A Rayleigh–Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices

A procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse Hermitian matrices. The procedure, a modification of an inverse subspace iteration procedure, uses adaptively determined Chebyshev polynomials to approximate the required application of the inverse operator on the subspace. The method is robust, converges with acceptable rapidity, and can easily handle operators with eigenvalues of multiplicity greater than one. Numerical results are shown that demonstrate the utility of the procedure.