DDtBe for Band Symmetric Positive Definite Matrices

We present a new parallel factorization for band symmetric positive definite (s.p.d) matrices and show some of its applications. Let A be a band s.p.d matrix of order n and half bandwidth m. We show how to factor A as A =DDt Be using approximately 4nm2 jp parallel operations where p =21: is the number of processors. Having this factorization, we improve the time to solve Ax = b by a factor of m using approximately 4nmjp parallel operations. Our factorization is better for parallel computations than the Cholesky factorization A = LL t , since the parallel solution of the band triangular system Lx = b requires O(nm2 ) operations and is numerically unstable [SB77]. The total number of messages passed in the algorithm is 3p, each containing no more than 3m2 numbers, implying a low communication overhead. The algorithm can be adapted to any number of processors and implemented on many existing architectures. We specifically demonstrate its implementation on a k-dimensional hyper-cube. There are more applications to our algorithm, such as calculating det(A), using iterative refinement methods, or computing eigenvalues by the inverse power method. Numerical experiments indicate that our results are as good as LINPACK!.