The Weighted Arnoldi Method For Large Nonsymmetric Matrix Eigenvalue Problems

Large scale eigenvalue problems are of increasing importance in scientific and engineering computing. Considerable progress has been made over the past decades towards the numerical solution of large scale nonsymmetric problems. However, there is still a great deal to be done. The work of this thesis focuses on the following.First, we introduce the background of large scale eigenproblems and basic numerical algorithms for solving them. We also review the state of the art of this subject briefly.Arnoldi method is very suitable for calculating extreme eigenvalues of the matrix. However, this method is often ineffective for the interior eigenproblems of large matrices arising from the practical problems. The shift-and-invert Arnoldi method is popular for computing the interior eigenvalues of large matrices and achieve convergence rapidly. We need decompose the large matrix in the calculation process, on the one hand, it would undermine the original structure of the sparse matrix, increase the storage; on the other hand, in the interative process, the Arnoldi method is very sensitive to the matrix's disturbance, while the inverse process is difficult to achieve the reliable accuracy, so this method is not practical. Now, the harmonic Arnoldi method proposed by Morgan is considered the most effective method for solving the interior eigenvalues of large matrices. In order to reduce the storage of this method, speed up the rate of convergence, improve the accuracy, in Chapter 2 we propose the weighted harmonic Arnoldi method and consider the choice of weight matrix. Numerical experiments illustrate that the new algorithms are more powerful and attractive than other algorithms.The vectors produced by the Arnoldi algorithm form an orthonormal basis of the Krylov subspace, in this process, we consider a new vector and all the basis vectors have produced orthogonal, while we also comput all the eigenpairs of an m -order Hessenberg matrix, the storage is large and even impossible. Therefore, the dimension m of Krylov subspace should not be too much from memory, convergence and computational considerations. For m of selected out, the Ritz values and Ritz vectors computed by Arnoldi method may not satisfy the required precision. In order to overcome the difficulty, we use a Ritz vector or a linear combination of some Ritz vectors instead of the initial vector to re-use Arnoldi method. In Chapter 3, we discuss several restarted Arnoldi method, and present a new algorithm by the idea of weight. Then, numerical experiments verify the new algorithm's convergence and accuracy.