How do I solve problems with iteration?
- To solve equations using iteration two things are required:
- The equation f(x) = 0 needs to be rewritten into the form x = g(x)
- Some equations may need rearranging to get to the f(x) = 0 stage too!
- There may be more than one way to rearrange to get the form x = g(x) but a question will state which one to show
- An initial value (x0) (or starting value)
- This is the first estimate of a solution to the equation f(x) = 0
- A solution to the equation f(x) = 0 is also called a root
- On a graph, a root is where the curve crosses the x-axis
- On one side of the root the value of f(x) will be positive
- On the other side of the root the value of f(x) will be negative
- This is called the change of sign rule
- Two values, a and b, chosen appropriately, lead to f(a) and f(b) having different signs
- It does not matter which one is positive and which is negative, they just need to have different signs
- In exam questions a and b are usually given but can be hidden
Look for phrases such as “… solution between …”, “… root in the interval …”
- This information can then be used to find an initial estimate (x0) of a solution
- Once x = g(x) and x0 are known, the iterative process can begin
- See Revision Notes Iteration – Using a Calculator
When writing down an iterative formula, write it down without the n and n+1
Go back and add these in afterwards.