Dominant eigenvalue

Figure 12.2. Real part of the dominant eigenvalue versus Lewis number for steady state E of Figure 12,1, (From Hite and Jackson CSl].)...

The curious shape of the computed parts of the curves suggests that there is, in each case, a discontinuity of slope. However, examination of the results shows that there is, in fact, a switch in the dominant eigenvalue as the Lewis number changes. Above a certain value of the Lewis number this is real and moves to the right as decreases. But eventually it crosses with a pair of complex eigenvalues moving to the left and these, which become the dominant eigenvalues for smaller values of, cause... 

Fig. 27. Convergence pattern of dominant eigenvalues, type I conditions.

The principle of this method, proposed originally by Davison (1966), is to neglect eigenvalues of the original system that are farthest from the origin (the nondominant modes) and retain only dominant eigenvalues and hence the dominant time constants of the system. If we consider the solution of the linearized model... 

Fig. 29. Dominant eigenvalues of the full and reduced models, type II conditions.

Figure 3. Model prediction for dominant eigenvalue dependence on polymerization temperature for perfect (curve 1) and imperfect (curve 2) mixing assumptions. Model parameters as in Figure 2.

Orbach, O. Crowe, C. M., "Convergence Promotion in the Simulation of Chemical Processes with Recycle - The Dominant Eigenvalue Method", Can. J. of Chem. Eng. (1971) 49 503-513. 

Consequently, classical PCR (CPCR) starts by mean-centering the data. Then, in order to cope with the multicollinearity in the x-variables, the first k principal components of Xnp are computed. As outlined in Section 6.5.1, these loading vectors ppJl = (p,... ) are the k eigenvectors that correspond to the k dominant eigenvalues of the empirical covariance matrix Sj = xrX. Next, the -dimensional scores of each data point t are computed as j. = In the final step, the centered response variables y. are... 

Since dead sites have zero activity, the overall activity of the catalyst is a(n) = a (n), where a is the vector giving the activity levels of the active states. In order to see how overall activity a changes over time, first consider what happens if one starts with a "quasi-steady-state" distribution of active states, i.e., let v(0) = Cj, the eigenvector of Pn corresponding to the dominant eigenvalue Aj of Pn. In this case Piiei=AiCj, so s(n) = Pn"ei = Ai"ei = Aj"s(0). Thus the relative proportions of sites in active states remain unchanged over time there is simply an overall exponential decrease in the total population of active sites. Similarly, in this quasi-steady-state case we have a(n) = a s(n) = A "a ei = Ai"a(0) i.e., the overall activity decreases exponentially. The decay constant Aj is very close to 1 since the columns of Pu all have a sum very close to 1. In fact, if the columns of Pu all have identical sums P, then Aj = P this corresponds to the situation where the probability of sudden death is the same from each active state, namely b = 1-P. 

This may be proven analytically, at least in the special case where each column of P2i(0) has the same sum p. That is to say, when each mono-ion has the same probability P of transiting to a di-ion. In this case all the columns of Pu(0) have the same sum (1 - P), and hence Pn(0) has dominant eigenvalue (1 - P). Let q be the eigenvector of Pu(0) corresponding to this dominant eigenvalue, so that ... 

The Wegstein and Dominant Eigenvalue methods listed in Figure L.3 are useful techniques to speed up convergence (or avoid non-convergence) of the method of... 

Theorem 1 highlights the strong relation between a decomposition of the state space into metastable subsets and a Perron cluster of dominant eigenvalues close to 1. ft states that the metastability of an arbitrary decomposition d cannot be larger than I-I-A2+... - -Ato, while it is at least 1- -K2 2+- c, which is close to the upper bound whenever the dominant eigenfunctions V2,..., Vm are almost constant on the metastable subsets Pi,..., Dm implying Kj fn 1 and c 0. The term c can be interpreted as a correction that is small whenever a 0 or Kj 1. It is demonstrated in [23] that the lower and upper bounds are sharp and asymptotically exact. 

However, whenever there is a gap in the spectrum of the transfer operator after m dominant eigenvalues, then the results of, e.g., [16,21] tell us that any decomposition into more than m sets will be associated with a significantly larger drop in metastability as measured by the function meta. 

It is instructive to compare the eigenvalues of transition matrices obtained for different lag times t. That is, we do not count transitions on a timescale of O.lps which means to observe transitions from one instance of the time series to the next, but count transitions on a timescale of, say, ps which is between every tenth step in the global Viterbi path. For all time lags. Fig. 5 clearly indicates two dominant eigenvalues after which we find a gap, followed by other gaps after 4, 9, or 16 eigenvalues. This yields 2, respectively 4, 9, or 16 metastable sets. To avoid confusion we call these metastable sets (molecular) conformations. 

W. Huisinga and B. Schmidt (2005) Metastability and dominant eigenvalues of transfer operators. In New Algorithms for Macromolecular Simulation (C. Chipot, R. Fiber, A. Laaksonen, B. Leimkuhler, A. Mark, T. Schlick, C. Schiitte, and R. Skeel, eds.), vol. 49 of Lecture Notes in Computational Science and Engineering, Springer, to appear... 

The eigenvalue with the largest magnitude is called the dominant eigenvalue, the next the subdominant, and so on. 

This is what the joint spectral radius analysis does. The joint spectral radius of the two matrices L and R, is defined as the limit, as n tends to oo, of the value of the nth root of the largest dominant eigenvalue of any of the matrices formed by taking all possible product sequences of length n of L and R. 

If the dth divided difference scheme Sd of some power of a scheme S has a dominant eigenvalue greater than or equal to the arity of that power, then the (d—l)th divided difference scheme has an eigenvalue greater than or equal... 

Examination of Eq. 5.6.19 shows that the initial trajectory will be dictated by the dominant eigenvalue, say 2 and the initial path will lie along ( 2)- As equilibrium is approached, the equilibration path will lie along the slow eigenvector (e ) as illustrated in Figure 5.11. 

BSM First Rank Correlation Times (Left Column), Dominant Eigenvalues (Right Column) and Some of the... 

These are calculated for D, = 1 (which defines the frequency scale) for increasing first rank potential coupling. For each dominant eigenvalue the relative weight is given (in parentheses). 

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