The document describes the Bairstow method to find the roots of a cubic function. It gives the steps to calculate the coefficients b and c, and then uses a system of equations to find the corrections Δr and Δs to the values of r and s. It performs three iterations, getting closer to the true roots each time. The final roots found are x1=2.29, x2=2.29, and x3=1.14956.

1. METODO DE BAIRSTOW
1. Sacar las raíces de la función:
𝑓( 𝑥) = −3.704𝑥3
+ 16.3𝑥2
− 21.97𝑥 + 9.34
Utilizando:
𝑏 𝑛 = 𝑎 𝑛 𝑐 𝑛 = 𝑏 𝑛
𝑏 𝑛−1 = 𝑎 𝑛−1 + 𝑟𝑏 𝑛 𝑐 𝑛−1 = 𝑏 𝑛−1 + 𝑟𝑐 𝑛
Para determinar los valores de b con:
𝑟 = 2
𝑠 = −0.5
SOLUCION:
𝑓( 𝑥) = −3.704𝑥3
+ 16.3𝑥2
− 21.97𝑥 + 9.34
𝑏3 = −3.704
𝑏2 = 16.3 + (2)(−3.704) = 8.892
𝑏1 = −21.97 + (2)(8.892) + (−0.5)(−3.704) = −2.334
𝑏0 = 9.34 + (2)(−2.334)+ (−0.5)(8.892) = 0.226
𝑐3 = −3.704
𝑐2 = (8.892)+ (2)(−3.704) = 1.484
𝑐1 = −2.334 + (2)(1.484) + (−0.5)(−3.704) = −2.3346
Obteniendo el ∆𝑟 𝑦 𝑒𝑙 ∆𝑠:
𝑐2∆𝑟 + 𝑐3∆𝑠 = −𝑏𝑖
1.484∆𝑟+ (−2.334)∆𝑠 = 2.334
𝑐1∆𝑟 + 𝑐2∆𝑠 = −𝑏0
−2.334∆𝑟 + 1.484∆𝑠 = −0.226

2. Resolviendo el sistema:
∆𝑟 = −0.9047
∆𝑠 = −1.5752
Ahora obtenemos los valores de r y s:
𝑟 = 2 + (−0.9047) = 1.0953
𝑠 = (−0.5)+ (−1.5752) = −2.0752
Con los valores de r y s obtenemos el % de error
𝐸 𝑎,𝑟 =
|∆𝑟|
| 𝑟|
∗ (100) = 82.6%
𝐸 𝑎,𝑠 =
|∆𝑠|
| 𝑠|
∗ (100) = 75.9%
Entonces:
∆𝑟 = −0.179 𝑟 = 2.05
∆𝑠 = −0.042 𝑠 = −1.08
Aplicando una tercera iteración:
∆𝑟 = −0.053 𝑟 = 2.103
∆𝑠 = −0.0165 𝑠 = −1.096
Calculando las raíces para cada caso
𝑋 𝑛 =
𝑟𝑛 + √(𝑟𝑛
2
+ 4𝑠 𝑛)
2
𝑥1 = 2.29
𝑥2 = 2.29
𝑥3 = 1.14956

3. 2.