A problem that most students should be familiar with from ordinary algebra is that of finding the root of an equation *f*(*x*) = 0, i.e., the value of the argument that makes *f* zero. More precisely, if the function is defined as *y* = *f*(*x*), we seek the value α such that

The precise terminology is that α is a *zero* of the function *f*, or a *root* of the equation *f*(*x*) = 0.^{1} Note that we have not yet specified what kind of function *f* is. The obvious case is when *f* is an ordinary real-valued function of a single real variable *x*, but we can also consider the problem when *f* is a vector-valued function of a vector-valued variable, in which case the expression above is a system of equations; this more complicated case is discussed in Chapter 7.

In this chapter we consider only the simple case where *f* is a scalar real-valued function of a single real-valued variable. We will discuss three basic methods for finding the point α: the bisection method, Newton’s method, and the secant method. We then consider a broad class of ideas coming under the heading of *fixed-point theory*, which will enable us to broaden and extend our understanding of iterations in general, whether applied to root-finding problems or not. Finally, we will discuss some variants of Newton’s method and other advanced topics.

Bisection is a marvelously simple idea that is based on little ...

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