Iteration (Edexcel GCSE Maths)
Revision Note
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MarkExpertise
Maths
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
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,
- Replace the x on the left with xn+1 (meaning the "next" value of x) and any x's on the right with xn (meaning the "current" value of x)
- 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
- 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
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
- 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
- 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
- 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
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