%% Function Handles and Function Functions

%%
% The name of an anonymous function is known as a _function handle_.
%
% We can define
sqr = @(x) x.^2

%%
% Note that this function will operate elementwise on vectors or matrices.
% For example
sqr(5)

%%
% or
sqr(1:5)

%%
% Another example with a random $2\times 2$ matrix
A = floor(10*rand(2,2))
A_squared = sqr(A)

%%
% MATLAB includes many so-called ''function functions''.
% These are functions that take function handles as input.
% Examples are 
%
% * |quad| (for numerical integration),
% * certain calculus-related routines (such as |fminbnd| or |fzero|),
% * and many commands for the solution of differential equations (such as
% |ode45| or |ode15s|).
%
% Of course, you can also write your own function functions.

%%
% We now give three examples.

%% Numerical Integration:

%%
% To integrate the function $f(x)=x^2$ on the interval $[0,1]$ we can pass
% the function handle |sqr| defined above to the MATLAB |quad| function
quad(sqr,0,1)

%% Minimization:

%%
% To find the minimum of the function $f(x)=(x-2)^2-5$ on the interval
% $[-3,3]$ we can define a new function handle and then pass it to
% |fminbnd|.
sqr2 = @(x) (x-2).^2-5
disp('The minimum occurs at')
fminbnd(sqr2,-3,3)
disp('and its value is')
sqr2(ans)

%% Find zeros:

%% 
% To find the zeros of the function $f(x)=(x-2)^2-5$ on the interval
% $[-3,3]$ we can use the function handle just defined together with
% |fzero|.
fzero(sqr2,-3,3)

%% 
% Here is a plot of this function
ezplot(sqr2,[-3,3]); grid on

%%
% or - using the standard plot command:
x = linspace(-3,3,100);
figure; plot(x,sqr2(x)); grid on