%% MATLAB MINI-TUTORIAL 
%

%% Starting MATLAB
% From the menu or write "matlab&" in a shell window

%% Quit MATLAB
% Write "exit" at the prompt

%% Help

help linspace

%% Building matrices
% The basic datatype is a matrix

a = rand(4)            % Random matrix
disp(a')
b = a'*a               % Matrix multiplication
h=1/100;               % Step length
x=linspace(h,1-h,100); 

A=[2 1
    1 2]               % Not the same as a 

%% Variables
% Matlab has built-in variables like pi, eps, and ans
% and special matrices like rand, ones, zeros, eye

pi
eps
ans

eye(3)                 % Identity matrix

%% Array sections: colon notation
% 
a(:,1)           % First column

a(2,:)           % Second row

a(:,5)=ones(4,1)        % Matrix of ones
a(5,:)=zeros(1,5)       % Matrix of zeros


%% See more digits

a

format long

a

format short             % Default

%% For loops

n=5;       % Dimension of a
for i=1:n 
    a(i,i)=a(i,i)+100;
end

a

%% While loops 
% What is computed here?  To see which logical and other operators there
% are, type "help >" for instance.

s=1;
term=1; k=0;
while term > 1e-15
    k=k+1;
    term=term/k;
    s=s+term;
end



%% Scripts
% This file is a script: a file with code with the extension .m


%% Functions
% You can define your own functions: 

disp('My exp') 
disp(exp(2)-myexp(2))

%% Plotting
% Try "help plot" to see what can be done

x=linspace(0,1,100);         
y=exp(x);              % The functions is evaluated with a vector as argument

plot(x,y,'-')
pause
clf
plot(x,y,'x')

%% Plot two functions

plot(x,y,'-',x,sin(x),'--')
title('exp and sin')

%% Mesh plots
% Plot the elements of a matrix

mesh(randn(100))

%% Mesh a function of two variables

x=linspace(0,2*pi,100);
[X,Y]=meshgrid(x,x);  
F=sin(X).*sin(Y);       % Evaluate sin(x)*sin(y) over a square
                        % .* is elementwise multiplication
mesh(F)

%% Matrix functions
% Common matrix functions are standard functions in MATLAB

n=4;
A=rand(n)
b=ones(n,1);
x=A\b                  % Solve  linear system Ax=b

[Q,R]=qr(A)            % QR decomposition A = Q*R

[U,S,V]=svd(A)         % Singular value decomposition A = U*S*V'

[P,D]=eig(A)           % Eigenvalue decomposition A*P=P*D

%% Sparse matrices
% Do not store zeros of most of the elements are zero

speye(3)         % Sparse identity matrix

n=50;
B=2*speye(n);

for i=1:n-1
    B(i,i+1)=-1;
    B(i+1,i)=-1;
end

B(1:4,1:4)

spy(B)

%% Matrix functions work also for sparse matrices

c=ones(n,1); 

x=B\c;

plot(x)

%%