Summary
In this paper the eigenvalue problemAx=λBx is considered, whereA andB are symmetric matrices so large that they cannot be stored in the high speed storage of a computer. A general theory of the conjugate gradient method for minimizing the Rayleigh quotient is described and extended to a more general functional for computing simultaneously a few of the largest or smallest eigenvalues and the corresponding eigenvectors. Abandoning the advantage of the basic algorithm, which does not require the inversion or the factorization of one of the given matrices, the procedure can be used to solve the problem more effectively than by simultaneous iteration, especically in the case of clustered eigenvalues in the end of the spectrum.
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Döhler, B. Ein neues Gradientenverfahren zur simultanen Berechnung der kleinsten oder größten Eigenwerte des allgemeinen Eigenwertproblems. Numer. Math. 40, 79–91 (1982). https://doi.org/10.1007/BF01459077
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DOI: https://doi.org/10.1007/BF01459077