Analysis of fixing nodes used in generalized inverse computation

Abstract

In various fields of numerical mathematics, there arises the need to compute a generalized inverse of a symmetric positive semidefinite matrix, for example in the solution of contact problems. Systems with semidefinite matrices can be solved by standard direct methods for the solution of systems with positive definite matrices adapted to the solution of systems with only positive semidefinite matrix. One of the possibilities is a modification of Cholesky decomposition using so called fixing nodes, which is presented in this paper with particular emphasise on proper definition of fixing nodes. The generalised inverse algorithm consisting in Cholesky decomposition with usage of fixing nodes is adopted from paper [1]. In [1], authors choose the fixing nodes using Perron vector of an adjacency matrix of the graph which is only a sub-optimal choice. Their choice is discussed in this paper together with other possible candidates on fixing node. Several numerical experiments including all candidates have been done. Based on these results, it turns out that using eigenvectors of Laplacian matrix provides better choice of fixing node than using Perron vector.