Tamtam Proceedings - lamsin

508 Labidi et al.1. IntroductionThere are several methods for computing eigenpairs of a large matricial eigenvalueproblem (cf. [3], [4]). The best known techniques are the Krylov subspace methods, suchus Arnoldi, which is stable numerically and converges faster than a subspace iterationtechnique (cf. [3]). However, it requires explicit orthogonalization against all previouslycomputed basis vectors. Hence, both the cost and the storage increase as the methodproceeds. Moreover, the Arnoldi method requires the calculation of the eigenpairs of aHessenberg matrix of order m, at the cost of O(m 3 ) operations (cf. [4]), which becomesprohibitive for large m. Therefore, the restarting of initial vector provides the solution,but slows down the convergence. There are several approaches to restart the Arnoldi algorithm.One particular technique, called the Implicitly Restarting Arnoldi Method IRAMand developed by Sorensen at 1992, stands among other restarting algorithms. This maybe viewed as a truncated form of the powerful implicitly shifted QR technique (cf. [4]).IRAM provides a mean to approximate a few user-specified eigenpairs with storage spaceproportional to nk. k is the number of eigenvalues sought, and n is the problem size.In this paper, our aim is to recall the main features of the IRAM technique. One particularproblem for which we sought the use of IRAM is the computation of guided modes in anoptical fibre. The numerical results obtained in this case are particularly interesting.2. The Implicitly Restarted Arnoldi MethodThe Implicitly Restarted Arnoldi Method IRAM is particularly useful for solvinglarge-scale matricial eigenvalue problems (cf. [3], [4]). IRAM is applicable for standard,as well as generalized, matricial eigenvalue problems. A generalized eigenvalue problemAx = λBx can be rewritten, using a spectral transformation, as a standard eigenvalueproblem Ax = λx (cf. [3], [4]). That is why we recall in the following the idea behind theIRAM method in the simple case of a standard matricial eigenvalue problem.2.1. Arnoldi MethodFor a given positive integer m and A ∈ C n×n , we define an m-step Arnoldi factorizationof A as the following relationship (cf. [3], [4])AV m = V m H m + f m e T mwhere V m ∈ C n×m has orthonormal columns, V ∗ mf m = 0, and H m ∈ C m×m is an upperHessenberg matrix with a non-negative subdiagonal.If eigenpairs of H m approximate, to a given precision, all the desired eigenpairs of A, wehave convergence of the Arnoldi technique. However, the desired eigenvalues may not allfigure in the spectrum of H m , at the selected precision. Hence, for a given integer m andinitial vector v 0 , we are not always assured of the convergence of the Arnoldi technique.TAMTAM –Tunis– 2005