%% Annotated Matlab transcript -- 2/13
%% CS 323 - Doug DeCarlo

% We define a function f(x) like this:

>> type f.m
function value = f(x)
value = x.^5 - 0.5;
end

% And we can just use it as follows:
% (since we used .^ above, it can operate on vectors)

>> x = 0:0.01:1;
>> plot(x,f(x))

% Here we plot it with x=0 so we can see the root (around 0.9)
>> plot(x,f(x),x,0)

% The root is actually:
>> (1/2)^(1/5)
ans = 0.87055056329612


% Now we write a simple version of bisection that calls f directly:
% we pass in the interval [a,b] and epsilon value

>> type bisect.m
function c = bisect(a, b, ep)
c=(a+b)/2;
while (b-c >= ep)
    if sign(f(b))*sign(f(c)) <= 0
        a=c;
    else
        b=c;
    end
    c=(a+b)/2;
end

% Let's use the interval [0,1] as there is a zero crossing there
% and try different epsilons:
>> bisect(0,1,0.1)
ans =  0.81250000000000

>> bisect(0,1,0.01)
ans =  0.86718750000000

>> bisect(0,1,0.001)
ans =  0.87011718750000

>> bisect(0,1,0.0001)
ans =  0.87054443359375

>> bisect(0,1,0.1e-5)
ans =  0.87055110931396

>> bisect(0,1,0.1e-12)
ans =  0.87055056329615