%% Lecture 7
%
% Last week.
% .) Newton's method.
% .) Stopping conditions.
%
% This week, what if you can't compute the derivative of the function
% analytically?

% estimating the derivative.
%   why you might want to "smooth" first.

% This continues the exploration of
% "Finding the roots of an equation", finding the
% values x such that f(x) = 0.

% Last time, we had a function in matlab:

function x = newton(x0)
%
% Special case coded for function y = x.^2 + 2x - 14
x = x0;
y = x.^2 + 2*x - 14;
% next class, when y = foo(x);
while abs(y) > 0.01
    % solve for value y = f(x0);
    y = x.^2 + 2*x - 14;
    % solve for value of y' = df(x0)/dx
    yprime = 2*x + 2;
    % solve for offset = -y / yprime
    offset = -y / yprime;
    x = x + offset
end

% 1) which line here computes the derivative of the function?

% 2) Suppose we don't know the derivative, what could we do?

% 3) "estimate derivative" procedure
%%%  what parameters should it take?

% 4) Could we make a function that does this generically (for any function?)  
%     the "feval" function in matlab

%  picture on the board of the numberical derivative?
% 5) Write "function dydx = numericalDerivative(foo,x,step);

%%%%%%%%%%%%% Numerical Integration

% how could we do generic numerical integration?
%  Trapezoidal rule?
%  "left side of the rectangle rule?

%%%%%%%%%%%%% Why do we work so hard to think about generic function?

% to give insight in how matlab uses them.  For many things:


% find x s.t. f(x) = 0

%fsolve

%fminbnd
%fminsearch

%"differential equation solvers..."