%% Code sections start with two percent signs. Section titles go on the same line.
% Comments start with a percent sign.

%{
Multi line comments look
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% Two percent signs denote the start of a new code section
% Individual code sections can be run by moving the cursor to the section followed by
% either clicking the "Run Section" button
% or     using Ctrl+Shift+Enter (Windows) or Cmd+Shift+Return (OS X)

%% This is the start of a code section
%  One way of using sections is to separate expensive but unchanging start-up code like loading data
load myFile.mat y

%% This is another code section
%  This section can be edited and run repeatedly on its own, and is helpful for exploratory programming and demos
A = A * 2;
plot(A);

% commands can span multiple lines, using '...':
 a = 1 + 2 + ...
 + 4

% commands can be passed to the operating system
!ping google.com

who % Displays all variables in memory
whos % Displays all variables in memory, with their types
clear % Erases all your variables from memory
clear('A') % Erases a particular variable
openvar('A') % Open variable in variable editor

clc % Erases the writing on your Command Window
diary % Toggle writing Command Window text to file
ctrl-c % Abort current computation

close all % Closes all figures

edit('myfunction.m') % Open function/script in editor
type('myfunction.m') % Print the source of function/script to Command Window

profile on  % turns on the code profiler
profile off     % turns off the code profiler
profile viewer  % Open profiler

help command    % Displays documentation for command in Command Window
doc command     % Displays documentation for command in Help Window
lookfor command % Searches for command in the first commented line of all functions
lookfor command -all % searches for command in all functions


% Output formatting
format short    % 4 decimals in a floating number
format long     % 15 decimals
format bank     % only two digits after decimal point - for financial calculations
fprintf('text') % print "text" to the screen
disp('text')    % print "text" to the screen

% pressing the up key shows you a history of previous commands

% Variables & Expressions
myVariable = 4  % Notice Workspace pane shows newly created variable
myVariable = 4; % Semi colon suppresses output to the Command Window
4 + 6       % ans = 10
8 * myVariable  % ans = 32
2 ^ 3       % ans = 8
a = 2; b = 3;
c = exp(a)*sin(pi/2) % c = 7.3891

% Logicals
1 > 5 % ans = 0
10 >= 10 % ans = 1
3 ~= 4 % Not equal to -> ans = 1
3 == 3 % equal to -> ans = 1
3 > 1 && 4 > 1 % AND -> ans = 1
3 > 1 || 4 > 1 % OR -> ans = 1
~1 % NOT -> ans = 0

% Logicals can be applied to matrices:
A > 5
% for each element, if condition is true, that element is 1 in returned matrix
A( A > 5 )
% returns a vector containing the elements in A for which condition is true

% Strings
a = 'MyString'
length(a) % ans = 8
a(2) % ans = y
[a,a] % ans = MyStringMyString


% Cells
a = {'one', 'two', 'three'}
a(1) % ans = 'one' - returns a cell
char(a(1)) % ans = one - returns a string

% Structures
A.b = {'one','two'};
A.c = [1 2];
A.d.e = false;

% Variables can be saved to .mat files
save('myFileName.mat') % Save the variables in your Workspace
load('myFileName.mat') % Load saved variables into Workspace

% Vectors
x = [4 32 53 7 1]
x(2) % ans = 32
x(2:3) % ans = 32 53
x(2:end) % ans = 32 53 7 1

x = [4; 32; 53; 7; 1] % Column vector

x = [1:10] % x = 1 2 3 4 5 6 7 8 9 10
x = [1:2:10] % Increment by 2, i.e. x = 1 3 5 7 9

% Matrices
A = [1 2 3; 4 5 6; 7 8 9]
% Rows are separated by a semicolon; elements are separated with space or comma
% A =

%     1     2     3
%     4     5     6
%     7     8     9

A(2,3) % ans = 6, A(row, column)
A(6) % ans = 8
% (implicitly concatenates columns into vector, then indexes into that)


