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.
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
QR algorithm NERFINISHED ⓘ 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 NERFINISHED ⓘ parallel Jacobi method ⓘ |
Referenced by (1)
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