Bisection and Newton root finding · Bisección y búsqueda de raíces de Newton
| English | Español |
|---|---|
| iteration/ˌɪtəˈreɪʃn/ | iteración |
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? · ¿Qué descripción define correctamente la iteración?
A repeated update in which each new approximation is calculated from the previous one. · Una actualización repetida en la que cada nueva aproximación se calcula a partir de la 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 · Bisección y búsqueda de raíces de Newton
For a continuous function with a sign change, bisect a root bracket and keep the half with opposite endpoint signs · Para una función continua con un cambio de signo, bisecte un intervalo raíz y conserve la mitad con signos de extremo opuestos.
Compare the model with the worked case and explain one change. · Compara el modelo con el caso resuelto y explica un cambio.
Find the midpoint of [1,2]. · Encuentre el punto medio de [1,2].
Midpoint=(1+2)/2=1.5. · Punto medio=(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. · Después de dos bisecciones descritas anteriormente, encuentre el extremo inferior.
After midpoints 1.5 then 1.25, the retained bracket is [1.25,1.5]. Its lower endpoint is 1.25. · Después de los puntos medios 1.5 y luego 1.25, el intervalo retenido es [1.25,1.5]. Su extremo inferior es 1.25.
Every sign change proves a root, including one across a discontinuity. · Cambio de signo garantiza una raíz, incluido uno a través de una discontinuidad.
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. · Un cambio de signo debe ocurrir en un intervalo donde la función es continua. Newton puede fallar si su derivada es cero, o si sus iteraciones salen del dominio 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. · Encuentre el extremo superior de ese intervalo.
The retained upper endpoint is 1.5, where f is still positive. · El extremo superior retenido es 1.5, donde f sigue siendo positivo.
Match each part of a complete solution to its purpose. · Emparejar cada parte de una solución completa con su propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Una suposición justifica el modelo; una comprobación verifica el resultado; la interpretación lo conecta con la pregunta.
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.