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Iteration
Some equations cannot be solved exactly. Iteration homes in on a solution by repeating a formula.
Change of sign
If f(x) changes sign between two values, there is a root between them (for a continuous graph).
x3 − x − 5: at x = 1 it is −5; at x = 2 it is 1. A root lies between 1 and 2.
x3 − x − 5: at x = 1 it is −5; at x = 2 it is 1. A root lies between 1 and 2.
Iterative formula
Rearrange the equation into x = (something).
x3 − x − 5 = 0 becomes x = ∛(x + 5).
This gives the formula xn+1 = ∛(xn + 5).
x3 − x − 5 = 0 becomes x = ∛(x + 5).
This gives the formula xn+1 = ∛(xn + 5).
Using it
Start with x0, put it in to get x1, then put x1 in to get x2, and so on. On a calculator, type the start value, press =, then type the formula using ANS and keep pressing =.
Use xn+1 = ∛(xn + 5) with x0 = 2 to find x1 to 4 decimal places.
- x1 = ∛(2 + 5) = ∛7
- = 1.9129
Answer: 1.9129
A sign change shows a root lies between two values. Rearrange to x = ..., then repeatedly substitute to approach the solution.
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