Lanczos algorithm
E697941
The Lanczos algorithm is an iterative numerical method used to approximate eigenvalues and eigenvectors of large sparse matrices, particularly in scientific computing and numerical linear algebra.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Lanczos algorithm canonical | 1 |
| SciPy sparse.linalg.eigsh | 1 |
| block Lanczos method | 1 |
| thick-restart Lanczos method | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7921599 — 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: Lanczos algorithm Context triple: [Jacobi matrix, usedIn, Lanczos algorithm]
-
A.
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.
-
B.
Schmidt orthogonalization
Schmidt orthogonalization is a mathematical procedure, also known as the Gram–Schmidt process, that converts a set of linearly independent vectors into an orthonormal set spanning the same subspace.
-
C.
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.
-
D.
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.
-
E.
Schur algorithm
The Schur algorithm is a recursive procedure in complex analysis and operator theory used to construct and analyze Schur functions, playing a key role in interpolation problems and system 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: Lanczos algorithm Target entity description: The Lanczos algorithm is an iterative numerical method used to approximate eigenvalues and eigenvectors of large sparse matrices, particularly in scientific computing and numerical linear algebra.
-
A.
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.
-
B.
Schmidt orthogonalization
Schmidt orthogonalization is a mathematical procedure, also known as the Gram–Schmidt process, that converts a set of linearly independent vectors into an orthonormal set spanning the same subspace.
-
C.
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.
-
D.
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.
-
E.
Schur algorithm
The Schur algorithm is a recursive procedure in complex analysis and operator theory used to construct and analyze Schur functions, playing a key role in interpolation problems and system theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Krylov subspace method
ⓘ
iterative method ⓘ numerical algorithm ⓘ |
| application |
electronic structure calculations
ⓘ
graph partitioning ⓘ information retrieval ⓘ model reduction ⓘ principal component analysis ⓘ quantum many-body problems ⓘ structural engineering ⓘ vibration analysis ⓘ |
| basedOn | Krylov subspace ⓘ |
| computes | orthonormal basis of Krylov subspace ⓘ |
| field |
computational chemistry
ⓘ
computational physics ⓘ numerical linear algebra ⓘ scientific computing ⓘ |
| hasVariant |
block Lanczos method
ⓘ
linked to:
Lanczos algorithm
implicitly restarted Lanczos method ⓘ thick-restart Lanczos method ⓘ
linked to:
Lanczos algorithm
|
| implementedIn |
ARPACK
ⓘ
linked to:
arpack
MATLAB eigs function ⓘ SLEPc ⓘ SciPy sparse.linalg.eigsh ⓘ
linked to:
Lanczos algorithm
|
| inputType |
Hermitian matrix
ⓘ
large sparse matrix ⓘ symmetric matrix ⓘ |
| namedAfter | Cornelius Lanczos ⓘ |
| outputType |
Ritz values
ⓘ
Ritz vectors ⓘ tridiagonal matrix ⓘ |
| property |
matrix-free
ⓘ
memory efficient for large sparse problems ⓘ sensitive to loss of orthogonality ⓘ short-term recurrence ⓘ well-suited for extreme eigenvalues ⓘ |
| purpose |
approximation of eigenvalues of large sparse matrices
ⓘ
approximation of eigenvectors of large sparse matrices ⓘ reduction of large symmetric matrices to tridiagonal form ⓘ |
| relatedTo |
Arnoldi iteration
ⓘ
linked to:
Arnoldi method
Rayleigh–Ritz method ⓘ
linked to:
Galerkin method
conjugate gradient method ⓘ
linked to:
Conjugate Gradient method
power method ⓘ |
| requires |
initial starting vector
ⓘ
matrix-vector products ⓘ |
| uses | three-term recurrence ⓘ |
| yearIntroduced | 1950 ⓘ |
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: Lanczos algorithm Description of subject: The Lanczos algorithm is an iterative numerical method used to approximate eigenvalues and eigenvectors of large sparse matrices, particularly in scientific computing and numerical linear algebra.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lanczos algorithm
linked to: Lanczos algorithm
linked to: Lanczos algorithm