Generic information, not Eigen-specific | Eigen-specific | |||||||
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Decomposition | Requirements on the matrix | Speed | Algorithm reliability and accuracy | Rank-revealing | Allows to compute (besides linear solving) | Linear solver provided by Eigen | Maturity of Eigen's implementation | Optimizations |
Invertible | Fast | Depends on condition number | - | - | Yes | Excellent | Blocking | |
- | Slow | Proven | Yes | - | Yes | Excellent | - | |
- | Fast | Depends on condition number | - | Orthogonalization | Yes | Excellent | Blocking | |
- | Fast | Good | Yes | Orthogonalization | Yes | Excellent | Soon: blocking | |
- | Slow | Proven | Yes | Orthogonalization | Yes | Average | - | |
Positive definite | Very fast | Depends on condition number | - | - | Yes | Excellent | Blocking | |
Positive or negative semidefinite1 | Very fast | Good | - | - | Yes | Excellent | Soon: blocking | |
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JacobiSVD (two-sided) | - | Slow (but fast for small matrices) | Excellent-Proven3 | Yes | Singular values/vectors, least squares | Yes (and does least squares) | Excellent | R-SVD |
Self-adjoint | Fast-average2 | Good | Yes | Eigenvalues/vectors | - | Good | Soon: specializations for 2x2 and 3x3 | |
Square | Slow-very slow2 | Depends on condition number | Yes | Eigenvalues/vectors | - | Average | - | |
Square and real | Average-slow2 | Depends on condition number | Yes | Eigenvalues/vectors | - | Average | - | |
Square | Fast-average2 | Depends on condition number | - | Generalized eigenvalues/vectors | - | Good | - | |
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Square and real | Average-slow2 | Depends on condition number | Yes | - | - | Average | - | |
Square | Slow-very slow2 | Depends on condition number | Yes | - | - | Average | - | |
Self-adjoint | Fast | Good | - | - | - | Good | Soon: blocking | |
Square | Average | Good | - | - | - | Good | Soon: blocking |
Notes:
A selfadjoint matrix is positive semi-definite if
for any non zero vector
. In the same vein, it is negative semi-definite if
for any non zero vector