Numerical iteration for equation roots · Higher
導入済み| English | 意味 |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | 反復計算 |
Will repeated calculation settle down?
- An equation has no convenient factorisation. Can repeated calculation locate a solution without guessing random values?
- This lesson studies iteration 迭代: Repeat a rule, using the previous output as the next input.
Choose the mathematical structure
- Rearrange an equation as x=g(x), choose x₀ and use x_(n+1)=g(x_n). Record sufficient working precision. A stable-looking sequence must still be checked in the original equation; not every rearrangement converges. A sign change across continuous inputs can check a rounded root.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
From x₀=3, the rearrangement x next=x²-2 diverges while x next=√(x+2) converges. Why?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The same equation x²-x-2=0 gives many forms x=g(x). Near a fixed point, convergence needs |g′|<1; x²-2 fails this at the root, so its iterates blow up while √(x+2) settles to 2.
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, use x next=√(x+2) with x₀=1: x₁=√3≈1.73205, x₂≈1.93185, x₃≈1.98289. The positive fixed point is 2 because 2=√4 and 2²-2-2=0. The rearrangement only seeks a nonnegative root; the original equation also has root -1. Using x next=x²-2 from 3 instead gives 7 then 47, so a different rearrangement can diverge.
Numerical iteration for equation roots
Rearrange an equation as x=g(x), choose x₀ and use x_(n+1)=g(x_n)
Compare the model with the worked case and explain one change.
Find x₁ from x next=√(x+2), x₀=1.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
√(1+2)=√3.
Test a tempting shortcut
- Use the previous iterate, not the starting value every time. Keep more digits than the final answer. A sign change needs continuity; a jump across a vertical asymptote does not guarantee a root.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every rearrangement of an equation converges from every starting value. This claim is false. Explain which definition or assumption it violates.
Find the positive fixed point.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Solve L²-L-2=0 and retain the nonnegative solution.
Every rearrangement of an equation converges from every starting value.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Use the previous iterate, not the starting value every time. Keep more digits than the final answer. A sign change needs continuity; a jump across a vertical asymptote does not guarantee a root.
Interpret a new situation
- AQA A20 Higher uses numerical iteration and suffix notation. Use the calculator to follow the stated rule, then report the requested rounding with a check. Newton derivatives are outside this GCSE lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the original-equation residual at x=2.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
2²-2-2=0.
Match each iteration step to its purpose.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
An apparent limit must satisfy L=g(L) AND the original equation; checking residuals catches divergent or wrong-root rearrangements.
Use this in your course
- 8300 · Higher · 3.2. Match the target tier and specification before assigning extensions.
- 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.
Repeat a rule, using the previous output as the next input. Choose the relationship, show the method, check its assumptions and interpret the result.