Largest eigenvalue

In our experiment we used thresholing value 6 = 0.025, yielding m = 25 and a ratio between tbe smallest and the largest eigenvalue 4 = 0.01. 

This is an important general result which relates the free energy per particle to the largest eigenvalue of the transfer matrix, and the problem reduces to detennining this eigenvalue. 

I - FH) r on a quadratic surface. Here I is the unit matrix and denotes the nth cycle. The ultimate convergence rate is governed by the magnitude of the largest eigenvalue of the matrix (I - FH). This will be... 

A simultaneous computation of the four largest eigenvalues Ai,..., A4 leads to the following table ... 

Fig. 7. Eigenmeasure V2 of the Frobenius-Perron operator to the second largest eigenvalue A2 = 0.9963 for the test system (15) with 7 = 3. iV2 was computed via our new subdivision algorithm (cf. Section 4).

For stability reasons, the micro-step-size 5t has to be chosen smaller than the inverse of the largest eigenvalue of the (scaled) truncated quantum operator % This can imply a very small value of 5t compared to... 

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

Both methods are time-reversible. For A = 0, they reduce to SHAKE and RATTLE. In contrast to SHAKE and RATTLE, the time step is not restricted by the largest eigenvalue of A. 

This scheme requires the exponential only of matrices that are diagonal or transformed to diagonal form by fast Fourier transforms. Unfortunately, this matrix splitting leads to time step restrictions of the order of the inverse of the largest eigenvalue of T/fi. A simple, Verlet-like scheme that uses no matrix splitting, is the following ... 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

This matrix must be diagonalized to obtain the largest eigenvalue and its eigenvector, which allows the partial coverages and the correlation functions to be obtained. This is trivial for no interactions (yi = yi = 3 12 = 1) and gives... 

If there is a largest eigenvalue of Z, say A, then the first sequence must terminate, and this can happen only if... 

The + sign of the radical is chosen to ensure that A is indeed the largest eigenvalue. Similar reasoning, starting with Eq. (7-30), which in this instance is multiplied by X + iY from the left, leads to... 

In this equation, the eigenvectors A are the target yields, and the eigenvectors, E t), are the optimal fields. The eigenvector associated with the largest eigenvalue is the globally optimal field, in the weak response limit. 

The largest eigenvalue fmax of F is the maximum product )deld (flux) and the corresponding eigenvector is the set of coefficients which define... 

The type of the most favorable second-order bond distortion is given by the eigenvector corresponding to the largest eigenvalue. 

If u, and V, represent the eigenvectors corresponding to the largest eigenvalue then according to eq. (31.4) we have ... 

The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations... 

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