CIE A Level Maths: Pure 3复习笔记6.1.3 x = g(x) Iteration

x = g(x) Iteration

What does xn + 1 = f(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 = f(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 = f(x)


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 = f(x) from your iterative formula


Exam 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