Modeling of linear algebra system with the Fourier transform

Solving dense linear system represented by square matrix of size N requires O(N³) operations. We developed procedures for modeling and computation of approximate solutions for problems which are too large to be solved routinely and for which an approximate solution is acceptable. The original dense system is Fourier transformed and then pruned off by removing the insignificant frequencies in accord with the rules of matrix algebra. This approach reduces the original system size and leads to a fast, approximate solutions of eigenvalue or linear system of equations problems. In order to increase flexibility of this approach, the partitioning of input matrix was also tested. The partitioning allows the use of mixed approach where part of the matrix is transformed and reduced in size, while other part is kept intact and in original domain. The examples analyzed were from theoretical chemistry and structural engineering calculations. The quality of solutions for practical problems is demonstrated.