Bisection and Newton root finding · Bisseção e Newton para busca de raízes
| English | Português |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | iteração |
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? · Qual descrição define corretamente iteração?
A repeated update in which each new approximation is calculated from the previous one. · Uma atualização repetida em que cada nova aproximação é calculada a partir da anterior.
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 · Bisseção e Newton para busca de raízes
For a continuous function with a sign change, bisect a root bracket and keep the half with opposite endpoint signs · Para uma função contínua com mudança de sinal, bissecte um intervalo raiz e mantenha a metade com sinais opostos nas extremidades
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
Find the midpoint of [1,2]. · Encontre o ponto médio de [1,2].
Midpoint=(1+2)/2=1.5. · Ponto médio=(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. · Após duas bisseções descritas acima, encontre a extremidade inferior.
After midpoints 1.5 then 1.25, the retained bracket is [1.25,1.5]. Its lower endpoint is 1.25. · Após os pontos médios 1.5 depois 1.25, o intervalo retido é [1.25,1.5]. Sua extremidade inferior é 1.25.
Every sign change proves a root, including one across a discontinuity. · Toda mudança de sinal comprova uma raiz, incluindo uma através de uma descontinuidade.
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. · Uma mudança de sinal deve ocorrer em um intervalo onde a função é contínua. Newton pode falhar se sua derivada for zero ou se seus iterativos saírem do domínio útil.
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. · Encontre a extremidade superior desse intervalo.
The retained upper endpoint is 1.5, where f is still positive. · A extremidade superior retida é 1.5, onde f ainda é positiva.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
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.