Authors: Feng. Wu, Jiqiang Hu, Li Zhu, Wanxie Zhong
Computation of matrix inversions has been an important topic in many fields, such as finite-element and network analysis. In many important problems in mechanics and engineering, the matrix inversions are dense, but contain a large number of entries that are very close to zero. For these matrix inversions, there is still a lack of efficient calculation methods. By using the real bandwidth and effective bandwidth, the sparsity of the matrix inversion and the matrices involved in the Newton iterative method are studied. It is found for each approximate inversion computed during the Newton iterative process, a corresponding sparse approximate inversion exists. An adaptive filtering method in terms of the convergence analysis is proposed to obtain this sparse approximate inversion. Combining the adaptive filtering with the Newton method, an efficient matrix inversion solver is developed. Some matrices derived from function approaching, finite element and network analysis are used to test the proposed method. Numerical results show that the execution time of the proposed method is of O (nnz(Y)^1.147), where nnz (Y) is the number of non-zero entries in the matrix inversion computed using the proposed method.