Finding the inverse and connections of a type of large sparse matrix
Martin Hershkowitz, Stedman B. Noble · Naval Research Logistics Quarterly · 1965
Abstract Much interest is being shown in methods for determining structure of matrices in order to simplify computation. This paper takes a slightly different approach, giving an algorithm that determines the structure of a matrix while inverting it. The algorithm applies to matrices with dominant diagonal or similar matrices that can be inverted using the C. Neumann series. This series is decomposed into terms which are products of connected coefficients: the procedure lends itself to an efficient method of truncation that obtains an approximation to the inverse matrix. While this algorithm might not be efficient for some matrices, it is quite efficient for the 430‐order Minkowski‐Leontief matrix upon which it was tested. Furthermore, the computation enumerates the connections in the matrix; hence the model can be “tested” by comparing the connections of the model with those known to exist in the underlying phenomena.