Recursive steps

This construction requires one matrix-vector multiplication with S and two inner products in each recursive step. Therefore, it is not necessary to store S explicitly as a matrix. The Lanczos process yields the approximation [21, 7, 12]... 

Continue the recursive steps until the solution settles down (when k = 50, or kT = 5 seconds) and hence determine the steady-state value of the feedback matrix K(0) and Riccati matrix P(0). What are the closed-loop eigenvalues ... 

In simple cases, you can start with the assumption that every type identified in the specification model will be implemented directly as a separate class for large components you may go through a recursive step of subcomponent partition and specification. We determine intended responsibilities of each class and then build interaction diagrams to design their interactions to realize the specified behaviors. 

However, there is a price to pay in a spectral transform Lanczos algorithm At each recursion step, the action of the filter operator onto the Lanczos vectors has to be evaluated. In the original version, Ericsson and Ruhe update the Lanczos vectors by solving the following linear equation ... 

The convergence of the eigenvalues obtained by LSFD can be checked by varying L and K. For practical purposes, L is typically in the range of 100. The number of recursion step (K) is proportional to the average density of states and the spectral range. The error of a particular eigenvalue can be estimated from its dispersion (H2 — E2S)b , where [H2L, = can also be... 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

After performing M Lanczos recursion steps, we will have computed the diagonal and off-diagonal elements in the M X M tridiagonal matrix T. From T, we can then quickly compute M eigenvalues and residues. This, in turn, permits an M-term approximation to the lineshape function in Eq. (56) ... 

This completes step n we now have the next recursion vector U tl, along with the two new elements of the tridiagonal matrix, an, and bn+l. From this point onward, the recursion method just loops over the same steps. After M recursion steps, we have the M X M tridiagonal matrix T. In the next step, we will extract the residues and eigenvalues from T. 

Table 3 Number of Filtered Lanczos Recursion Steps Required to Obtain an Energy with an Error Smaller than 0.1 cm and a Residue Known with Three Significant Digits...

How are the Bj k(t) functions computed An efficient way is to use the Cox-DeBoor algorithm [18,19] which is based on a recursive formulation of the Bj,k(t) functions. Using a cubic B-spline to illustrate, it is possible to write out the recursive steps explicitly ... 

Fig. 2. Convergence of residues of the model system with 301 zero-order states, as compared to direct diagonalization. The number of recursive steps 101

Fig. 4. Convergence of the ground state survival probability for the model system for various number of recursion steps, as in Fig. 2.

In Fig. 3, we compare time development of the ground state survivability obtained for the hierarchy of reduction of irrelevant information for the present system (the number of recursion steps M exceeds the number N of zero-order states, in order to extend the spectrum of non-ghosts eigenvalues (5,13)). These results are compared with our reference obtained from direct diagonalization. Obviously, the irreducible (and partly the linked) moment calculations do reproduce the results obtained from the reference calculations. 

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