ABSTRACT
The Lanczos recursive algorithm is one of the most widely used eigenvalue solvers for large matrices. This chapter focuses on the resolvent operator of the Lanczos recursive algorithm and performs a parallel study of the spectra of evolution and resolvent operators. One of the possibilities for obtaining the spectrum of the evolution operator is to diagonalize directly an eigenvalue problem in a conveniently chosen complete orthonormal basis. There is a flexibility in the Lanczos algorithm, which permits the recursion to be carried out with tensors, matrices, operators or scalar. The monic Lanczos states are orthogonal, but unnormalized, as opposed to orthonormalized Lanczos states. In the Lanczos algorithm, a sufficiently large number is not only necessary for obtaining accurate eigenvalues, but also for arriving at good eigenvectors if they are needed in the analysis. The Lanczos algorithm is a low-storage method, as opposed to the corresponding Gram–Schmidt orthogonalization which uses all states at each stage of the computation.
