This article discusses a more general and numerically stable Rybicki Press algorithm, which enables inverting and computing determinants of covariance matrices, whose elements are sums of exponentials. The algorithm is true in exact arithmetic and relies on introducing new variables and corresponding equations, thereby converting the matrix into a banded matrix of larger size. Linear complexity banded algorithms for solving linear systems and computing determinants on the larger matrix enable linear complexity algorithms for the initial semi-separable matrix as well. Benchmarks provided illustrate the linear scaling of the algorithm.