Krylov subspace

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 ... 

V.L. Druskin and L.A. Knizhnerman Krylov subspace approximation of eigen-pairs and matrix functions in exact and computer arithmetics. Num. Lin. Alg. Appl., 2 (1995) 205-217... 

M. Hochbruck and Ch. Lubich On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34 (1997) (to appear)... 

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... 

Krylov subspace methods (such as Conjugate Gradient CG, the improved BiCGSTAB, and GMRES) in combination with preconditioners for matrix manipulations aimed at enhanced convergence, and... 

The power method uses only the last vector in the recursive sequence in Eq. [21], discarding all information provided by preceding vectors. It is not difficult to imagine that significantly more information may be extracted from the space spanned by these vectors, which is often called the Krylov subspace- 0,14... 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

Scaling Laws and Motivation for Recursive Diagonalization 291 Recursion and the Krylov Subspace 292... 

For the RDE, the operating range of rotation frequency is between approximately 1 and 50 Hz and a typical radius is 0.25 cm. Dimensionless rate constants were interpolated from working curves generated from a fully implicit simulation using preconditioned Krylov subspace methods (Alden, unpublished work). 

An effective method to acetderato the convttrgerice of the MRM algorithm is based on the Krylov-subspace method (Kleinman and van den Berg, 1993). VVe introduced the Krylov subspace in Chapter 2 as the finite dirncnsiorial subsjtace A, of the Hilbert space M, spanned by the vec.tors r , Lr . lAr,.. 

Note that the dimension, s, of the Krylov subspace is always less than or... 

The Krylov-subspace method is based on approximating the iteration step, Am , in the recursive formula (4.7) by an element of the Krylov subspace... 

The difference between the iteration procedure (4.58)-(4.59) and (4.7) is that now we move from one iteration to another not only in the direction of one residual vector r , but along a multidimensional Krylov subspace spanned by the vectors Vn,Lrn,.L Tn] ... 

We note here in passing that Chebyshev propagation is related to several other recursive methods based on the Krylov subspace = span i//(, Hy/(, ...,H Wq) ... 

There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

Diagonalizing this matrix gives the eigenvalnes e and the expansion coeffieients C" of the eigenvectors in the Krylov subspace Ig fc >.k = L , j such that... 

In the work of Lindborg et al [119], the resulting linear equation systems were solved with preconditioned Krylov subspace projection methods [166]. The Poisson equation was solved by a conjugate gradient (CG)-solver, while the other transport equations were solved using a bi-conjugate gradient (BCG)-solver which can handle also non-symmetric equations systems. The solvers were preconditioned with a Jacobi preconditioner. 

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