Approximate solutions for large transfer matrix problems (Q1122305)

In physical lattice models, one is interested in the largest eigenvalue and the corresponding eigenvector for a very large and extremely sparse transfer matrix. However, usually only a small number of the elements in the eigenvector are large, and only these large elements are relevant for the physical description of the model. In order to cut down dramatically the computational effort - as well as storage requirements - in the power method, a modified power method is developed which seeks to compute only the largest elements of the left and right eigenvectors. The idea is to use a sparse representation for the eigenvectors in which only a small fraction of the elements are actually stored, combined with a strategy to select these nonzero elements in such a way that no significant physical information in the model is lost. The such computed sparse eigenvectors are satisfactory in the sense that some physical quantities, such as spin configuration probabilities and partition functions, are determined with high accuracy. However, the corresponding eigenvalue (which describes the free energy) is not so accurate, basically because not all the information present in the sparse left and right eigenvectors are used to compute the eigenvector.