Bisection and Newton root finding
| English | Français |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | itération |
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?
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 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
For a continuous function with a sign change, bisect a root bracket and keep the half with opposite endpoint signs
Compare the model with the worked case and explain one change.
Find the midpoint of [1,2].
Midpoint=(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.
After midpoints 1.5 then 1.25, the retained bracket is [1.25,1.5]. Its lower endpoint is 1.25.
Every sign change proves a root, including one across a discontinuity.
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.
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.
The retained upper endpoint is 1.5, where f is still positive.
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 pure 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.