Function root-finding

Examples of this chapter can be found in the Vol3 Chapterl directory in the WileyVol3. zip file available at www.chem.polimi.it/homes/gbuzzi. 

This chapter deals with the problem of finding the value t = ts that zeroes a function in the one-dimensional space  

We will, for instance, find the value of tj that zeroes the function y(t) = — 2t — 5 

Only real problems involving the variable f R are considered. 

we describe the main iterative methods used to numerically solve this problem these are referred to as iterative because they provide a series of values ti, t2. t such that the value of U for a sufficiently large i approximates the solution ts with the required precision. 

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The zero-crossing problem consists of function root finding during the integration of a differential system. In other words, during the integration of ODE/DAE systems, there may be a need to calculate the value of the independent variable t at which a certain fimction of the dependent variables y is zeroed. 

The first strategy is the only one of any use when the interval of uncertainty is unknown. If the function is monotone, it is always possible to find an interval of uncertainty and, consequently, to use ad hoc techniques to find its root Algorithms for function root-finding with a known interval of uncertainty are discussed below, whereas the other situation is considered in Section 1.6. 

Inverse interpolation with rational functions is very efficient and is the ideal basis for the development of a general function root-finding program, even though it is slightly more complex than the other algorithms. 

The inverse rational interpolation method is used as the basic method when more than three iterations of the BzzMath classes for function root-finding have been performed. 

Before ending this chapter, we should emphasize several points crucial to the function root-finding problem. 

One-dimensional optimization has an additional disadvantage with respect to function root-finding. 

Function root-finding to reduce the interval of uncertainty, simply analyze the sign of the function evaluated in a single point. 

If the prediction, ts, is iteratively used to replace the worst of the three previous points and if the procedure converges, this method has a convergence order of 1.3 (Luenberger and Ye, 2008). The convergence is not quadratic (contrary to certain methods for function root-finding) but is faster than the linear one. 

When none of the previous conditions is satisfied, the minimum is in [tcits]. Since a single point is enough to discriminate the subintervals, the direct method is as efficient as Bolzano s method at function rooting-finding (Chapter 1). 

Stop criteria for one-dimensional optimization problems are very similar to function root-finding and are discussed at exhaustive length in Chapter 1. 

The values of tTolAhs and tTolRd allow absolute and relative errors, respectively, to be evaluated. As previously stressed, there is an important difference with respect to function root-finding. 

In function root-finding, the value of tTolRel can be in the order of macheps here, it has to be in the order of the square root of macheps. 

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