Krylov space

Another approach involves starting with an initial wavefimction Iq, represented on a grid, then generating // /q, and consider that tiiis, after orthogonalization to Jq, defines a new state vector. Successive applications //can now be used to define an orthogonal set of vectors which defines as a Krylov space via the iteration (n = 0,.. ., A)... 

For the numerical study of the whole spectrum (for g R fixed), [79] uses a spectral tau-Chebychev discretization in y and the Arnold method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids. 

Definition 16 The finite dimensional subspace Ki of the Euclidean space E, spanned by the vectors c. Be,., ..B c, is called a Krylov space ... 

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... 

We consider first the tridiagonalization process using Krylov space of dimension N Kl) =span c, Ac, A c. In this case, according to the definition, the matrix Q v is orthogonal, Therefore, the reduction formula (E.21)... 

W. PoUard and R. Eriesner (1993) Efficient Fock matrix diagonalization by a Krylov-space method. J. Chem. Phys. 99, p. 6742... 

The matrix elements /i, y of the upper Hessenberg representation of L are thus automatically generated during the construction of the vectors vy. The essence of the short-iterative-Arnoldi propagator is to form an explicit representation of the exponential operator in the n-dimen-sional Krylov space based on the initial density matrix, cr t). 

An alternative solution to this problem is provided by fast Krylov-space algorithms. " These techniques construct a small subspace of orthogonal vectors which contains a good approximation to the true eigenvector. This Krylov subspace 5p, . ..,... 

The recursive construction of the desired Krylov space starts with the initialization,... 

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]. 

Gutknecht, M. H., and Strakos, Z. [2000] Accuracy of two three-term and three two-term recurrences for Krylov space solvers, SIAM J. Matrix Anal Appl, 22, 213-229. 

Two popular methods for solving the N x N linear system Ax = b are based on constructing the Krylov representation of A [24, 25]. The Krylov space of dimension p is defined as the span of the set of vectors... 

Since Avj span(Vj+i), and v,- G spanCVj+i) only if z < j +1, A hsis the important property that Aij = 0 for z > j +1. This kind of matrix, known as upper-Hessenberg, is usually much easier to manipulate than A because A is almost upper triangular. Thus, the philosophy behind the Krylov space based methods (KSM) is to transform the original linear system into the simpler form Ax = b where x = V x and b = Vjb, which can be solved more easily. The computational effort is usually dominated by the construction and orthogonalization of the transformation matrix Vp, which requires a matrix vector multiply at each iteration. 

In a development which follows Wall and Neuhauser in spirit, it has recently become apparent that Krylov-space filtering can be extended in a straightforward manner to carry out FD calculations without the need to explicitly store the filtered states in core memory. The FD calculation is carried out... 

---