A new parallel factorization A = DD ^tBC for band symmetric positive definite matrices

Abstract

We present a new factorization for band symmetric positive definite (s.p.d) matrices which is more useful for parallel computations than the classical Choleskey decomposition method. Let A be a band s.p.d matrix of order n and half bandwidth m and let p = 2^k be the number of processors. We show how to factor A as A = DD ^tBC using approximately 4nm ²/p parallel operations, which is the maximum number of operations used by any processor. Having this factorization, we improve the time to solve Ax=b by a factor of m using approximately 4nm/p parallel operations. 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 which is a Fortran machine-independent software for numerical linear algebra.