A Fast Parallel Cholesky Decomposition Algorithm for Tridiagonal Symmetric Matrices

Abstract

In this paper we present a new parallel algorithm for computing the $LL^T$ decomposition of real symmetric positive-definite tridiagonal matrices. The algorithm consists of a preprocessing and a factoring stage. In the preprocessing stage it determines a rank-($p-1$) correction to the original matrix ($p=$ number of processors) by precomputing selected components $x_k$ of the L factor, $k=1\dt p-1$. In the factoring stage it performs independent factorizations of p matrices of order $n/p$. The algorithm is especially suited for machines with both vector and processor parallelism, as confirmed by the experiments carried out on a Connection Machine CM5 with 32 nodes. Let $\hat{x}_k$ and $\hat{x}'_k$ denote the components computed in the preprocessing stage and the corresponding values (re)computed in the factorization stage, respectively. Assuming that $\abs{\hat{x}_k/\hat{x}'_k}$ is small, $k=1\dt p-1$, we are able to prove that the algorithm is stable in the backward sense. The above assumption is justified both experimentally and theoretically. In fact, we have found experimentally that $\abs{\hat{x}_k/\hat{x}'_k}$ is small even for ill-conditioned matrices, and we have proven by an a priori analysis that the above ratios are small provided that preprocessing is performed with suitably larger precision.