This question comes from https://stats.stackexchange.com/questions/457375/recover-full-covariance-matrix-from-covariance-diagonal-and-precision-off-diagon where it have not found answers. So, let $\Sigma$ be an $N\times N$ covariance (that is, positive semidefinite) matrix, but we do only know its diagonal and the off-diagonal elements of its inverse $\Sigma^{-1}$ (known as precision matrix).

How can we find $\Sigma$ effectively?