Math 441

Newton’s Method

Newton’s Method

Suppose \(f \in \mathcal{C}^2([a,b])\) and \(f(p) = 0\) for some \(p \in (a,b)\).

Given \(p_n \in (a,b)\) such that \(f'(p_n) \neq 0\), define

\[p_{n+1} = p_n - \frac{f(p_n)}{f'(p_n)}.\]

Then

\[p - p_{n+1} = - \frac{f''\left(\xi\left(p,p_n\right)\right)}{2f'(p_n)}(p - p_n)^2\]

where \(\xi(p,p_n)\) is a number between \(p\) and \(p_n\).

Exercise

Suppose \(f \in \mathcal{C}^2([a,b])\) and \(p_n \in (a,b)\) such that \(f'(p_n) \neq 0\).

• Write down the equation of the tangent line to the graph of \(f\) at \((p_n,f(p_n))\).
• Find the \(x\)-intercept of that line.

Theorem

Let \(f \in \mathcal{C}^2([a,b])\), and let \(p \in (a,b)\) such that \(f(p) = 0\). Assume that \(f'(p) \neq 0\).

Then there exists \(\varepsilon > 0\) such that if \(\left\lvert p - p_0\right\rvert < \varepsilon\), the sequence \(p_0, p_1, p_2, \ldots\) generated recursively by the formula

\[p_{n+1} = p_n - \frac{f(p_n)}{f'(p_n)}\]

converges to \(p\), and

\[\lim_{n \to \infty}\frac{p = p_{n+1}}{(p - p_n)^2} = -\frac{f''(p)}{2f'(p)}.\]