Preserving Positive Definiteness in Hierarchically Semiseparable Matrix Approximations

Abstract

Given a symmetric positive definite (s.p.d.) matrix, two methods are proposed for directly constructing hierarchically semiseparable (HSS) matrix approximations that are guaranteed to be positive definite. The methods are based on a new, recursive description of the HSS approximation process where projection is used to compress the off-diagonal blocks. The recursive description also leads to a new error analysis of HSS approximations. By constructing an s.p.d. HSS approximation directly, rather than in a factored form, the approximation errors can be better understood. As could be expected, larger approximation errors are introduced in the new s.p.d. methods compared to those in existing HSS approximation methods where positive definiteness is not guaranteed. However, numerical tests using the approximations as preconditioners show that methods that preserve positive definiteness may be better than other methods even when those methods happen to generate a positive definite preconditioner. Like existing HSS construction algorithms, the new methods have quadratic computational complexity and can be implemented in parallel.