Parallel Numerical Algorithms for Symmetric Positive Definite Linear Systems

We give a matrix factorization for the solution of the linear system Ax = f , when coefficient matrix A is a dense symmetric positive definite matrix. We call this factorization as "WW T factorization". The algorithm for this factorization is given. Existence and backward error analysis of the method are given. The WDWT factorization is also presented. When the coefficient matrix is a symmetric tridiagonal matrix, a small modification to the WWT factorization is given and we call this factorization as "WZ factorization". In this factorization the inner (n — 2) x (n — 2) submatrices of W and Z are same as the inner (n — 2) x (n — 2) submatrices of W and WT respectively corresponding to the case when coefficient matrix is symmetric tridiagonal matrix. When combined with partitioning scheme, it renders a divide and conquer algorithm for the symmetric tridiagonal linear systems. We proved the existence of the factorization in the important cases which occur frequently in scientific computations; when the coefficient matrix A is (0 Symmetric positive definite matrix (ii) Symmetric diagonally dominant in addition to the nonsingularity. Solution procedure crucially hinges on the solution of the "reduced system". We proved that reduced system retains the original properties of the original matrix like symmetric positive definiteness, diagonally dominance. The backward error analysis is given. The WDZ algorithm is also designed for symmetric tridiagonal linear systems.