On inversion of some meromorphic matrices (Q1344083)

The authors discuss a problem of inversion of a real symmetric matrix meromorphically depending on a parameter \(\lambda\), i.e. of the form \(\chi (\lambda)= I- \sum^r_{k=1} {{f_{0k} f^T_{0k}} \over \lambda}+ \sum^\infty_{k=1} {{f_k f^T_k} \over {\lambda_k- \lambda}}\) where \(\lambda, \lambda_k>0\), \(f_{0k}, f_k\in \mathbb{R}^n\), and \(I\) is the identity matrix of order \(n\). Here it is also assumed that the inverse matrix \(\chi^{-1} (\lambda)\) is meromorphic and its poles are of the first order. Such a matrix arises in many applied problems. One such problem -- the problem of inverting a perturbed operator -- is considered in Section 2. In Section 3, they obtain an explicit formula for the inverse matrix \(\chi^{-1} (\lambda)\), the right side of which is represented in terms of some numbers and vectors related to the matrix \(\chi (\lambda)\). In Section 4, they give an equivalent representation for the inverse matrix via a series with accelerated convergence. Finally, in Section 5, they state some remarks on using formulas with accelerated convergence.