TITLE: The derivative of the Cholesky factor
QUESTION [9 upvotes]: Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am wondering if the derivative
$$
\frac{\mathrm{d}\operatorname{vech}\left(f(A)\right)}{\mathrm{d}\operatorname{vech}\left(A\right)}
$$
is known, where $\operatorname{vech}$ is the half-vectorization function.
(I am conjecturing that it is something like $L^{\top} \left(f(A)\otimes A^{-1}\right) L$, where $L$ is the elimination matrix, but I would need a reference or proof anyway.)

REPLY [7 votes]: I've written a relevant note on arXiv: http://arxiv.org/abs/1602.07527
I included the neat closed form solution pete gives in a comment, and also a messy expression (converted to the notation f = chol(A)):
$$
    \frac{\partial f_{ij}}{\partial A_{kl}} =
    \bigg(\sum_{m>j} f_{im}f_{mk}^{-1} + \tfrac{1}{2}f_{ij}f_{jk}^{-1}\bigg)f_{jl}^{-1}
    +
    (1-\delta_{kl})\bigg(\sum_{m>j} f_{im}f_{ml}^{-1} + \tfrac{1}{2}f_{ij}f_{jl}^{-1}\bigg)f_{jk}^{-1}.
$$
However, if you're interested in differentiating a larger expression, you can do that in $O(N^3)$, without computing all $O(N^4)$ derivatives in $\frac{\partial \mathrm{vech}(f)}{\partial \mathrm{vech}A}$. The note explains different ways to do that.
(pete: if you tell me who you are, I'll add a proper acknowledgement to my note in any future revision.)