Computing Interior Eigenvalues of Large Matrices

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A new technique for computing a few interior eigenvalues of a large matrix is proposd. It is assumed that the size and structure of the matrix make factorization impractical. The new technique is related to the Rayleigh-Ritz, it finds interior eigenvalues in an optimal way. The technique can be applied to the Lanczos algorithm for symmetric matrices and to the Davidson and GD methods for both symmetric and nonsymmetric matrices. While the cost per iteration increases, the number of iterations needed decreases. The new methods can be a great improvement. The methods must be tested and the proper implementations developed. It must be determined for what types of problems the new methods are better than current methods.