Dominant-eigenvalue method

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. 

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... 

In the previous subsection, the successive substitution and Wegstein methods were introduced as the two methods most commonly implemented in recycle convergence units. Other methods, such as the Newton-Raphson method, Broyden s quasi-Newton method, and the dominant-eigenvalue method, are candidates as well, especially when the equations being solved are highly nonlinear and interdependent. In this subsection, the principal features of all five methods are compared. 

In the dominant-eigenvalue method, the largest eigenvalue of the Jacobian matrix is estimated every third or fourth iteration and used in place of Sj in Eq. (4.20) to accelerate the method of successive substitutions, which is applied at the other iterations (Orbach and Crowe, 1971 Crowe andNishio, 1975). 

Crowe, C.M., and M. Nishio, Convergence Promotion in the Simulation of Chemical Processes—The Genera] Dominant Eigenvalue Method, A/CAE/, 21,3 (1975). 

Orbach, O., Crowe, C.M. (1971) Convergence promotion in the simulation of chemical processes with recycle-the dominant eigenvalue method. Can.J. Chem. wg. 49(4), 509-513. 

A very large array of methods have been proposed to accelerate the convergence rate of successive substitution. These include the use of the Dominant Eigenvalue Method of Crowe and Nishio (1975). In spite of significant increase in the rate of convergence, such methods may cause the loss of basic stability of successive substitution. 

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... 

The exponential operator (- ) is one of various alternatives that can be employed to compute the ground-state properties of the Hamiltonian. If the latter is bounded from above, one may be able to use 11 — , where x should be small enough that 0 = 1 — xE0 is the dominant eigenvalue of 11 — . In this case, there is no time-step error and the same holds for yet another method of inverting the spectrum of the Hamiltonian the Green function Monte Carlo method. There one uses ( — ) 1, where is a constant chosen so that the ground state becomes the dominant eigenstate of this operator. In a Monte Carlo context, matrix elements of the respective operators are proportional to transition probabilities and therefore... 

The predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

Experiments on transition for 2D attached boundary layer have revealed that the onset process is dominated by TS wave creation and its evolution, when the free stream turbulence level is low. Generally speaking, the estimated quantities like frequency of most dominant disturbances, eigenvalues and eigenvectors matched quite well with experiments. It is also noted from experiments that the later stages of transition process is dominated by nonlinear events. However, this phase spans a very small streamwise stretch and therefore one can observe that the linear stability analysis more or less determines the extent of transitional flow. This is the reason for the success of all linear stability based transition prediction methods. However, it must be emphasized that nonlinear, nonparallel and multi-modal interaction processes are equally important in some cases. 

In the Davidson method, one approximates A by the current iteration s eigenvalue, and H is assumed to be diagonally dominant so that 6 can be approxi-... 

As one would expect, if we describe ionization potentials with IP-EOM-CCSD, we will have similar behavior. We should do quite well for most principal ionizations where the eigenstate is dominated by single excitations, meaning linear combinations of i lO) determinants, with i a.y Q) playing most of the correlation and relaxation role but when the latter shake-up is dominant, then we would logically need i ayb k )Q in our space to do as well for the shake-up eigenvalues. The latter requires IP-EOM-CCSDT [156], and the complementary EA-EOM-CCSDT method [157] and higher [166]... 

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