In the calculation of absorption spectra one needs to compute an eigenvalue decomposition of a hermitian Hamiltonian which consists of the sum of two matrices: A dense hermitian matrix with known eigenvalues and eigenvectors and a structured hermitian matrix. The structured matrix represents a short range interaction and contains only elements in a small block in the upper left corner. To compute the eigenvalue decomposition I suggested
and implemented a prototype updating algorithm which can be found in the following report:
needed. The real problem was to solve linear systems of equations with the hermitian
Hamiltonian. Thus Gene Golub suggested a different approach which does not require the
eigenvalue decomposition of the matrix any more: One can use the Sherman-Morrison-Woodbury formula to invert low rank pertrurbed large matrices for which the original inverse is known, which is here the case, since the original eigenvalue decomposition is known. I implemented a prototype of this algorithm which gives very promising results. The real implementation on the Cray thus considered.
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