Block Lanczos method

New York, 1977, pp. 361-377. The Block Lanczos Method for Computing Eigenvalues. 

The natural extension of the simple scheme is the block or band Lanczos method. In exact arithmetic both variants are equivalent. Though being computationally more demanding than the single-vector Lanczos method, the block Lanczos method is particularly well adapted to the present purposes. The major (formal) differences with the simple scheme are that the block algorithm generates a set of orthonormal vectors at once in time instead of one, as does the simple scheme, and that the matrices resulting from projection are... 

The importance of the block Lanczos method for many-body GF calculations has recently been discussed and demonstrated. In the case of the one-particle GF it has been shown that the computational effort of the diagonalization can be substantially reduced using block Lanczos. The proposed procedure consists of a block Lanczos prediagonalization of the N + l)-particle block and a subsequent diagonalization of the resulting smaller secular matrices and quite naturally exploits the specific structure of the Dyson equation. 

Calculating the matrix elements of the Hamiltonian in this basis set gives a sparse, real, and symmetric M(N) x M(N) matrix at order N. By systematically increasing the order N, one obtained the lowest two eigenvalues at different basis lengths M(N). For example, M(N) = 946 and 20,336 at N = 20 and 60, respectively [11]. The symmetric matrix is represented in a sparse row-wise format [140] and then reordered [141] before triangularizations. The Lanczos method [142] of block-renormalization procedure was employed. 

Similarly to Lanczos method for solving the RPA problem eq 2.15, Davidson s algorithm needs to be modified to take into account the block paired structure of eq 2.18 and scalar product eq Bl. The first RPA algorithm has been developed by Rettrup " and later improved by Olsen. The method has been further refined in ref 79, combined with TDDFT technique, and incorporated into Gaussian 98 package. We will follow ref 79 to describe this method. 

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