The eigenvalue problem (A − λB)x = 0 for symmetric matrices of high order

Vibrational problems of complex structures treated by the method of finite elements lead to the general eigenvalue problem $\text{(A ? λB)x = 0}$, where $\text{A}$ and $\text{B}$ are symmetric and sparse matrices of high order. The smallest eigenvalues and corresponding eigenvectors of interest are usually computed by a variant of the inverse vector iteration. Instead of this, the smallest eigenvalue can be computed as the minimum of the corresponding Rayleigh quotient for instance by the method of the coordinate relaxation of Faddejew/Faddejewa. The slow convergence of this simple algorithm can however be sped up considerably in analogy to the successive overrelaxation method by a systematic overrelaxation. Numerical experiments indicate indeed a rate of convergence of this coordinate overrelaxation as a function of the relaxation parameter which is comparable to that of the usual seccessive overrelaxation for linear equations. In comparison with known procedures for the solution of the general eigenvalue problem there result some important computational advantages with regard to the amount of work. Finally, the higher eigenvalues can be computed successively by minimizing the Rayleigh quotient of a modified eigenvalue problem based on a deflation process.