Krylov basis

Krylov Approximation of the Matrix Exponential The iterative approximation of the matrix exponential based on Krylov subspaces (via the Lanczos method) has been studied in different contexts [12, 19, 7]. After the iterative construction of the Krylov basis ui,..., Vn j the matrix exponential is approximated by using the representation A oi H(g) in this basis ... 

This finding, which was mentioned in Section 4, exhibits a remarkably regular factorability of the general result for H (c0) as a direct consequence of a judicious combination of symmetry of the Hankel determinant (46) and the Lanczos orthogonal polynomials. Thus, given either the set /3 or qVi0, the Hankel determinant H (c0) or equivalently the overlap determinant detS in the Schrodinger or Krylov basis rj can be constructed at once from Eq. (175). This is very useful in practical computations. 

We here describe the alternative of approximating <,c(S)b via Lanczos method. The Lanczos process [18, 22] recursively generates an orthonormal basis Qm = [qi,.., qm] of the mth Krylov subspace... 

The Lanczos method is based on generating the orthonormal basis in Krylov... 

The Lanczos method is based on generating the orthonormal basis in Krylov space Ki =span c, Ac, A c by applying the Gram-Schmidt orthogonaliza-tion process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix and also with the special properties of T/,. This reduction (called also QT decomposition) is described by the formula... 

The first term of Eq. 7 vanishes if feasible points (i.e. steady states) are computed at each iteration [6]. Clearly the calculation of the basis Z from Eq. 5 is expensive, as the large system Jacobians need to be constructed and inverted. Furthermore, in the case of input/output simulators Jacobians are not explicitly available to the optimisation procedure, or even to the solver itself as is the case of solvers using iterative linear algebra (e.g. Newton-Krylov solvers). In such cases the Jacobian can only be numerically approximated, with great computational expense in terms of CPU and memory requirements. For this purpose here we compute only reduced-order Jacobians... 

In order to see the difference in the two approaches, below I focus on the excitation energies, AE, of the Be states that are discussed here. The nice thing about atomic spectra of this type is that there is accurate experimental information with which one can compare the results of a theoretical method (See, Tables of NIST, USA, in the WWW). Specifically, I compare the AE from the Be Fermi-sea energies for which cancellations on subtraction of total energies are expected, with those obtained from methods that have used one of the known basis sets. I consider two such publications. The first is in 1986 by Graham et al. [105] where a (9s9p5d) contracted GTO basis (61 basis functions) was used for different types of computations. I keep the full Cl (FCI) results. The second is in 2003 by Sears, Sherrill and Krylov [106], who studied aspects of "spin-flip" methods and compared them with FCI using the same basis set, which is a 6-31G. 

The (n + 1)-dimensional Krylov space constructed in Eq. (C.5) spans locally over p(t) and the subsequent n actions of A(t). As an orthogonal but incomplete basis set, the Gaussian quadrature accuracy of order would be expected for the n-dimensional Krylov space approximation. It thus allows the time-local evolution, p t + St) exp[A(t)St]p t) be evaluated accurately with a fairly large St, The project-out error can be estimated similarly as that of the short-iterative-Lanczos Hilbert-space propagator [51]. 

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