This paper is concerned with eigenvalue approaches for the pole-zero calculations of the transfer function of a linear time-invariant network. Combinatorial and sparse matrix algorithms are fully used to increase numerical accuracy and computational speed in the two-sets-of-eigenvalues approach. Some matrix decomposition-reduction algorithms are presented to simplify and stabilize the numerical eigenvalue-finding procedures.

This paper deals with a partitioning strategy of sparse matrices. In particular, the problem of transforming a non-singular matrix by symmetric permutation to an optimal bordered Triangular Form is solved. It is shown that the problem is equivalent to the determination of a minimum essential set of a directed graph. An efficient algorithm is given for finding minimum essential sets of a digraph. The method depends on, as a preliminary step, graph simplication using local information at a vertex. A circuit-generation technique based on vertex elimination is then introduced. The algorithm is illustrated with a complete example.

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