Bisection and Newton root finding · 二分法与牛顿迭代法求根
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| 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
- For a continuous function with a sign change, bisect a root bracket and keep the half with opposite endpoint signs. Newton updates x to x-f(x)/f prime(x), provided the derivative is nonzero.
- 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? · 以下哪项描述正确定义了迭代法？
A repeated update in which each new approximation is calculated from the previous one. · 一种重复更新过程，其中每个新近似值均从前一个近似值计算得出。
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 f(x)=x²-2, f(1)=-1 and f(2)=2. At midpoint 1.5, f=0.25, so the new bracket is [1,1.5]. At midpoint 1.25, f=-0.4375, so the next bracket is [1.25,1.5]. Newton from 1.5 gives 1.4166667.
Bisection and Newton root finding · 二分法与牛顿迭代法求根
For a continuous function with a sign change, bisect a root bracket and keep the half with opposite endpoint signs · 对于存在符号变化的连续函数，对根所在区间进行二分，并保留端点符号相反的半个区间。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the midpoint of [1,2]. · 求区间 [1,2] 的中点。
Midpoint=(1+2)/2=1.5. · 中点=(1+2)/2=1.5。
Test a tempting shortcut
- A sign change must occur on an interval where the function is continuous. Newton can fail if its derivative is zero, or if its iterates leave the useful domain.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every sign change proves a root, including one across a discontinuity. This claim is false. Explain which definition or assumption it violates.
After two bisections described above, find the lower endpoint. · 根据上述两次二分过程，求区间的下界端点。
After midpoints 1.5 then 1.25, the retained bracket is [1.25,1.5]. Its lower endpoint is 1.25. · 依次求出中点 1.5 和 1.25 后，保留的区间为 [1.25,1.5]。其下界端点为 1.25。
Every sign change proves a root, including one across a discontinuity. · 每一次符号变化都证明存在一个根，包括跨越不连续点的符号变化。
A sign change must occur on an interval where the function is continuous. Newton can fail if its derivative is zero, or if its iterates leave the useful domain. · 在函数连续的区间内必须出现符号变化。若导数为零或迭代值超出有效定义域，牛顿法可能失效。
Interpret a new situation
- FP1 numerical methods concern roots: bisection, interpolation and Newton methods. Numerical integration is not included in this lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the upper endpoint of that bracket. · 求该保留区间的上界端点。
The retained upper endpoint is 1.5, where f is still positive. · 保留的上界端点为 1.5，此处函数 f 仍为正值。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL further mathematics; official unit FP1. 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.