Use all the methods that we have learned to solve this problem: Find the root of
ID: 664818 • Letter: U
Question
Use all the methods that we have learned to solve this problem:
Find the root of
f(x) = 0.95x3-5.9 x2+10.9x-6
For the numerical methods, stop doing the iterations if either you have done 5 iterations or the error is below 0.05%.
Summarize your results in the tables provided below. Notice the initial values already given to you in the table for the brackets and starting points for the numerical methods.
Graphical method:
Plot of the function (copy and paste the figure here):
MATLAB code to plot the function (copy and paste the code here):
Estimate of the root:
Bisection:
Iteration
xl
f(xl)
xu
f(xu)
xr
f(xr)
f(xl)*f(xr)
|ea|
0
3
4
1
2
3
4
5
False Position:
Iteration
xl
f(xl)
xu
f(xu)
xr
f(xr)
f(xl)*f(xr)
|ea|
0
3
4
1
2
3
4
5
Simple fixed point Iteration:
Iteration
x0
xr
f(xl)
f(xr)
f(xl)*f(xr)
|ea|
0
3.5
1
2
3
4
5
Newton-Raphson:
Iteration
xi
f(xi)
f’(xi)
|ea|
0
3.5
1
2
3
4
5
Secant:
Iteration
xi-1
f(xi-1)
xi
f(xi)
|ea|
0
2.5
3.5
1
2
3
4
5
Solution using MATLAB “fzero”:
X=
Code to create solution in MATLAB (please copy and paste here the MATLAB code):
Iteration
xl
f(xl)
xu
f(xu)
xr
f(xr)
f(xl)*f(xr)
|ea|
0
3
4
1
2
3
4
5
Explanation / Answer
f = @(x) 0.95*x*x*x - 5.9*x*x + 10.9*x - 6;
eps_step = 1e-5;
eps_abs = 1e-5;
N = 5;
xi1 = 1.0;
xi = 1.1;
fprintf('%-12s%12s%12s%12s%12s%12s ', 'Iteration', 'x(i-1)', 'f(x(i-1))', 'x(i)', 'f(x(i)', '|e(a)|');
for i=1:N
xn = (xi * f(xi1) - xi1 * f(xi)) / (f(xi1) - f(xi));
if abs(xi - xn) < eps_step && abs(f(xn)) < 0.0005
break;
elseif i == N
error('error');
end
fprintf('%-12d%12f%12f%12f%12f%12f ', i, xi1, f(xi1), xi, f(xi), xn);
xi1 = xi;
xi = xn;
end
xn
title('Plot of error')
plot(xn)
xlabel('Number of iterations')
ylabel('Error')
grid on;
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