Estimation of the Local Error

It is known that the local error estimate is obtained to the lower-order solution. This is applied, also, in our case of the local error estimate, i.e. the local error estimate is obtained for j>J+1. However, if this error estimate is acceptable, i.e. less than the bound acc, we consider the widely used local extrapolation technique. Thus, although we are controlling an estimation of the local error in the lower-order solution y%+l, we use the higher-order solution y f+l at each accepted step. 

An estimation of the local error is also provided for each Gauss rule family (that depends on the function r x) and the values of a and b). For instance, in the important Gauss-Legendre method, we have the following error estimation (Kahaner, Moler, and Nash, 1989) ... 

In modern programs, the integration step is modified to adapt it to the local requirements of the problem. Usually, h selection is carried out to check the local error E of the algorithm. Later, we will consider the problem of the estimation of the local error for different algorithms and study a strategy to vary h according to E. 

They provide a trivial estimate of the local error. 

To control the step size adaptively we need an estimate of the local truncation error. With the Runge - Kutta methods a good idea is to take each step twice, using formulas of different order, and judge the error from the deviation between the two predictions. Selecting the coefficients in (5.20) to give the same a j and d values in the two formulas at least for some of the internal function evaluations reduces the overhead in calculation. For example, 6 function evaluations are required with an appropriate pair of fourth-order and fifth-order formulas (ref. 5). 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

The integration process is carried out, from f 2 to t , first in two steps of step size h, then in a single step of step size 2h. From the difference between the corresponding values xand x h an estimate of the local discretization error can be computed, viz. 

Propagation error Since the exact value y t ) is unknown in t , we know only its approximation y and thus we have a second source of error independent of the local error. This error can be estimate by means of the secants theorem Given a function d>(a ) and two values 0( va) and 0(xb) obtained in xa and Xb, there is a point within the interval [xa,Xb] where the function s derivative is equal to the secant obtained with the previous points ... 

Another method of controlling the step size is to obtain an estimation of the truncation error at each interval. A good example of such an approach is the Runge-Kutta-Fehlberg method (.see Table 5.2), which provides the estimation of the local truncation error. This error estimate can be easily introduced into the computer program, and let the program automatically change the step size at each point until the desired accuracy is achieved. 

A further objective is the evaluation in group (c) of the local polarization state by taking account of IR errors due to direct currents. Here Eq. (3-28) and the further explanations in the second half of Section 3.3.1 are relevant. In practical application, the error effect of A /<,ff must be estimated [2]. When foreign fields are present, it is necessary to substitute for the At/ value the average of the measurements made on both sides of the pipeline [2,52]. Figure 3-30 gives an example of... 

The error of approximation on a grid. So far we have considered the local difference approximation meaning the approximation at a point. Just in this sense we spoke about the order of approximation in the preceding section. Usually some estimates of the difference approximation order on the whole grid are needed in various constructions. 

Figure 6. The Bohr-Sommerfeld phase corrections t)(8, ) for k = 0, 1, and 2. The ratio r z,k)lK estimates of the error of primitive Bohr-Sommerfeld eigenvalues as a fraction of their local vibrational spacing.

A tighter and local estimate on the generalization error bound can br derived by observing locally the maximum encountered empirical error Consider a given dyadic multiresolution decomposition of the input spac( and, for simplicity, let us assume piecewise constant functions as approxi mators. In a given subregion of the input space, let p be the set o... 

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