x = g(x) Iteration (CIE A Level Maths: Pure 3)

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x = g(x) Iteration

What does xn + 1 = g(xn) mean? 

  • When an equation cannot be solved using the usual analytical methods, we can still find approximate solutions to a certain degree of accuracy
  • Iteration is one way to do this, by repeatedly using each answer as the new starting value for a function, we can achieve an ever more accurate answer
  • Iterations are shown using the notation xn + 1 = g(xn)
  • This is a recurrence relation where, starting with a number (xn), we will get an answer xn + 1 which we can then reuse in the original function
  • Equations need to be rearranged into an iterative formula – ie. the form x = g(x)

new-10-1-3-x-gx-diagram-1-new

Iteration diagrams

  • Iterations can be shown on diagrams called staircase or cobweb diagrams
  • These can be drawn by plotting the graphs of y = x against y = g(x) from your iterative formula

10-1-3-x-gx-diagram-2

Examiner Tip

  • You must show all your steps when rearranging an equation into an iterative formula
  • Working backwards can often be helpful to figure out how an equation has been rearranged but you must write your answer as if you worked forwards
  • Use ANS button on your calculator to calculate repeated iterations
  • Keep track of your iterations using x2, x3… notation
  • Iteration may be part of bigger numerical methods questions

Worked example

x = g(x) Example Solution, AS & A Level Maths revision notes

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Lucy

Author: Lucy

Expertise: Head of STEM

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.