Errors functions for

This may be reduced to an elementary function and error functions for b > o and an elementary function and Dawson s Integral for b < o (13,15). Calculations are easily performed since... 

Integrated error functions can be expressed in terms of error functions. For example, integrating by part, we can find... 

Stopping criteria. A rule used to terminate the iterative training process for neural network learning or function minimization. To prevent overtraining, the stopping criteria may not be based solely upon the error function for example performance on a validation set is often used to stop training. 

P. Zoltowski, "The Error Function for Fitting of Models to Immitance Data," Journal of Electroanalytical Chemistry, 178 (1984) 11-19. 

The following two results are even more impressive. Figure 4 depicts the radial distribution functions, and conjugate error functions, for a system of... 

Note, the inclusion of a fixed value choice in this formula to avoid the singularity in the Error function for small values of its argument. 

After suitable working the integral and all the others needed for a full calculation reduce to forms involving derivatives of the Error function. For example, equation 6.45 becomes... 

Let us now consider the third strategy. The adjacent elements have one single common support point and in this point the adjacent elements have the same first derivative. In this case also, it is possible to find the internal points such that the resulting polynomial (which exploits the two derivatives at the extremes in this case) minimizes the maximum polynomial error function. For instance, in the case of sixth-order polynomial valid in the interval... 

The function erf is the error function, for which tables exist (Abramowitz and Stegun 1968), and which can be numerically computed... 

This equation can be approximated in terms of error function for the range S>40 as... 

The function erf is the error function, for which tables exist [16], and which can be numerically computed (see the function ERF discussed in AppendixE). The solution, (2.35), is shown in Fig. 2.4 for three values of t, increasing as the curves go to the right. 

The error function cannot be expressed in closed form (as a formula with a finite number of terms) for values of z other than 0 or oo. The following asymptotic formula gives values of the error function for large arguments ... 

Table 4.2 Values of the error function for several kL values...

Ideally, the value for the error function for the new parameter set will be lowered. Under certain circumstances, however, particularly when one or more eigenvalues of the Hessian are very small, the improved parameter set can give rise to a larger error function. Evaluation of the error function is computationally very costly, and an alternative method of updating the parameter set has been developed. 

Often in diffusion problems, it is convenient to be able to approximate the error function for very small or very large arguments. For such purposes, the following relations are useful ... 

Learning involves thus designating the miiumum of the error function. For this purpose, we usually apply gradient methods (conjugate gradients), based on the Hessian matrix (Newton, Levenberg-Marquardt methods), or on approximation of the inverse of the Hessian matrix (quasi-Newton methods) [2]. The MLP neural network is sensu stricte a static model, but it is possible to introduce dynamics... 

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