Problem

Determine the highest real root of f (x) = 0.95x3 – 5.9x2 + 10.9x – 6:(a) Graphically.(b)...

Determine the highest real root of f (x) = 0.95x3 – 5.9x2 + 10.9x – 6:

(a) Graphically.


(b) Using the Newton-Raphson method (three iterations, xi = 3.5).


(c) Using the secant method (three iterations, xi–1 = 2.5 and xi = 3.5).


(d) Using the modified secant method (three iterations, xi = 3.5, δ = 0.01).


(e) Determine all the roots with MATLAB.

Step-by-Step Solution

Solution 1

a. Type the following code on the console.

Code:

>> x = linspace(0,4);

>> y = 0.95*x.^3-5.9*x.^2+10.9*x-6;

>> plot(x,y);grid

Output:

Highest real root ≈ 3.3

b. Newton-Raphson Method

Perform the iterations to get the following table

i

x i

f ( x ) f ' ( x ) εa

0

1

2

3

3.5

3.365 6 51

3.345 1 12

3.344 6 45

0.60625 4.5125

0.071249 3.468997 3.992%

0.001549 3.318537 0.614%

7.92E-07 3.315145 0.014%

c. Secant Method

Perform the iterations to get the following table

i x i 1 f(xi1)

x i

f ( x i ) εa

0 2.5 -0.78125

1 3.5 0.60625

2 3.063063 -0.6667

3 3.291906 -0.16487

3.5

3.063 0 63

3.291 9 06

3.367 0 92

0.60625

-0.6667 14.265%

-0.16487 6.952%

0.076256 2.233%

d. Modified secant method

Perform the iterations to get the following table

i

x

x + dx f ( x ) f ( x + d x ) f ' ( x ) εa

0

1

2

3

3.5

3.3698

3.346161

3.344702

3.535 0.60625 0.76922 4.6563

3.403498 0.085704 0.207879 3.6256 3.864%

3.379623 0.005033 0.120439 3.4489 0.706%

3.378149 0.000187 0.115181 3.4381 0.044%

e. Write the following on the console

>> a = [0.95 -5.9 10.9 -6];

>> roots(a)

ans =

3.3446

1.8391

1.0268

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Solutions For Problems in Chapter 6