Krylov vectors

Unless the initial vector is already an eigenvector, the Krylov vectors are linearly independent and they eventually span the eigenspace of H ... 

The vectors generated by the Lanczos recursion differ from the Krylov vectors in that the former are mutually orthogonal and properly normalized, at least in exact arithmetic. In fact, the Lanczos vectors can be considered as the Gram-Schmidt orthogonalized Krylov vectors.27 Because the orthogonalization is performed implicitly along the recursion, the numerical costs are minimal. 

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)... 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

But then is the characteristic polynomial of A, and its coefficients are the elements of / and can be found by solving Eq. (2-11). This is essentially the method of Krylov, who chose, in particular, a vector et (usually ej) for vx. Several methods of reduction of the matrix A can be derived from applying particular methods of inverting or factoring V at the same time that the successive columns of V are being developed. Note first that if... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

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. 

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 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] ... 

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). 

Note that even through projection, the input Z and output Y vectors of the system remain unchanged. For large-scale system, an effective and prevalently used projection subspace is the Krylov subspace [5]. 

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, . ..,... 

A simple recursive procedure allows one to build a set of orthogonal vectors spanning the same Krylov subspace. Finding each new vector wwi only requires the two previous vectors Wm and w,... 

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