Gauss–Seidel method
E29368
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.
Aliases (1)
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm
→
iterative method → linear solver → numerical method → |
| advantage |
often faster than Jacobi method per iteration
→
|
| alternativeName |
Liebmann method
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|
| appliesTo |
linear systems Ax = b
→
square matrices → |
| basedOn |
fixed-point iteration
→
|
| category |
direct and iterative methods for linear systems
→
|
| characteristic |
low memory requirements
→
simple to implement → stationary iterative method → uses latest available values in iteration → |
| convergesUnder |
strict diagonal dominance of matrix A
→
symmetric positive definite matrices → |
| dateIntroduced |
19th century
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|
| disadvantage |
inherently sequential
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less suitable for parallelization than Jacobi method → may fail to converge for some matrices → |
| field |
computational engineering
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numerical analysis → scientific computing → |
| generalizedBy |
Successive Over-Relaxation method
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|
| hasComponent |
iteration matrix
→
splitting of matrix A into L and U → |
| matrixCondition |
convergence depends on spectral radius of iteration matrix
→
|
| namedAfter |
Carl Friedrich Gauss
→
Philipp Ludwig von Seidel → |
| relatedTo |
Jacobi method
→
Richardson iteration → Successive Over-Relaxation → conjugate gradient method → |
| requires |
initial guess for solution vector
→
|
| stoppingCriterion |
residual norm below tolerance
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small change between successive iterates → |
| updateType |
sequential update
→
|
| usedFor |
discretized partial differential equations
→
iterative refinement of linear system solutions → solving large sparse linear systems → solving systems of linear equations → |
| usedIn |
computational fluid dynamics
→
electromagnetic field simulation → finite difference methods → finite element methods → structural analysis → |
Referenced by (2)
| Subject (surface form when different) | Predicate |
|---|---|
|
Gauss–Seidel method
("Liebmann method")
→
|
alternativeName |
|
Carl Friedrich Gauss
→
|
hasConceptNamedAfter |