%% Theory0:
% MSc MATLAB Crash Course: Basic operations with the command window.
%
% Originally written by Nick Hale, Oct. 2013.
% Extended by Asgeir Birkisson, Oct. 2014, 2015.
% Modified by Behnam Hashemi and Hadrien Montanelli, Sep. 2016.

%% First steps
5 + 10
3 - 2
3*2
3/2
3^2
exp(3)
sqrt(9)
factorial(5)
sin(3)
sin(pi)
sind(90)

%% Get help
help sin
doc sin

%% Initialize vectors 
a = [1 3 5]      % Row vector
a = [1, 3, 5]    % The same
size(a)          % Size of a
length(a)        % max of the above
a = [1 ; 3 ; 5]  % Column vector
size(a)          % Size of a
a = [1+1i 3 5]   % Column vector with complex entries
a = [1+1i 3 5].' % .' gives the transpose
a = [1+1i 3 5]'  % ' gives the conjugate transpose

%% Simple commands
clc             % clear command window
a
max(a)          % Maximum value
min(a)          % Minimum value
sum(a)          % Sum of entries
mean(a)         % Average value

%% Addition and multiplication
b = [2 6 10]';  % Another column vector
c = a + b
d = 4*a
e = 3*a + 5*b;

%% Modifying a vector
a
a(2) = 11       % Modify second entry
a = [a; 4]      % Add an entry at the end
a = [7; a]      % Add an entry at the start
a(3) = []       % Remove the third entry

%% Vector syntax
1:100
1:5:101
10:-2:0
linspace(0, 1, 51)

%% Initialise a matrix
A = [1 8; 5 2]  % 2x2 matrix
A'              % (Conjugate) Transpose of the matrix
size(A)
length(A)

%% Simple commands -- acting columnwise
max(A)
min(A)
sum(A)
mean(A)

%% Simple commands -- acting rowwise
% Notice extra arguments to the function
max(A, [], 2)
min(A, [], 2)
sum(A, 2)
mean(A, 2)

%% Addition and multiplication
B = [4 5; 9 3];   % Another 2x2 matrix
C = 3*A + B

%% Matrix syntax
A(1, 2)
A(:, 2)
A(1, :)
D = diag(A)        % Diagonal elements
det(A)

%% Useful commands
A = rand(3, 3)  % matrix with random elements
A = rand(3)     % the same
O = ones(3)     % matrix with ones
Z = zeros(3)    % matrix with zeros

%% Factorizations
A = rand(5)
[V, D] = eig(A)    % Eigenvectors and eigenvalues
[L, U, P] = lu(A)     % LU decomposition
[Q, R] = qr(A)     % QR factorisation
Q*Q'

%% Solve a linear system -- Ax = b

% Solve 
%     x1  + 2*x2 = 1
%    5*x1 + 8*x2 = 2
A = [1 2; 5 8];
b = [1 2]';
x = A\b             % Use backslash for solving
x = inv(A)*b;       % This is not good -- numerical instabilities

% Solve with random coefficients and right-hand side:
A = rand(2, 2);
b = rand(2, 1);
x = A\b

%% Formats
pi
format long
pi
% Format does NOT affect how Matlab computations are done, just the display
format short
a = sqrt(2)
format long
b = sqrt(2)
a - b

% Get rid of extra linespaces
format compact
a - b

% Reintroduce the extra linespaces
format loose
a - b

%% Basic plotting

x = linspace(-1, 1, 100);
plot(x, sin(4*pi*x))
%%
hold on
plot(x, exp(cos(10*x)), 'r')
hold off

%% Tidying up
close
who
whos
clc
a = 4; b = 5;
clear a
clear all