Chebyshev acceleration techniques for large complex non hermitian eigenvalue problems

The computation of a few eigenvalues and the corresponding eigenvectors of large complex non hermitian matrices arises in many applications in science and engineering such as magnetohydrodynamic or electromagnetism [6], where the eigenvalues of interest often belong to some region of the complex plane. If the size of the matrices is relatively small, then the problem can be solved by the standard and robust QZ algorithm [9]. The QZ algorithm computes all the eigenvalues while only a few eigenvalues may be of interest, requires a high computational cost and does not exploit the sparsity of the matrices and is therefore only suitable for small matrices, When the size of the matrices becomes large, then Krylov subspace methods are a good alternative since these methods take into account the structure of the matrix and approximate only a small part of the spectrum. Among these methods, Arnoldi's algorithm appears to be the most suitable. Arnoldi's method builds an orthogonal basis in which the matrix is represented in a Hessenberg form whose spectrum, or at least a part of it, approximates the sought eigenvalues. Saad [11] has proved that this method gives a good approximation to the outermost part of the spectrum. However, convergence can be very slow especially when the distribution of the spectrum is unfavorable. On the other hand, to ensure convergence, the dimension of the Krylov subspace must be large, which increases the cost and the storage. To overcome these difficulties, one solution is to use the method iterativety, that is, the process must be restarted periodically with the best estimated eigenvectors associated with the required eigenvalues. These eigenvectors can also be improved by choosing a polynomial that amplifies