LinearAlgebra: extend eigen with left eigenvectors and errorbounds
#48657
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| (The `k`th eigenvector can be obtained from the slice `F.vectors[:, k]`.) | ||
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| Iterating the decomposition produces the components `F.values` and `F.vectors`. | ||
| Iterating the decomposition produces the components `F.values`, `F.vectors`, `F.vectorsl`, | ||
| `F.eerrbd`, `F.verrbd`. |
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Doesn't this mean that values, vectors = eigen will not work anymore?
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No - the lhs tuple needs not to be exhaustive. You can even do a, = [1,2,3] to set a to 1.
| Iterating the decomposition produces the components `F.values`, `F.vectors`, `F.vectorsl`, | ||
| `F.eerrbd`, `F.verrbd`. | ||
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| The matrix of left eigenvectors is empty, if not opted in by `eigen( ; left=true)`. |
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That's annoying. Can we default to inv(Rvectors)?
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Or just compute them unconditionally? Are they much more expensive than the right?
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Calculation of left eigenvectors comes with an effort comparable to that of deriving it from the right vectors. That is approximately 10-15% for n in 100-1000 for random matrices. Actually vl = inv(vr)' ./ cn where cn is to normalize the columns.
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| The matrix of left eigenvectors is empty, if not opted in by `eigen( ; left=true)`. | ||
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| The error bounds are empty, if not opted in by `eigen( ; eerror=true, verror=true)`. |
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xerrbd are both vectors. You propose to give them size n. Why not Inf? For me a full vector of NaN or Inf doesn't look good.
| Eigen{T,V,S,U}(values::AbstractVector{V}, vectors::AbstractMatrix{T}) where {T,V,S,U} = | ||
| new(values, vectors) | ||
| vectorsl::S | ||
| eerrbd::R |
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These names are very cryptic. We're not constrained by Fortran or the need for short identifiers, might as well give them proper names
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The names are taken from the LAPACK docu. What do you think about valbounds and vecbounds.
stdlib/LinearAlgebra/src/eigen.jl
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| # Generalized eigenvalue problem. | ||
| """ | ||
| GeneralizedEigen <: Factorization | ||
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| Matrix factorization type of the generalized eigenvalue/spectral decomposition of | ||
| Matrix factorization type of the generalized eigenvalue/apply decomposition of |
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That was an edit error, should be unchanged.
This is a remake of #38483, this time with documentation and test cases included.
The changes (added features) are:
struct Eigenandstruct GeneralizedEigenhave new matrix fieldvectorslto keep optional left eigenvectors.struct Eigenhas additional vector fieldseerrbdandverrbdto keep optional error bounds for eigenvalues and eigenvectors.eigenhas new boolean keyword argumentsleft,eerror,verrorto opt in the calculation of the new output fields.The data are obtained by use of the existing
LAPACKAPI. See also user's guide