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

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b[]

1.0 -1.0 2.010002 -3.191

-0.1 1.100001

-0.1

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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

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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

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b[]

1.0 -6.7684536 0.124308825 0.41970253

4.2315464 -10.735041

-3.2346656

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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

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b[]

1.0 -6.732 0.001 0.004

4.267569 -10.518969

-3.2675886

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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.