Eigenvalue smallest

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. 

If some of the reactions of (A3.4.138) are neglected in (A3.4.139). the system is called open. This generally complicates the solution of (A3.4.141). In particular, the system no longer has a well defined equilibrium. However, as long as the eigenvalues of K remain positive, the kinetics at long times will be dominated by the smallest eigenvalue. This corresponds to a stationary state solution. 

The smallest eigenvalue is independent of Ax (it is DK /L ) so that the ratio of largest to smallest eigenvalue is proportional to /Ax. Thus, the problem becomes stiff as /Ax approaches zero (Ref. 106). 

The maximum absolute eigenvalue corresponds to the component with the largest iC value Kif) and the tray with the smallest holdup. Therefore, if the derivative term and any variation in Lj, Vj, and Kij are neglected. 

FIG. 11 Eigenvalues of the radius of gyration tensor (dots largest, squares middle triangles smallest) of micelles vs aggregation number N in an oif-lattiee model of H2T2 surfaetants. The mieelle size distribution for this partieular system has a peak at 28. (From Viduna et al. [144].)... 

The smallest and greatest eigenvalues of the operator A are given by the formulas... 

The constants h and A remain as yet unknown. Since 5 i.s the smallest eigenvalue, it is not difficult to show that... 

The ratio of the largest to the smallest eigenvalue of the Hessian matrix at the minimum is defined as the condition number. For most algorithms the larger the condition number, the larger the limit in Equation 5.5 and the more difficult it is for the minimization to converge (Scales, 1985). 

The condition number is always greater than one and it represents the maximum amplification of the errors in the right hand side in the solution vector. The condition number is also equal to the square root of the ratio of the largest to the smallest singular value of A. In parameter estimation applications. A is a positive definite symmetric matrix and hence, the cond ) is also equal to the ratio of the largest to the smallest eigenvalue of A, i.e.,... 

Given the governing differential equations (the model) and the initial conditions, xo> determine the optimal inputs, u(t), so that the determinant (for the volume criterion) or the smallest eigenvalue (for the shape criterion) of matrix Ane v (given by Equation 12.18) is maximized... 

Layer True Permeability Estimated Permeability Standard Deviation (%) Eigenvector of Smallest Eigenvalue... 

Step 6. When a converged solution has been achieved or at the minimum of the objective function check the variances of the parameters and the eigenvectors corresponding to the smallest eigenvalues to identify any highly correlated zones. Combine any adjacent zones of high variance. 

The relation between the modulus of elasticity and the smallest nonzero eigenvalue of the connectivity matrix established here lends support to the theory3 that has been developing in recent years. By utilizing some of the techniques that have been applied to the solid state one gains an important relation between macroscopic theory and statistical mechanics. 

As the matrix ATA is positive definite, i.e., it has positive eigenvalues, for c2 to be minimum, the solution x must be the eigenvector associated with the smallest eigenvalue Al of this matrix. 

---