principal component analysis
ⓘ
quantum mechanics eigenproblems ⓘ vibration analysis ⓘ |
| complexity | O(n^3) for an n-by-n matrix ⓘ |
| computes |
eigenvalues
ⓘ
eigenvectors ⓘ |
| convergenceType | global convergence for symmetric matrices ⓘ |
| coreOperation |
Jacobi rotations
ⓘ
linked to:
Givens rotations
plane rotations ⓘ |
| disadvantage |
not optimal for large-scale problems
ⓘ
relatively slow compared to modern methods ⓘ |
| field | numerical linear algebra ⓘ |
| goal | diagonalize a symmetric matrix ⓘ |
| historicalPeriod | 19th century origin ⓘ |
| implementation | used in some LAPACK routines historically ⓘ |
| lessSuitableFor |
sparse matrices
ⓘ
very large matrices ⓘ |
| matrixClass | normal matrices (via unitary version) ⓘ |
| methodType | iterative diagonalization ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| operatesOn |
Hermitian matrices
ⓘ
real symmetric matrices ⓘ |
| output |
diagonal matrix of eigenvalues
ⓘ
orthogonal matrix of eigenvectors ⓘ |
| pivotSelection | largest off-diagonal element strategy ⓘ |
| propertyPreserved |
eigenvalues
ⓘ
orthogonality of eigenvectors ⓘ symmetry of the matrix ⓘ |
| relatedTo |
Householder transformation methods
ⓘ
linked to:
Householder transformation
QR algorithm ⓘ power iteration ⓘ |
| requires | selection of pivot elements ⓘ |
| stability | numerically stable for symmetric problems ⓘ |
| stoppingCriterion | small off-diagonal elements ⓘ |
| suitableFor | small to medium size dense matrices ⓘ |
| transformationType | orthogonal similarity transformations ⓘ |
| uses | Givens rotations in some formulations ⓘ |
| variant |
blocked Jacobi method
ⓘ
cyclic Jacobi method ⓘ
linked to:
Jacobi eigenvalue algorithm
parallel Jacobi method ⓘ |
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 eigenvalue algorithm Description of subject: The Jacobi eigenvalue algorithm is an iterative numerical method for computing all eigenvalues and eigenvectors of a real symmetric matrix by applying a sequence of orthogonal similarity transformations.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Jacobi eigenvalue algorithm