Answer to Question #142402 in Quantitative Methods for Usman

Question #142402
Use Mu ̈ller’s method to determine the real and complex roots of
f(x)=2x^4 + 6x^2 + 8
1
Expert's answer
2020-11-05T13:57:32-0500

Let "f(x)=2x^4+6x^2+8"

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

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

"f(x_1)=f(0)=8"

"f(x_2)=f(1)=16"

"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{8-16}{1}=-8"

"\\delta_1=\\dfrac{f(x_2)-f(x_1)}{h_1}=\\dfrac{16-8}{1}=8"

"a=\\dfrac{\\delta_1-\\delta_0}{h_1+h_0}=\\dfrac{8-(-8)}{1+1}=8"

"b=a\\times h_1+\\delta_1=8\\times1+8=16"

"c=f(x_2)=f(1)=16"

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

"x_3=1+\\dfrac{-2(16)}{16+\\sqrt{16^2-4(8)(16)}}=i"


"x_0=-1, x_1=0, x_2=i"

Calling the function Muller with different parameters yields two complex roots of the equation "x_1\\approx 0.5+1.32288i, x_3\\approx-0.5+1.32288i."

Two other roots can be determined as their conjugate pairs "x_2\\approx0.5-1.32288i,"  "x_4\\approx-0.5-1.32288i."


"x_1\\approx 0.5+1.32288i"

"x_2\\approx 0.5-1.32288i"


"x_3\\approx -0.5+1.32288i"

"x_4\\approx -0.5-1.32288i"


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