Answer in matlab code please 6.3 Determine the highest real root of f(x)=x3-6x2

Question

Answer in matlab code please

6.3 Determine the highest real root of f(x)=x3-6x2 +1 1x-61 using the following techniques a) Graphically -don't forget to always add a title and axis labels. Add comment to your code identifying the solution, or add a text box to the figure b) Create a Newton-Raphson function and use it to solve this problem,with a starting value ofx 3.5. Assign a convergence factor ey such that the answer will have at least 3 significant figures c) Create a secant function and use it to solve this problem. Your two starting values should be x 2.5 and x-3.5. Assign a convergence factor ey such that the answer will have at least 3 significant figures d) Create a modified secant function and use it to solve this problem. The starting value of x should be 3.5 and 6-0.01. Assign a convergence factor, er such that the answer will have at least 3 significant figures. e) Use a built in MATLAB function, roots, to determine all the roots.I

Explanation / Answer

Find the root using Newton-Raphson method:

Code:

fun=@(x) x^3-6*x^2+11*x-6.1;

x=3.5;

c_factor=0.001;

d=@(x) 3*x^2-12*x+11;

for i=1:100

    x(i+1)=x(i)-((fun(x(i))/d(x(i))));

    e(i)=abs((x(i+1)-x(i))/x(i));

    if e(i)<c_factor

        break

    end

end

fprintf('The root of given equation is: %.4f ', x(i))

Find the root using secant method:

Code:

fun=@(x) x^3-6*x^2+11*x-6.1;

x(1)=2.5;

x(2)=3.5;

c_factor=0.001;

iter=0;

for i=3:1000

   x(i) = x(i-1) - (fun(x(i-1)))*((x(i-1) - x(i-2))/(fun(x(i-1)) - fun(x(i-2))));

    iter=iter+1;

    if abs((x(i)-x(i-1))/x(i))*100<c_factor

        fprintf('The root of given equation is: %.4f ',root=x(i))

        fprintf('The iteration is: %d ',iter)

        break

    end

end

Find the root using modified secant method:

Code:

fun=@(x) x^3-6*x^2+11*x-6.1;

x(1)=2.5;

x(2)=3.5;

delta=0.01;

c_factor=0.001;

iter=0;

for i=3:1000

x(i) = x(i-1)-((delta*fun(x(i-1)))/(fun(x(i-1)+delta*x(i-2))-fun(x(i-2))));

    iter=iter+1;

    if abs((x(i)-x(i-1))/x(i))*100<c_factor

        fprintf('The root of given equation is: %.4f ',root=x(i))

        fprintf('The iteration is: %d ',iter)

        break

    end

end

(e)

Code:

fun = [1 -6 11 -6.1];

fprintf('All roots in the given equation')

roots(fun)

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