Abstract
In solving parametric sparse linear systems, we want 1) to know relations on parametric coefficients which change the system largely, 2) to express the parametric solution in a concise form suitable for theoretical and numerical analysis, and 3) to find simplified systems which show characteristic features of the system. The block triangularization is a standard technique in solving the sparse linear systems. In this paper, we attack the above problems by introducing a concept of local blocks. The conventional block corresponds to a strongly connected maximal subgraph of the associated directed graph for the coefficient matrix, and our local blocks correspond to strongly connected non-maximal subgraphs. By determining local blocks in a nested way and solving subsystems from low to higher ones, we replace sub-expressions by solver parameters systematically, obtaining the solution in a concise form. Furthermore, we show an idea to form simple systems which show characteristic features of the whole system.
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