How can I escape this loop and draw the diagram? (Find minimum of a function using Bisection method)

I need to find the minimum of the function using Bisection method. And I'm a beginner and this is the code I created. Can you show me the mistakes of this please? I need to draw the graph also.

x = [0,1]

tolerance = E1 = 0.01

thank you.

%% Find the minimum of a function using Bisection method

clc

clear

% Define givens

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;

while b - a > n

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

L = [a , b];

plot (L,n*1,'black','LineWidth',6)

hold on

grid on

Answer 1

[Chandler Hall]

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)

[Janidu]

Understood!

Thank you very much.