A(2,3) = 42 % Update row 2 col 3 with 42
% A =

%     1     2     3
%     4     5     42
%     7     8     9

A(2:3,2:3) % Creates a new matrix from the old one
%ans =

%     5     42
%     8     9

A(:,1) % All rows in column 1
%ans =

%     1
%     4
%     7

A(1,:) % All columns in row 1
%ans =

%     1     2     3

[A ; A] % Concatenation of matrices (vertically)
%ans =

%     1     2     3
%     4     5    42
%     7     8     9
%     1     2     3
%     4     5    42
%     7     8     9

% this is the same as
vertcat(A,A);


[A , A] % Concatenation of matrices (horizontally)

%ans =

%     1     2     3     1     2     3
%     4     5    42     4     5    42
%     7     8     9     7     8     9

% this is the same as
horzcat(A,A);


A(:, [3 1 2]) % Rearrange the columns of original matrix
%ans =

%     3     1     2
%    42     4     5
%     9     7     8

A(1, :) =[] % Delete the first row of the matrix
A(:, 1) =[] % Delete the first column of the matrix

squeeze(A); % Removes singular dimensions ie. 2x1x3 -> 2x3

transpose(A) % Transpose the matrix, which is the same as:
A one

A' % Concise version of complex transpose
A.' % Concise version of transpose (without taking complex conjugate)

size(A) % ans = 3 3

% Element by Element Arithmetic vs. Matrix Arithmetic
% On their own, the arithmetic operators act on whole matrices. When preceded
% by a period, they act on each element instead. For example:
A * B % Matrix multiplication
A .* B % Multiple each element in A by its corresponding element in B

% There are several pairs of functions, where one acts on each element, and
% the other (whose name ends in m) acts on the whole matrix.
exp(A) % exponentiate each element
expm(A) % calculate the matrix exponential
sqrt(A) % take the square root of each element
sqrtm(A) %  find the matrix whose square is A

% Solving matrix equations (if no solution, returns a least squares solution)
% The \ and / operators are equivalent to the functions mldivide and mrdivide
x=A\b % Solves Ax=b. Faster and more numerically accurate than using inv(A)*b.
x=b/A % Solves xA=b

inv(A) % calculate the inverse matrix
pinv(A) % calculate the pseudo-inverse

% Common matrix functions
zeros(m,n) % m x n matrix of 0's
ones(m,n) % m x n matrix of 1's
diag(A) % Extracts the diagonal elements of a matrix A
diag(x) % Construct a matrix with diagonal elements listed in x, and zeroes elsewhere
eye(m,n) % Identity matrix
linspace(x1, x2, n) % Return n equally spaced points, with min x1 and max x2
inv(A) % Inverse of matrix A
det(A) % Determinant of A
eig(A) % Eigenvalues and eigenvectors of A
trace(A) % Trace of matrix - equivalent to sum(diag(A))
isempty(A) % Tests if array is empty
all(A) % Tests if all elements are nonzero or true
any(A) % Tests if any elements are nonzero or true
isequal(A, B) % Tests equality of two arrays
numel(A) % Number of elements in matrix
triu(x) % Returns the upper triangular part of x
tril(x) % Returns the lower triangular part of x
cross(A,B) %  Returns the cross product of the vectors A and B
dot(A,B) % Returns scalar product of two vectors (must have the same length)
transpose(A) % Returns the transpose of A
fliplr(A) % Flip matrix left to right
flipud(A) % Flip matrix up to down

% Matrix Factorisations
[L, U, P] = lu(A) % LU decomposition: PA = LU,L is lower triangular, U is upper triangular, P is permutation matrix
[P, D] = eig(A) % eigen-decomposition: AP = PD, P's columns are eigenvectors and D's diagonals are eigenvalues
[U,S,V] = svd(X) % SVD: XV = US, U and V are unitary matrices, S has non-negative diagonal elements in decreasing order
[Q, R] = qr(A) % if A is mxn, Q is mxm and R is mxn upper triangular

