Answer to Question #142403 in Quantitative Methods for Usman

Question #142403
Use Mu ̈ller’s method to determine the real and complex roots of
f (x) = x^4 − 2x^3 + 6x^2 − 2x + 5
1
Expert's answer
2020-11-05T13:56:49-0500

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"


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