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10 Root Finding

The root of a function f(x) (f:R → R) is simply some value r for which the function is zero, that is, f(r) = 0.

This topic is broken into two major sub-problems:

  1. Finding the root of a real-valued function of a single variable, and
  2. Finding the root of a vector-valued function of a many variables.

There are five techniques which may be used to find the root of a univariate (single variable) function:

  1. Bisection method
  2. False-position method
  3. Newton's method
  4. Secant method
  5. Müller's method

Given a vector-valued multivariate function f(x) (f:Rn → Rn), we will focus on a generalization of Newton's method to find a vector of values r such that each of the functions is zero, that is, f(r) = 0.