% Common vector functions
max     % largest component
min     % smallest component
length  % length of a vector
sort    % sort in ascending order
sum     % sum of elements
prod    % product of elements
mode    % modal value
median  % median value
mean    % mean value
std     % standard deviation
perms(x) % list all permutations of elements of x
find(x) % Finds all non-zero elements of x and returns their indexes, can use comparison operators, 
        % i.e. find( x == 3 ) returns indexes of elements that are equal to 3
        % i.e. find( x >= 3 ) returns indexes of elements greater than or equal to 3

% Plotting
x = 0:.10:2*pi; % Creates a vector that starts at 0 and ends at 2*pi with increments of .1
y = sin(x);
plot(x,y)
xlabel('x axis')
ylabel('y axis')
title('Plot of y = sin(x)')
axis([0 2*pi -1 1]) % x range from 0 to 2*pi, y range from -1 to 1

plot(x,y1,'-',x,y2,'--',x,y3,':') % For multiple functions on one plot
legend('Line 1 label', 'Line 2 label') % Label curves with a legend

% Alternative method to plot multiple functions in one plot.
% while 'hold' is on, commands add to existing graph rather than replacing it
plot(x, y)
hold on
plot(x, z)
hold off

loglog(x, y) % A log-log plot
semilogx(x, y) % A plot with logarithmic x-axis
semilogy(x, y) % A plot with logarithmic y-axis

fplot (@(x) x^2, [2,5]) % plot the function x^2 from x=2 to x=5

% Creates a meshgrid (2D grid) to calculate a function for every point in the grid
[X, Y] = meshgrid(x_min:step:x_max, y_min:step:y_max)

grid on % Show grid; turn off with 'grid off'
axis square % Makes the current axes region square
axis equal % Set aspect ratio so data units are the same in every direction

scatter(x, y); % Scatter-plot
hist(x); % Histogram
stem(x); % Plot values as stems, useful for displaying discrete data
bar(x); % Plot bar graph

z = sin(x);
plot3(x,y,z); % 3D line plot

pcolor(A) % Heat-map of matrix: plot as grid of rectangles, coloured by value
contour(A) % Contour plot of matrix
contourf(A) % Filled contour plot of matrix
mesh(A) % Plot as a mesh surface

h = figure % Create new figure object, with handle h
figure(h) % Makes the figure corresponding to handle h the current figure
close(h) % close figure with handle h
close all % close all open figure windows
close % close current figure window

shg % bring an existing graphics window forward, or create new one if needed
clf clear % clear current figure window, and reset most figure properties

% Properties can be set and changed through a figure handle.
% You can save a handle to a figure when you create it.
% The function get returns a handle to the current figure
h = plot(x, y); % you can save a handle to a figure when you create it
set(h, 'Color', 'r')
% 'y' yellow; 'm' magenta, 'c' cyan, 'r' red, 'g' green, 'b' blue, 'w' white, 'k' black
set(h, 'LineStyle', '--')
 % '--' is solid line, '---' dashed, ':' dotted, '-.' dash-dot, 'none' is no line
get(h, 'LineStyle')


% The function gca returns a handle to the axes for the current figure
set(gca, 'XDir', 'reverse'); % reverse the direction of the x-axis

% To create a figure that contains several axes in tiled positions, use subplot
subplot(2,3,1); % select the first position in a 2-by-3 grid of subplots
plot(x1); title('First Plot') % plot something in this position
subplot(2,3,2); % select second position in the grid
plot(x2); title('Second Plot') % plot something there

% Given
x1 = [-3:0.5:3];
x2 = x1;
y = randi(500, length(x1), length(x1));

% Show a 3-D plot
figure
subplot(2,1,1);
surf(x1,x2,y);
xlabel(’x_1’);
ylabel(’x_2’);

% Show contours
subplot(2,1,2);
contour(x1,x2,y);
xlabel(’x_{1}’);
ylabel(’x_{2}’);
axis equal

% Show a colour map
figure
imagesc(x1,x2,y)
xlabel(’x_{1}’);
ylabel(’x_{2}’);

% Calling Functions
% Standard function syntax:
load('myFile.mat', 'y')
% Command syntax:
load myFile.mat y   % no parentheses, and spaces instead of commas

% Calling a function from a script