Let $A$ be a $N\times N$ symmetric positive definite real matrix, and $f$ a column vector of length $N$. Moreover, let's call $C$ the inverse of $A$ and $\alpha$ the vector of solution of the linear system $$ A\alpha=f. $$
I am interested in computing, for each $k\in\{1,2,\dots,N\}$, the value $$ E_k:= {\alpha_k \over C^{(k)}_k}, $$ where $\alpha_k$ is the $k$-th element of $\alpha$, and $C^{(k)}_k$ is the $k$-th element of the $k$-th column of $C$ (that is, the $k$-th element of the diagonal of $C$).
I know that the matrix $A$ is badly conditioned, therefore the computation of $E_k$ using the above formula is not stable, because it require to invert the matrix~$A$. Is there a formula to compute the same value which is more stable from a computational point of view?