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Clarify pros, cons and limitations of Cholesky and LDLt #621

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May 8, 2025
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Clarify pros, cons and limitations of Cholesky and LDLt #621

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10 changes: 10 additions & 0 deletions src/solvers/cholmod.jl
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Expand Up @@ -1563,6 +1563,9 @@ Setting the optional `shift` keyword argument computes the factorization of
it should be a permutation of `1:size(A,1)` giving the ordering to use
(instead of CHOLMOD's default AMD ordering).

See also [`ldlt`](@ref) for a similar factorization that does not require
positive definiteness, but can be significantly slower than `cholesky`.

# Examples

In the following example, the fill-reducing permutation used is `[3, 2, 1]`.
Expand Down Expand Up @@ -1728,6 +1731,10 @@ To include the effects of permutation, it is typically preferable to extract
`P'*L`) and `LtP = F.UP` (the equivalent of `L'*P`).
The complete list of supported factors is `:L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP`.

Unlike the related Cholesky factorization, the ``LDL'`` factorization does not
require `A` to be positive definite. However, it still requires all leading
principal minors to be well-conditioned and will fail if this is not satisfied.

When `check = true`, an error is thrown if the decomposition fails.
When `check = false`, responsibility for checking the decomposition's
validity (via [`issuccess`](@ref)) lies with the user.
Expand All @@ -1737,6 +1744,9 @@ Setting the optional `shift` keyword argument computes the factorization of
it should be a permutation of `1:size(A,1)` giving the ordering to use
(instead of CHOLMOD's default AMD ordering).

See also [`cholesky`](@ref) for a factorization that can be significantly
faster than `ldlt`, but requires `A` to be positive definite.

!!! note
This method uses the CHOLMOD[^ACM887][^DavisHager2009] library from [SuiteSparse](https://github.com/DrTimothyAldenDavis/SuiteSparse).
CHOLMOD only supports real or complex types in single or double precision.
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