1. Determine the largest root in the interval [2, 5 of f(x) = e-r2 + sin x (a) b

Question

1. Determine the largest root in the interval [2, 5 of f(x) = e-r2 + sin x (a) by using MATLAB to plot y = f(x) over the given interval. As part of your answer include a printout of the graph. (b) using the Bisection method (by hand) and four iterations on the interval 3,3.4]. At each iteration give the maximum true error of the estimate of the root, as well as the absolute value of the approximate percent relative error (c) How many iterations would be required in part (b) to ensure that the true error is less than 10-15? (d) Verify your anser in (c) using MATLAB

Explanation / Answer

a) Matlab code to plot the function

f = @(x) (exp(-x^2) + sin(x));
fplot(f,[-10,10])
xlabel('x');
legend('f(x)')

Output plot:

b)

Actual answer is

So,

iter = 1

c = (a+b)/2 = 3.200000000000000

error = mod(c - ans) = 0.058355640025260

percentage_error = mod(c - ans)*100 = 5.835564002526006

iter = 2

c =3.100000000000000

error = 0.041644359974740

percentage_error = 4.164435997474003

iter =3

c = 3.150000000000000

error =0.008355640025260

percentage_error =0.835564002526024

In iter = 4

c = 3.125000000000000

error = 0.016644359974740

percentage_error = 1.664435997474012

root = 3.125000000000000

c) It will take arount 50 iterations

d)

format long;
f = @(x) (exp(-x^2) + sin(x));
a = 0;
b = 4;
c = a;
EPSILON = 10^(-15);
iter = 1;
while ((b-a) >= EPSILON)
c = (a+b)/2;
if ((f(c)) == 0.0)
break;
elseif ((f(c))*(f(a)) < 0)
b = (a+b)/2;
else
a = (a+b)/2;
end
iter = iter + 1;
end
root=c

Sample Output:

root =

3.141644359974742

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