Summary
The asymptotic convergence of the coordinate overrelaxation method for computing the smallest eigenvalue and corresponding eigenvector of the general eigenvalue problem (A−λB)x=0 is investigated. A certain analogy between the asymptotic behaviour of this algorithm and the successive overrelaxation method is developed in deriving the appropriate iteration matrix. By the aid of this matrix it is shown that asymptotic convergence of the process towards the eigenvector of the smallest eigenvalue is guaranteed for all relaxation factors ω ε (0,2).The optimal choice of ω can be discussed theoretically in case thatA andB have “propertyA”. The theoretical facts are illustrated by two examples originating from the problem of the vibrating string using spline functions.
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Schwarz, H.R. The method of coordinate overrelaxation for (A−λB)x=0. Numer. Math. 23, 135–151 (1974). https://doi.org/10.1007/BF01459947
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DOI: https://doi.org/10.1007/BF01459947