Preconditioning Method For Generalized Eigenvalue Problems

The large generalized eigen-value problems are often met in engineering mechanics. At present, the solution to the problems is divided into two types: the transformation method and vector iteration method. The transformation method deals with original matrix directly by a series of transformation so that it can be reduced to a simple form, these methods include Jacobi method, QR method and Givens method, et al. Because the transformation method needs to store the elements of matrix, the large memory is required for large-size matrix. Such a method can only be used in small-size matrices. Usually the method is used by combining with the vector iteration method. Another class of methods is trying to get eigenvalues and eigenvectors with a series of matrix and vector multiplicative. Because the vector iteration method can use the compressed technology of data, it is suitable to solve the large-size eigen-value problems, especially in large-size sparse eigen-value problems. At present, the computational methods of large symmetrical eigen-value problems include subspace iteration method and Lanczos method, etc. The subspace iteration method can get several extreme eigenvalues and eigenvectors. But the greatest shortcoming of subspace iteration method is that the speed of convergence is slow, especially when eigenvalues are densely distributed. The Ritz vector iteration method is also a vector iteration method, it constructs a group of Ritz vector from the initial vector space, then the projection on this group of Ritz vectors of original eigen-value problems leads to a small-size eigenvalue problem. Then the accuracy is improved with iteration. Similar to subspace iteration method, we construct a group of standard Ritz vectors, and convert the generalized eigen-value problems of subspace iteration to standard eigen-value problems. The operations of the Ritz vector iteration method are bigger than that from the subspace iteration method, but its convergence speed is faster than that of subspace iteration method. Generally the Ritz vectoriteration method is more efficient than the subspace iteration method. Because the construction of the iteration vector of the Ritz vector iteration method is similar to the subspace iteration method, it also has weakness that when eigenvalues are densely distributed, the convergence of the method is slow. The preconditioning technology originated from solving system of linear equations. It changes the condition number of the original coefficient matrix by multiplying preconditioned matrix, thus accelerate the solution of original system. Since the late of 1980's, the preconditioning for eigen-value problems has been being the hotspot in matrix computations. Now the generalized Davidson method and the preconditioning Lanczos method are the effective methods of solving large eigen-value problems. Studies of Davidson method and preconditioning Lanczos method have been done and some theorems on convergence property have been provided.