Newton-Raphson Method (Edexcel International AS Further Maths: Further Pure 1): Revision Note

Newton-Raphson Method

How do I apply the Newton-Raphson method?

• The Newton-Raphson method is a process for finding roots of equations

  • Equations must be rearranged into the form $\mathrm{f}\left(x\right)=0$

• If $x=\alpha$ is a root of the equation $\mathrm{f}\left(x\right)=0$, then you need to choose $x_0$, the initial (first) approximation

  • This is usual given in the question (and is a value near to the root)

• You can then find successive approximations $x_1$, $x_2$, $x_3$, ... using the Newton-Raphson formula:

  • $x_{n+1}=x_n-\frac{\mathrm{f}\left(x_n\right)}{\mathrm{f}\text{'}\left(x_n\right)}$

  • This is an iterative formula

  • $\mathrm{f}\text{'}\left(x\right)$ is the derivative of $\mathrm{f}\left(x\right)$

    • Work this out beforehand

• Increasing the number of approximations improves the accuracy

  • The Newton-Raphson method usually converges quickly to the root compared to other methods

How do I apply the Newton-Raphson method on my calculator?

• If your calculator has a recursion function

  • Input the formula, start value and number of steps

  • The output is a table showing all the approximations

• Alternatively:

  • Type in the value of $x_0$ and press =

    • This stores it in the "Ans" (answer) button

  • Type $\mathrm{Ans}-\frac{\mathrm{f}\left(\mathrm{Ans}\right)}{\mathrm{f}\text{'}\left(\mathrm{Ans}\right)}$ into your calculator and press =

    • This finds $x_1$

  • Without pressing any other button, press = again

    • This finds $x_2$

  • Repeatedly pressing = gives further approximations

    • The more you press it, the closer to the root it becomes

How does the Newton-Raphson method work geometrically?

• The method works by drawing a tangent to the curve $y=\mathrm{f}\left(x\right)$ at $x=x_0$

  • then finding where the tangent cuts the x-axis

  • This x-intercept is the new approximation, $x_1$

• This process is then repeated

  • Draw the tangent to $y=\mathrm{f}\left(x\right)$ at $x=x_1$

  • Find its x-intercept and call it $x_2$

• The approximations get closer and closer to the root

How do I know if the Newton-Raphson method will fail?

• The Newton-Raphson method fails if the initial value, $x_0$, satisfies $\mathrm{f}\text{'}\left(x_0\right)=0$

  • Algebraically, this is because the formula $x_{n+1}=x_n-\frac{\mathrm{f}\left(x_n\right)}{\mathrm{f}\text{'}\left(x_n\right)}$ cannot divide by zero

  • Geometrically, this is because $x_0$ is at a stationary point on the curve $y=\mathrm{f}\left(x\right)$

    • A tangent drawn at a stationary point is horizontal

    • This will never intersect the x-axis

• The Newton-Raphson method also fails if the sequence $x_1$, $x_2$, $x_3$, ... diverges

  • This can happen if $x_0$ is chosen:

    • too far away from the root

    • or at a point where the gradient is very small

• The Newton-Raphson method is sometimes avoided when $\mathrm{f}\left(x\right)$ is too tricky to differentiate