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Iteration

Iteration

What is iteration?

  • Some equations do not have “nice” solutions
    • They are not integers (whole numbers), fractions or simple decimals
    • Instead, they can be irrational decimal solutions that go on forever with no pattern 
  • Iteration is a repeated process used to solve such equations
    • the process starts with an initial value (starting value)
    • after each stage of the process (after each "iteration"), a solution is produced
    • the solutions get more and more accurate as more and more iterations are performed
      • these solutions are called estimates
  • Scientific calculators allow us to perform iterations very quickly using the ANS button
    • Iteration questions will only be asked in the calculator exam

It Calc Notes fig1, downloadable IGCSE & GCSE Maths revision notes

How do I make an iterative formula?

  • Find the equation you would like to solve using iteration
    • for example,  x3 + x = 7
  • Rearrange this equation into the form x = f(x) by making any x the subject of the equation
    • for example, x equals cube root of 7 minus x end root
  • Replace the x on the left with xn+1 (meaning the "next" value of x) and any x's on the right with x(meaning the "current" value of x)
    • x subscript n plus 1 end subscript equals cube root of 7 minus x subscript n end root
    • This called the iterative formula
      • n and n+1 are just counters: n+1 is simply one more than n
      • n starts at 0 so the process starts with x0, the initial value
      • x1 is your first estimate, x2 is your second estimate, etc

How do I use my calculator to do iteration?

  • Find a good initial (starting) value (x0) near to the solution
    • This is often given in the question, for example x0 = 2
  • Store x0 = 2 into your calculator (by typing 2 and pressing the "=" button)
    • 2 is now stored under the "Ans" button
  • Type in the right-hand side of the iteration formula with "Ans" instead of xn
    • cube root of 7 minus Ans end root
  • Press "=" to find x1 (be careful to only press "=" once)
    • x1 = 1.709975...
  • Without pressing any other button, press "=" again to find x2
    • x2 = 1.742418...
  • Press "=" again to find x3
    • x3 = 1.738849...
  • Repeat as many times as required
    • the more you do, the closer the estimates get to the true solution

It Calc Notes fig2, downloadable IGCSE & GCSE Maths revision notes

What do x1, x2, x3, ... represent?

  • x1, x2, x3 ... etc are estimates to the solution of x = f(x)
    • for example, x1 = 1.709975..., x2 = 1.742418..., x3 = 1.738849... are estimates to x equals cube root of 7 minus x end root
  • They are also estimates of solutions to any rearrangements of x = f(x)
    • such as the original equation trying to be solved,  x3 + x = 7
  • This makes x1, x2, x3 ... estimates to the solution of the original equation
  • The more times you perform the iteration, the better the estimates get to the real solution

How do you show that there is a solution in a given interval?

  • To find x0 (the initial / starting value), you are often asked to show that there is a solution in an interval
  • For example, show that there is a solution to x3 + x = 7 between 1 and 2
    • Method 1: Leave a constant term (e.g. the 7) on the right, substitute x = 1 and x = 2 into the left and show that this gives values below and above 7
      • 13 + 1 = 2 and 23 + 2 = 10 which are below and above 7
      • A solution therefore lies between 1 and 2
    • Method 2: Use "0" as your constant term on the right (by rearranging the equation into "... = 0"), then substitute in x = 1 and x = 2, showing this gives values below and above 0, i.e. negative and positive
    • this is called a change of sign between 1 and 2
      • x3 + x - 7 = 0
      • Substitute x = 1 into the left-hand side: 13 + 1 - 7 = -5 (negative)
      • Substitute x = 2 into the left-hand side:  23 + 2 - 7 = 3 (positive)
      • A solution lies between 1 and 2 as there is a change of sign
  • Knowing an interval that contains the solution helps to find x0
    • If the solution is between 1 and 2 then you could choose either x0 = 1 or x0 = 2

Exam Tip

  • Be careful to not press =/EXE or "Ans" more than once at a time. If you do the best thing to do is to restart from the beginning.
  • Iteration questions always require working with a lot of decimal places, so write down all digits from your calculator display for x1, x2, etc. and round them at the end if necessary

Worked example

It Calc Example fig1 sol, downloadable IGCSE & GCSE Maths revision notes

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