Inverting sparse matrices by tree partitioning

Abstract

This pape r s tudies the tree partitioning of a graph whose de finition is base d o n th e patt e rn of zero e le m ent s prese nt in a given matrix.Thi s partitionin g the n indi cat es a parti cul a rly advantageo us strate gy for e mploying bloc k Gaussian eliminat ion over a certain class of matrices.The strategy is exploite d for matrix inve rsion, where it is espec ially appropr iat e for proble ms whic h require findin g only selected s ubmatrices of th e inverse.A graph-theoretic algorithm is give n for a utomati cally generating tree partitionings fo r any matrix.Combinatorial properties of this pro cedure are al so di sc usse d.