LDLT v LLT for cholesky decompose

[Robert Trangucci]

@bgoodri it seems like a lot of our multivariate code uses stan::math::ldlt_factor rather than LLT. Is there any reason to use LLT for cholesky rather than LDLT? It'd be good for consistency.

We have a function to check a Stan::math::ldlt_factor for positive definite ness called check_ldlt_factor in math/prim/mat/err. If we used ldlt internally we wouldn't need a check_pos_definite function to take an Eigen::LLT type.

Rob

[Ben Goodrich]

LDLT in Eigen is not so good for Stan, except for checking positive definiteness and solving systems of equations. Eigen does pivoting so it is really a

Sigma = PLDL'P'

decomposition. Also, the pivoting is discontinuous, so it is not great for HMC.

Ben

[Michael Betancourt]

Autodiffing through LDLT is a bad idea, but it wouldn’t necessarily

be bad given analytic derivatives.  My understanding is that LDLT

is more stable than LLT but also slower.

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[Ben Goodrich]

We could, in the future, have a decomposition that returned P * L and D. But I think we should leave cholesky_decompose as is because it returns an actual lower-triangular matrix. All these matrix decompositions have analytic derivatives, but they are really tedious. The first-order derivatives of LDLT involve inverses of principal submatrices of the original matrix.

Ben

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