Spectral transform Lanczos algorithm

However, there is a price to pay in a spectral transform Lanczos algorithm At each recursion step, the action of the filter operator onto the Lanczos vectors has to be evaluated. In the original version, Ericsson and Ruhe update the Lanczos vectors by solving the following linear equation ... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

The formal properties of operator L eq 2.18 (known as the symplectic structure ) allow the introduction of a variational principle eq D3, " a scalar product (eq Bl), and ultimately to reduce the original non-Hermitian eigenvalue problem (eq 2.18) to the equivalent Hermitian problem which may be solved using standard numerical algorithms (Appendices B—E). For example, F is a Hermitian operator. Lowdin s symmetric orthogonalization procedure " " leads to the Hermitian eigenvalue problem as well (eq E5), which may be subsequently solved by Davidson s algorithm (Appendix E). The spectral transform Lanczos method developed by Ruhe and Ericsson is another example of such transformation. 

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