Chapter 18 Numerical Methods
In general, many mathematical problems do not have algebraic solution methods. Instead, we need to resort to numerical methods. This usually results in an approximate solution: we get an answer that is close to the true solution \(x\), but might have an error \(\varepsilon\), so that we obtain some value in the interval \(x-\varepsilon\) to \(x+\varepsilon\). For many practical applications an approximate solution is “good enough”.
18.1 Numerical Root Finding
We have seen how to solve linear and quadratic equations algebraically. We mentioned in section 2.2 that there also exist formulae for finding roots of cubic and quartic polynomial equations, but not quintic or higher.
We often need to find the roots (values of \(x\) for which \(f(x)=0\)) of some function