A = @(x) [[1,2,3,x];...
          [4,5,x,6];...
          [7,x,8,9];...
          [x,10,11,12]];

f = @(x) det(A(x))-1000;


% plot the function to "visually" bracket the root
X=linspace(9,10,100);
plot(X,arrayfun(f,X));

tol = 1.0E-06;
a = 9; b = 10;

fa = f(a);
fb = f(b);
assert(fa*fb<0);

iter = 0;
while b-a > tol,
    iter = iter + 1;
    
    c  = (b+a)/2;
    fc = f(c);
    
    fprintf('Iter: %4d, a: %e, f(a): %e, c: %e, f(c): %e, b: %e, f(b): %e\n',...
            iter,a,fa,c,fc,b,fb);
    
    if fc == 0.0  % in reality this should be fc < tol_fun
        break;
    end
    
    if fa*fc < 0
        b  = c;
        fb = fc;
    else
        a  = c;
        fa = fc;
    end
end

fprintf('Backward error: %e\n', fc);