Solving a positive definite system of linear equations via the matrix exponential

We present a new direct algorithm for solving a system of linear equations with a positive definite matrix by discretizing a continuous-time dynamical system for a large sampling time. The obtained algorithm is highly fine-grain parallelizable and its computational complexity grows logarithmically with respect to the condition number of the system of linear equations. When the parallelism is fully exploited, the algorithm is shown to be more efficient in terms of computational speed in comparison to other popular methods for solving a positive definite system of linear equations, especially for large and ill-conditioned problems.