Algorithms to Solve Hierarchically Semi-separable Systems

‘Hierarchical Semi-separable’ matrices (HSS matrices) form an important class of structured matrices for which matrix transformation algorithms that are linear in the number of equations (and a function of other structural parameters) can be given. In particular, a system of linear equations Ax = b can be solved with linear complexity in the size of the matrix, the overall complexity being linearly dependent on the defining data. Also, LU and ULV factorization can be executed ‘efficiently’, meaning with a complexity linear in the size of the matrix. This paper gives a survey of the main results, including a proof for the formulas for LU-factorization that were originally given in the thesis of Lyon [1], the derivation of an explicit algorithm for ULV factorization and related Moore-Penrose inversion, a complexity analysis and a short account of the connection between the HSS and the SSS (sequentially semi-separable) case. A direct consequence of the computational theory is that from a mathematical point of view the HSS structure is ‘closed’ for a number operations. The HSS complexity of a Moore-Penrose inverse equals the HSS complexity of the original, for a sum and a product of operators the HSS complexity is no more than the sum of the individual complexities. Mathematics Subject Classification (2000). Primary 65F; Secondary 05