This is useful in case where the values are very close to zero or
possibly even slightly negative. This can quite easily happen due to
numerical errors.
A common strategy is to replace these values with a small epsilon value
that keeps the matrix SPD. Some libraries even do this by default (such
as https://github.com/STOR-i/GaussianProcesses.jl/issues/1).
We point the user to `LU` decomposition and also make it clearer that
the method is basically a hack. The public method no longer takes an
`Option` which didn't really make sense. A private method is used to not
repeat implementation in `new`.
This is useful if there the data for the matrix comes from elsewhere,
such as a custom algorithm. One such use-case can be seen
https://github.com/nestordemeure/friedrich/issues/43 where we do special
handling for the try_sqrt or is_zero cases that would normally fail the
decomposition in `Cholesky::new()`.
The previous implementation was correct only for real elements. The
Cholesky decomposition is `L L^H`, so the determinant is `det(L) *
det(L^H)`. Since `L` is a triangular matrix, `det(L)` is the product
of the diagonal elements of `L`. Since `L^H` is triangular and its
diagonal elements are the conjugates of the diagonal elements of `L`,
`det(L^H)` is the conjugate of `det(L)`. So, the overall determinant
is the product of the diagonal elements of `L` times its conjugate.
Most call sites still invoke UB through `assume_init`. Said call sites instead invoke `unimplemented!()` if the `no_unsound_assume_init` feature is enabled, to make it easier to gradually fix them.
Progress towards #556.
Mateusz Kowalczyk
Sébastien Crozet
Violeta Hernández
Malte Tammena
Malte Tammena
Jim Turner
Avi Weinstock
Michael Stevens
Nestor Demeure