OCR A Level Maths: Pure

Revision Notes

10.1.4 Newton-Raphson

Newton-Raphson

The Newton-Raphson method

  • The Newton-Raphson method finds roots of equations in the form f(x) = 0
  • It can be used to find approximate solutions when an equation cannot be solved using the usual analytical methods
  • It works by finding the x-intercept of tangents to f(x) to get closer and closer to a root

Newton-Raphson Diagram 1, AS & A Level Maths revision notes

Using the Newton-Raphson method

  • The formula for Newton-Raphson uses the same xn + 1 = f(xn) notation as used in iteration and other recurrence relations
  • After using differentiation to find f’(x) the formula uses iteration to come to an ever more accurate solution

 

Newton-Raphson Diagram 2, AS & A Level Maths revision notes

Can the Newton-Raphson method fail?

The Newton-Raphson method can fail when:

  • the starting value x0 is too far away from the root leading to a divergent sequence or a different root
  • the tangent gradient is too small, where f’(x) close to 0 leading to a divergent sequence or one which converges very slowly
  • the tangent is horizontal, where f’(x) = 0 so the tangent will never meet the x‑axis
  • the equation cannot be differentiated (or is awkward and time-consuming to do)

 Newton-Raphson Diagram 3, AS & A Level Maths revision notes

Exam Tip

  • The formula for the Newton-Raphson method is given in the formula booklet.
  • Use ANS button on your calculator to calculate repeated iterations.
  • Keep track of your iterations using x2, x3… notation.
  • Newton-Raphson questions may be part of bigger numerical methods questions.

Worked example

Newton-Raphson Example Solution, AS & A Level Maths revision notes

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Lucy

Author: Lucy

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.