Fixed-point iteration
| English | 中文 | Pinyin |
|---|---|---|
| iteration/ˌɪtəˈreɪʃn/ | 迭代 | dié dài |
How reliable is an approximation?
- A nonlinear equation has no convenient exact solution. An approximation still needs evidence that it is reliable.
- This lesson studies iteration 迭代: A repeated update in which each new approximation is calculated from the previous one.
Choose the mathematical structure
- Rearrange the equation as x=g(x), choose a starting value and iterate. A limit L must satisfy L=g(L). Near a fixed point, |g prime(L)|<1 provides a local convergence check; a sign bracket can validate the reported rounded root.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines iteration?
A repeated update in which each new approximation is calculated from the previous one.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x²-x-2=0 choose x next=√(2+x), starting at x₀=1. Then x₁=√3≈1.73205, x₂≈1.93185 and the positive fixed point is 2. At 2, g prime=1/(2√4)=1/4, so small local errors shrink. The square-root rearrangement seeks the positive root; it does not give the negative root -1.
Fixed-point iteration
Rearrange the equation as x=g(x), choose a starting value and iterate
Compare the model with the worked case and explain one change.
Find x₁ from x next=√(2+x), x₀=1.
Substitute x₀=1: x₁=√(2+1)=√3.
Test a tempting shortcut
- Different rearrangements can have different convergence behaviour. Check the iterates, domain and original equation; a square-root update cannot reach a negative fixed point.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every rearrangement of an equation converges to the same root from every starting value. This claim is false. Explain which definition or assumption it violates.
Find the positive fixed point.
At a positive fixed point L=√(2+L), so L²-L-2=0 and L=2.
Every rearrangement of an equation converges to the same root from every starting value.
Different rearrangements can have different convergence behaviour. Check the iterates, domain and original equation; a square-root update cannot reach a negative fixed point.
Interpret a new situation
- P3 uses equation rearrangement and numerical iteration. This lesson excludes Newton and numerical integration, which belong to other units.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For x=2, evaluate x²-x-2.
The residual is 2²-2-2=0.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL mathematics; official unit P3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A repeated update in which each new approximation is calculated from the previous one. Choose the relationship, show the method, check its assumptions and interpret the result.