Question

Use Mu ̈ller’s method to determine the real and complex roots of
f (x) = x^4 − 2x^3 + 6x^2 − 2x + 5

Answer 1

Let $f(x)=x^4-2x^3+6x^2-2x+5$

Use initial guesses $x_0=-1, x_1=0, x_2=1$

$f(x_0)=f(-1)=16$

$f(x_1)=f(0)=5$

$f(x_2)=f(1)=8$

$h_0=x_1-x_0=0-(-1)=1$

$h_1=x_2-x_1=1-0=1$

$\delta_0=\dfrac{f(x_1)-f(x_0)}{h_0}=\dfrac{5-16}{1}=-11$

$\delta_1=\dfrac{f(x_2)-f(x_1)}{h_1}=\dfrac{8-5}{1}=3$

$a=\dfrac{\delta_1-\delta_0}{h_1+h_0}=\dfrac{3-(-11)}{1+1}=7$

$b=a\times h_1+\delta_1=7\times1+3=10$

$c=f(x_2)=f(1)=8$

$$x_3=x_2+\dfrac{-2c}{b\pm\sqrt{b^2-4ac}}$$

$$x_3=1+\dfrac{-2(8)}{10+\sqrt{10^2-4(7)(8)}}=$$ $$=0.285714+0.795395i$$

$x_0=-1, x_1=0, x_2=0.285714+0.795395i$

Calling the function Muller with different parameters yields two complex roots of the equation $x_1\approx i, x_3\approx1+2i.$ Two other roots can be determined as their conjugate pairs $x_2\approx-i, x_4\approx1-2i.$

$$x_1\approx i$$

$$x_2\approx -i$$

$$x_3\approx 1+2i$$

$$x_4\approx 1-2i$$