A parallel Cholesky algorithm for the solution of symmetric linear systems

For the solution of symmetric linear systems, the classical Cholesky method has proved to be difficult to parallelize. In the present paper, we first describe an elimination variant of Cholesky method to produce a lower triangular matrix which reduces the coefficient matrix of the system to an identity matrix. Then, this elimination method is combined with the partitioning method to obtain a parallel Cholesky algorithm. The total serial arithmetical operations count for the parallel algorithm is of the same order as that for the serial Cholesky method. The present parallel algorithm could thus perform with efficiency close to 1 if implemented on a multiprocessor machine. We also discuss the existence of the parallel algorithm; it is shown that for a symmetric and positive definite system, the presented parallel Cholesky algorithm is well defined and will run to completion.