Superlinear

In the above equation, the norm is usually the Euclidean norm. We have a linear convergence rate when 0 is equal to 1. Superlinear convergence rate refers to the case where 0=1 and the limit is equal to zero. When 0=2 the convergence rate is called quadratic. In general, the value of 0 depends on the algorithm while the value of the limit depends upon the function that is being minimized. 

Warren GL, Davis JE, Patel S (2008) Origin and control of superlinear polarizability scaling in chemical potential equalization methods. J Chem Phys 128(14) 144110... 

As you can see from the third and fourth columns in the table the rate of convergence of Newton s method is superlinear (and in fact quadratic) for this function. 

The actual dose-response curve falls toward zero risk more slowly than is indicated by the LNT (it is a so-called superlinear model). 

Also, when the constraints are linear, the method always remains in the subspace of the constraints (for a feasible starting point) and dy remains zero. Consequently, for this case the Z BY matrix is unimportant and only the n-m)x n- m) Z BZ matrix needs to be supplied or calculated. While this is not true for nonlinear constraints, it is still convenient to deal only with Z BZ and set Z BY to zero. Moreover, as long as dy remains relatively small, a reasonable superlinear rate of convergence can still be maintained for this decomposition (Nocedal and Overton, 1985). 

A second scheme (22) utilizes a blend of sublinear and super-linear phosphors. At low current, the sublinear phosphor dominates the emission. As the beam current is increased, the emission of the superlinear phosphor begins to surpass that of the... 

IPC [292]. The data are normalized to the transit time and shifted along the current axis for clarity. However, as in the case of CT, there are some difficulties in formulating a simple interpretation of the results obtained due to the impracticability of the Gaussian statistics because the necessary superlinear dependence of the transit time on the sample thickness was not observed. 

Measurements. Evidence for many-body processes beyond the intercolli-sional dip is presented in Fig. 3.6 which shows the variation of the moment yi = f a(v)dv with the product of the densities [329], gig2- For strictly binary interactions, a straight line is expected and is indeed observed at the lower values of the product of densities. Above a certain threshold that is different for each system shown, superlinear dependences are observed that indicate the emergence of spectral components arising from higher than binary interactions. 

Does linearity mean that the response factor is constant for different sample sizes—or that response, when plotted against sample size on log-log paper gives a straight line The latter definition allows sublinear or superlinear behavior— where response follows a power law with an exponent different than one. 

This method is called regula falsi, and it has superlinear convergence, which means that 70 > 1. In this one-dimensional case, 70 = 1.618. 

Note that quadratic convergence implies superlinear convergence. 

Quadratic convergence means that eventually the number of correct figures in Xc doubles at each step, clearly a desirable property. Close to x Newton s method Eq. (3.9) shows quadratic convergence while quasi-Newton methods Eq. (3.8) show superlinear convergence. The RF step Eq. (3.20) converges quadratically when the exact Hessian is used. Steepest descent with exact line search converges linearly for minimization. 

On the contrary, UT is not valid at low concentrations, where it misses the saturation effect. On the other hand, it is better than the two others at high concentrations, where it reproduces the nonlinear increase of r 1 with c, originating from the nonstationary and static energy quenching. Such a superlinearity is usually attributed to the nonlinear corrections to IET available with DET, UT, and a number of other theories (see Section XII) [54,197,198,136]. Fortunately, this effect is very easy to distinguish from the saturation phenomenon at higher concentrations of acceptors their depletion by electron transfer is removed, while the non-linearity related to nonstationary quenching is enhanced. 

Fig. 11.19 Structures and data showing how (5max (2PF method) for dendritic-type bis(diarylamino)stilbene chromophores increase in a superlinear fashion with the...

Very recently Sargent 88 has shown that, although the Lagrangian function is not an exact penalty function, it can indeed be used in a descent test to force global convergence. Moreover the test is satisfied without step-reduction in the final stages, so the convergence is superlinear. Thus the penalty parameter and its associated problems are eliminated. 

At least superlinear convergence rates as the solution is approached. 

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