Question

For each of the following polynomials, apply either the Muller's method or the Bairstow's method to find all real or complex roots. Analyze and contrast performance between the two methods. Plot the functions to choose root guesses appropriately. You can apply any simplifications you deem appropriate prior to applying the numerical methods.1. f(x) = x³ - x² + 2x - 2
2. f(x) = 2x⁴ + 6x² + 8
3. f(x) = -2 + 6.2x - 4x² + 0.7x³
4. f(x) = x⁴ - 2x³ + 6x² - 2x + 5

Answer 1

Answer & Explanation:

Solutions for given polynomial expressions.

1. f(x) = x3 - x2 + 2x - 2

Bairstow's method divides the polynomial by a quadratic function.

Let initial values p = 0.1 q = 0.1

[0]         [1]          [2]           [3]

a[]

1.0         1.0         2.0         2.0

-0.1        1.11        -2.010004        -2.010004

-0.1        1.11  

__________________________________________

b[]

1.0        -1.0        2.010002        -3.191

-0.1        1.100001

-0.1  

__________________________________________

c[]

1.0        -11.200001        34.030003  

dp = -1.100001 dq = 1.9070003

p =   -1.6770535 q = 2.0070002

 

a[]

1.0        -11.0        32.0        -22.0

2.6770535        -22.280973        20.645508        -15.478039

-2.0070002        16.704155  

_________________________________________

b[]

1.0        -8.322947        7.7120266        15.349663

2.6770535        -15.114358

-2.0070002  

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c[]

1.0        -5.645893        -9.409331  

dp = -1.2019181 dq = 0.9261256

p =   -3.8789716 q = 2.9331257

a[]

1.0        -11.0        32.0        -22.0

3.8789716        -27.622267        5.6035914        -4.2372155

-2.9331257        20.886871  

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b[]

1.0        -7.1210284        1.4446075        4.4904633

3.8789716        -12.575847

-2.9331257  

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c[]

1.0        -3.2420568        -14.064364  

dp = -0.35257483 dq = 0.3015399

p =   -4.2315464 q = 3.2346656

a[]

1.0        -11.0        32.0        -22.0

4.2315464        -28.641026        0.52601856        -0.4020975

-3.2346656        21.893684  

__________________________________________

b[]

1.0        -6.7684536        0.124308825        0.41970253

4.2315464        -10.735041

-3.2346656  

__________________________________________

c[]

1.0        -2.5369072        -13.845398  

dp = -0.036022577 dq = 0.032922886

p =   -4.267569 q = 3.2675886

a[]

1.0        -11.0        32.0        -22.0

4.267569        -28.731113        0.0055390946        -0.004241169

-3.2675886        21.998816  

_________________________________________

b[]

1.0        -6.732        0.001        0.004

4.267569        -10.518969

-3.2675886  

_________________________________________

c[]

1.0        -2.4648619        -13.785259  

dp = -3.8030953E-4 dq = 3.605403E-4

Final solution

p = -4.2679496 , q = 3.267949

Now

  -p + Ö(p2- 4q)  

root1 =          = 3.26

  2  

  -p - Ö(p2- 4q)  

root2 =          = 1

  2  

Now the deflated polynomial is

Qn-2(x)* = x - 6.732    Þ  root3 =  6.732

*(Qn-2(x) is nothing but the first n-2 elements of b[ ]. )

the roots of the polynomial equation are 3.26, 1.00 and 6.732.