AQA A Level Maths: Pure复习笔记10.1.4 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


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


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)


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