Cholesky Decomposition
A sequential cholesky decomposition with run time O(n^3). The routine identifies when the matrix appears to be singular and adds epsilon jitter to the diagonal of these rows. Epsilon is defined by the double type in c++.
The reason our covariance matrices often appear singular is because two or more rows (or linear combinations of these) are identical to machine precision. To avoid this it is common to add iid noise to the entire covariance matrix. However this is an unnecessary hack to a numerical precision issue.
The code provided identifies this numerical issue occurs and adds the minimum amount noise to the diagnol to create a numerically stable result.