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Chapter 7Nonlinear equations

Nonlinear equations (NLEs) in more than one unknown parameter are the subject of this chapter. This is the multiparameter version of root-finding from Chapter 4. Generally, I have NOT found this to be, of itself, a prominent problem for statistical workers, but there are some important uses. Unfortunately, people sometimes try to use NLEs methods to find extrema of nonlinear functions. For such problems, my experience suggests that it is almost always better to use an optimization tool.

As we have already dealt with one-parameter problems in Chapter 4, we will be dealing with two or more equations in an equal number of unknowns. If we consider that these functions are like residuals in the nonlinear least squares problem and we write these equations in the parameters x as

7.1

then clearly a least squares solution that has a zero sum of squares is also a solution to the NLEs problem, so nonlinear least squares methods can be considered for NLEs problems, but we must be careful to check that a solution has indeed been found, and there may be efficiencies in using methods that are explicitly intended for the NLE problem, as a least squares approach in some sense squares quantities that should be considered in their natural scale. However, my opinion is ...

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