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Given Cholesky decomposition of matrix A = LDL$^{T}$ = $A^{T}$ provide a possibly most efficient method to calculate upper left element of $A^{-1}$. I was thinking that this could be solved by using Gaussian algorithms, but couldn't really get anything out of it.

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You want to compute $$ e_1^TL^{-T}D^{-1}L^{-1}e_1=(L^{-1}e_1)^TD^{-1}(L^{-1}e_1) $$ which means you have to solve the lower left triangular system $Lv=e_1$ and then compute $v^TD^{-1}v=\sum d_k^{-1}v_k^2$.

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