A couple issues with the exit condition for the while loop: the algorithm ultimately produces an interval [a, b] with a width of 2n, but will only exit if the interval has width n. More subtly, the proper comparison
b - a > 2n
is susceptible to floating point precision limitations. So, while mathematically speaking the interval [(a+b)/2-n, (a+b)/2+n] will always converge to an interval of width 2n, when represented as floating point values the interval may be wider after "converging". Matlab provides a function eps(x) to represent the smallest increment from the floating point value x to the next highest representable value. Values farther from zero are inherently less precise. You should use:
while b - a > 2*(n + max(eps(a), eps(b)))
to ensure the loop actually reaches its exit condition.
Regarding plotting the function, you will need to define your x-interval before 'a' and 'b' are clobbered by the while loop and using the colon operator to specify the spacing of values between the endpoints:
f = @(x) exp((x.^4 + x.^2 -x + sqrt(5))./5) + sinh((x.^3 + 21.*x + 9)./(21.*x + 6)) -3 ;
a = 0;
b = 1;
E1 = 0.01;
n = E1/10;
xs = a:E1:b;
ys = f(xs);
while b - a > 2*(n + max(eps(a), eps(b)))
X1 = (a + b)/2 - n ;
X2 = (a + b)/2 + n ;
if f(X1) >= f(X2)
a = X1 ;
else
b = X2 ;
end
Xmin = (X1+X2)/2;
end
plot (xs, ys,'black','LineWidth',6)
hold on
grid on
% plot a circle marking the minimum value
plot(Xmin, f(Xmin), 'black', 'marker', 'o', 'markersize', 20)
