Integration local error

The most sophisticated differential equation solver considered in this book and discussed in the next section includes such step size control. In contrast to most integrators, however, it takes a full back step when facing a sudden increase of the local error. If the back step is not feasible, for example at start, then only the current step is repeated with the new step size. 

In the case of the ultramicroelectrodes such as the disk electrode, it is necessary to integrate over the surface, and sometimes there will be unequally spaced points along the surface, as for example, in direct discretisation on an unequal grid in the example program UME DIRECT. As mentioned in Chap. 12, it is found that due to the errors in the computed concentration values, the local fluxes are so inaccurate that any integration method better than the simple trapezium method is not justified. The routine U TRAP is thus recommended here. It integrates local current densities, precalculated by using the above routine U DERIV. 

The true (global) error of a computed solution is its deviation from the true solution of the initial value problem. Practically all present-day integrators, including this one, control the local error for each step and do not attempt to control the global error directly. 

If the local phase-lag error is bounded by acc and the step size of the integration used for the nth step length is h , the estimated step size for the (n + l)st step, which will give a local error bounded by acc, must be... 

A number of characteristics are desirable for user interfaces. Roberts and Moran (1982) identify the most important attributes of text editors as functionality of the editor, learning time required, time required to perform tasks, and errors committed. To this might be added the cost of evaluation. Heurison and Hix (1989) identify usability, completeness, extensibility, escapabUity, integration, locality of definition, structured guidance, and direct manipulation as well in their more general study of user interfaces. They also note a number of tools useful for interface development, as does Lee (1990). 

When the points are evenly spaced, it is usual to express the order m of local error of the algorithm as a junction of the integration step h Oljf). 

The Newton-Cotes formulae use a constant distance between the points within the integration interval. They can be close, open, or semiopen and they allow us to obtain the expression of the local error depending on h. 

Knowing how to calculate the local error of a specific algorithm is not important here. It is enough to remark that it depends on a known power of the integration step. For example, the Runge-Kutta algorithm (2.9)-(2.11) has a local error of 0 h ). 

Theoretically, algorithms with higher orders have smaller local errors and allow larger integration steps. 

In practice, it is reasonable to select a good compromise between the local error and the order pt the latter cannot be too small to avoid large local errors and hence too small integration steps p cannot be too large to prevent... 

In the previous procedure, we did not account for the effect of round-off errors on the global solution. In the most of cases, this error is negligible with respect to the local error, but it can become significant when the number of integration steps is quite large. 

When the method has a large local error (i.e., the Euler method), there is the need to use a very small integration step and, hence the number of steps can become very large. 

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. 

Although qualitative, this discussion shows that controlling only the local error is usually a good way to keep the integration step within the region of algorithm stability since is overestimated as it exits such a region. 

Since the alternative methods of evaluating the local error can exploit a common strategy to automatically control the integration step, we discuss the latter point first. 

The point at which each couple of curves crosses indicates where an order has a smaller local error than the other one with the same integration step. Consequently, Figure 2.7 shows that if the acceptable error is larger than Ci2, the first-order method is preferred if it is within ei2 and 623, the second-order method is preferred and if it is smaller than 623, the third-order method is preferred. 

Thus, the global error of the Trapezoid rule formula carried out over n intervals is of order of h, while the local error (which is the error of integration of one interval) is of order of h. ... 

The above factors contribute a local error at each step of the calculation. As we move from point to point, the local errors combine to produce a global error. Good integration packages estimate the amount of error as the calculation moves along. The user specifies a tolerance that defines the magnitude of local error that... 

Bieniasz LK (2008) An adaptive Huber method with local error control, for the numerical solution of the first kind Abel integral equations. Computing 83 25-39... 

Integrating with MATLAB s integrator ODE45 leads to the critical steps given in Tab. 6.1. ERR/TOL is the estimated local error increment divided by the scaled required tolerance and thus should be less than one. Fig. 6.1 shows some simulation results. 

More accurately, as the inverse problem process computes a quadratic error with every point of a local area around a flaw, we shall limit the sensor surface so that the quadratic error induced by the integration lets us separate two close flaws and remains negligible in comparison with other noises or errors. An inevitable noise is the electronic noise due to the coil resistance, that we can estimate from geometrical and physical properties of the sensor. Here are the main conclusions ... 

---