Bartels–Stewart algorithm
E695944
The Bartels–Stewart algorithm is a numerical linear algebra method that efficiently solves certain matrix equations, particularly Sylvester and Lyapunov equations, using Schur decompositions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bartels–Stewart algorithm canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7833262 — 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: Bartels–Stewart algorithm Context triple: [Lyapunov equation, solvedBy, Bartels–Stewart algorithm]
-
A.
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.
-
B.
LinearAlgebra
LinearAlgebra is Julia’s standard library module providing core functionality for vectors, matrices, and advanced linear algebra operations.
-
C.
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.
-
D.
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.
-
E.
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.
- 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: Bartels–Stewart algorithm Target entity description: The Bartels–Stewart algorithm is a numerical linear algebra method that efficiently solves certain matrix equations, particularly Sylvester and Lyapunov equations, using Schur decompositions.
-
A.
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.
-
B.
LinearAlgebra
LinearAlgebra is Julia’s standard library module providing core functionality for vectors, matrices, and advanced linear algebra operations.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm in numerical linear algebra
ⓘ
matrix equation solver ⓘ numerical algorithm ⓘ |
| advantage |
avoids explicit Kronecker product formulation
ⓘ
reduces Sylvester equation to sequence of simpler triangular problems ⓘ |
| appliesTo |
A X A^T - X + Q = 0
ⓘ
AX + X B^T + Q = 0 ⓘ AX + XB = C ⓘ |
| basedOn |
back substitution
ⓘ
triangular matrix equations ⓘ |
| category | direct method for matrix equations ⓘ |
| complexity | O(n^3) for n×n matrices (up to constants depending on sizes) ⓘ |
| field |
computational mathematics
ⓘ
numerical linear algebra ⓘ |
| implementedIn |
LAPACK
NERFINISHED
ⓘ
MATLAB lyap function ⓘ SciPy scipy.linalg.solve_continuous_lyapunov ⓘ SciPy scipy.linalg.solve_discrete_lyapunov NERFINISHED ⓘ SciPy scipy.linalg.solve_sylvester NERFINISHED ⓘ |
| input |
matrix A
ⓘ
matrix B ⓘ matrix C ⓘ |
| namedAfter |
George W. Stewart
NERFINISHED
ⓘ
Richard H. Bartels NERFINISHED ⓘ |
| originalAuthors |
G. W. Stewart
NERFINISHED
ⓘ
R. H. Bartels NERFINISHED ⓘ |
| originalJournal | Communications of the ACM NERFINISHED ⓘ |
| originalPublication | Solution of the matrix equation AX + XB = C ⓘ |
| output | matrix X ⓘ |
| property |
backward stable in typical implementations
ⓘ
exploits triangular structure ⓘ numerically stable ⓘ |
| relatedTo |
Hessenberg–Schur method
NERFINISHED
ⓘ
Lyapunov equation solvers in control libraries ⓘ |
| requirement | matrices A and B have no common eigenvalues for unique solution of Sylvester equation ⓘ |
| solves |
Sylvester equation
NERFINISHED
ⓘ
continuous Lyapunov equation ⓘ discrete Lyapunov equation ⓘ |
| step |
compute Schur decomposition of A
ⓘ
compute Schur decomposition of B ⓘ solve resulting triangular Sylvester equations ⓘ transform Sylvester equation into Schur basis ⓘ |
| usedIn |
control theory
ⓘ
model reduction ⓘ solution of Riccati equations via Lyapunov subproblems ⓘ stability analysis ⓘ |
| uses |
Schur decomposition
NERFINISHED
ⓘ
complex Schur decomposition ⓘ real Schur decomposition NERFINISHED ⓘ |
| yearIntroduced | 1972 ⓘ |
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: Bartels–Stewart algorithm Description of subject: The Bartels–Stewart algorithm is a numerical linear algebra method that efficiently solves certain matrix equations, particularly Sylvester and Lyapunov equations, using Schur decompositions.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.