Perturbations of M-matrices via ABS methods and their applications to input-output analysis

摘要

Some properties of M-matrices are investigated in order to analyze the closure of the class with respect to perturbations introduced in the matrix. An algorithm is derived by means of the implicit Gauss-Cholesky algorithm in the ABS class for linear systems: a lower estimation of the solution of linear system Ax = b, where A is an M-matrix, and the solutions of the leading principal subsystems are available at each iteration step for a total number of multiplications of order O(n33). The algorithm also tests whether a Z-matrix is an M-matrix and the effects of changes in the entries of the coefficient matrix during the execution are investigated as well. The number of the extra multiplications, in general, is less than those needed by the Gaussian elimination. Finally, an application to the open Leontief static input-output (I-O) model is presented together with numerical results on a real case using the IBM 3090 200VF computer.