The Lanczos Type Method For Computing The Extreme Eigenvalue Problems Of Large Sparse Symmetric Matrices

Matrix eigenvalue problems, an essential part in the field of algebra eigenvalue, is of great significance both in theory and in practice. In the engineering technology and other subjects, it is usually necessary to calculate certain extreme eigenpairs of the large sparse symmetric matrix. One of the effective methods to solve the problem is the Lanczos method, which was first proposed by Lanczos in 1950s. Because the tridiagonal process of the Lanczos method has not destroyed the sparseness of the original matrix, it is very suitable for solving the eigenvalue problems of the large sparse symmetric matrix. Nevertheless, when using the Lanczos method to solve the dense or multiple eigenvalue problem, the corresponding validity and the reliability is decreased. Consequently, Underwood proposed a new method, block Lanczos method, which is suitable for calculating the dense eigenpairs in the 1970s. Whereas, the convergence rate of both the Lanczos method and the block Lanczos is very slow. To enhance the convergence rate of these two Lanczos, Dai Hua and Zhou Shuquan proposed the Chebyshev-Lanczos iteration and the block Chebyshev-Lanczos iteration. The use of the Chebyshev iteration and the block Chebyshev iteration greatly enhances the convergence rate of the Lanczos method and the block Lanczos method.The main work of this article is: firstly, these four Lanczos type methods are introduced, meanwhile, the Chebyshev iteration and the block Chebyshev iteration is introduced; secondly, the floating point error analysis, taking the block Lanczos method as the example, are given; Finally, in the numerical experiment, these method were compared through the massive number-calculated examples, and drew the following conclusions: the block Lanczos method and the block Chebyshev-Lanczos method can compute dense or multiple eigenvalue but when using the Lanczos method or the Chebyshev-Lanczos method to solve dense or multiple eigenvalue, the corresponding validity and the reliability will decrease; the iterative Chebyshev-Lanczos method enhance the convergence rate of the Lanczos method, furthor more, the superiority of the Chebyshev-Lanczos method is more obvious when the order of the matrices are getting bigger; the block Chebyshev-Lanczos method not only overcome the shortcoming of the Lanczos method and the Chebyshev-Lanczos method, which is the validity and the reliability of the Lanczos method and the Chebyshev-Lanczos method decreasing when using those solving dense eigenvalues or multiple eigenvalues, but also enhance the co-nvergence rate of the block Lanczos method.