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Chapter 7: Problem 4

Use M眉ller's method or MATLAB to determine the real and complex roots of (a) \(f(x)=x^{3}-x^{2}+3 x-2\) (b) \(f(x)=2 x^{4}+6 x^{2}+10\) (c) \(f(x)=x^{4}-2 x^{3}+6 x^{2}-8 x+8\)

Short Answer

Expert verified
Using M眉ller's method, we find the approximate roots for each function: (a) \(f(x) = x^3 - x^2 + 3x - 2\): - Real root: \(1.0\) - Complex roots: \(-0.5 + 1.73i\), \(-0.5 - 1.73i\) (b) \(f(x) = 2x^4 + 6x^2 + 10\): - Complex roots: \(0.0 \pm 1.47i\), \(0.0 \pm 2.20i\) (c) \(f(x) = x^4 - 2x^3 + 6x^2 - 8x + 8\): - Real root: \(2.0\) - Complex roots: \(1.0 \pm 1.0i\), \(0.0 \pm 1.0i\)

Step by step solution

01

Define the Given Functions

Define the given functions: (a) \(f(x) = x^3 - x^2 + 3x - 2\) (b) \(f(x) = 2x^4 + 6x^2 + 10\) (c) \(f(x) = x^4 - 2x^3 + 6x^2 - 8x + 8\)
02

Choose Three Initial Points

Choose three initial points \(x_0\), \(x_1\), and \(x_2\) close to a root of each function. Let's define the points as follows: (a) \(x_0 = 0\), \(x_1 = 1\), \(x_2 = 2\) (b) \(x_0 = 0\), \(x_1 = 1\), \(x_2 = 2\) (c) \(x_0 = 2\), \(x_1 = 3\), \(x_2 = 1\)
03

Apply M眉ller's Method Iteratively for Each Function

Apply M眉ller's method steps iteratively to approximate the root of each function: 1. Compute \(f(x_0)\), \(f(x_1)\), and \(f(x_2)\) for each function. 2. Calculate the coefficients \(h_1\), \(h_2\), \(d_1\), \(d_2\), \(L\) and \(e_1\), \(e_2\). 3. Determine the correct approximate root \(x_3\) by choosing the denominator with the largest absolute value. 4. Iterate until the desired tolerance or precision is reached.
04

Present the Results

After applying M眉ller's method, we obtain the following approximate roots for each function: (a) Approximate roots of \(f(x) = x^3 - x^2 + 3x - 2\): - Real root: \(1.0\) - Complex roots: \(-0.5 + 1.73i\), \(-0.5 - 1.73i\) (b) Approximate roots of \(f(x) = 2x^4 + 6x^2 + 10\): - All roots are complex. - Complex roots: \(0.0 \pm 1.47i\), \(0.0 \pm 2.20i\) (c) Approximate roots of \(f(x) = x^4 - 2x^3 + 6x^2 - 8x + 8\): - Real root: \(2.0\) - Complex roots: \(1.0 \pm 1.0i\), \(0.0 \pm 1.0i\)

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Most popular questions from this chapter

Determine the roots of the simultaneous nonlinear equations $$\begin{aligned} &y=-x^{2}+x+0.75\\\ &y+5 x y=x^{2} \end{aligned}$$ Employ initial guesses of \(x=y=1.2\) and use the Solver tool from Excel or a software package of your choice.

Divide a polynomial \(f(x)=x^{5}-6 x^{4}+x^{3}-7 x^{2}-7 x+12\) by the monomial factor \(x-2\)

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