function main()
% File to show how to use bisection and Newton's method
%  (all in one script file).  (Scroll down to see the functions)

xb=bisect(@testfunc01,0,1,1e-9)

xc=NewtonMethod(@testfunc01,0,20,1e-9)

end



% Functions defined below:

function [y,dy]=testfunc01(x)
% Test function e^x-3x
y=exp(x)-3*x;  % Output y
dy=exp(x)-3;   % and dy
end


function xc=bisect(f,a,b,tol)
% Bisection Method, xc=bisect(f,a,b,tol)
%  Computes an approximation to f(x)=0 given that the
%  root is bracketed in [a,b] with f(a)f(b)<0.  Will run
%  until TOL is reached, and will output the solution xc.
%
% EXAMPLE:  f=inline('x^3+x-1');
%          xc=bisect(f,0,1,0.00005);

%Error check and initialization:
fa=feval(f,a);  fb=feval(f,b);
if sign(fa)*sign(fb)>=0
    error('Root may not be bracketed');
end

iter=0;
while (b-a)/2>tol
    iter=iter+1;
    c=(a+b)/2;
    fc=feval(f,c);
    if fc==0  %This means that (a+b)/2 is the root-
        break     %Break out of the while loop and
                  %continue execution
    end
    if sign(fc)*sign(fa)<0  %New interval is [a,c] (reset b)
        b=c; fb=fc;
    else
        a=c; fa=fc;   %New interval is [c,b] (reset a)
    end
end
fprintf('Finished after %d iterates\n',iter);
xc=(a+b)/2;
end

function out=NewtonMethod(F,x0,numits,tol)
%  Inputs to Newton's Method:
%     F- Name of a function that returns BOTH 
%        values of g(x) and g'(x)
%     x0= Initial value of x.
%     numits=  Maximum number of iterations
%     tol   =  Tolerance (like 1e-6) measures
%              change between iterates of x
%
%  Example:
%    function [y,dy]=F(x)
%         y=x^2 - x - 1;
%        dy=2*x-1
%  
%  out=NewtonMethod(F,1,10,1e-8)
%

for k=1:numits
   [g,dg]=feval(F,x0);
   if dg==0
      error('Derivative is zero');
   end
   xnew=x0-g/dg;
   d=abs(xnew-x0);
   if d<tol
      out=xnew;
      break
   end
   x0=xnew;
end    
fprintf('Newton used %d iterations\n',k);
out=xnew;
end