% --------------------------------------------------------------------------------
%                           M A T L A B     T U T O R I A L
% --------------------------------------------------------------------------------
% 
% Originally, this tutorial was started as an HTML document, the idea was
% to create a comprehensive document which will introduce Matlab basics
% to the newcomers. Also, it supposed to be as compact as possible, so the
% reader would not be overwhelmed with unnecessary information. I really
% didn't want to make just another one tutorial, which everyone can find
% through the WEB. So, after I cut everything from my original document I
% ended up with just some Matlab statements and some comments. Finally, I
% decided to put this document into a different format, which, in my
% opinion is more suitable - .m file. Now, it is a Matlab script you can
% run and learn.
%
%                                                   Dmitry Yershov
%
% First what we will learn is comments. Every line, that starts with %
% character is a comment, and Matlab simply ignores it. So, you can see,
% that all my blah-blah by now is just a comments, and Matlab does not
% care about it, but I still hope that someone else do.
%
% Second and not the least important thing, is getting help from Matlab
% command line. Type help, and you will get a list of function
% categories, with a short description for all of them.

help

% Then, suppose you are interested in general purpose commands in Matlab,
% type help with a category name followed after white space, and you'll
% get a list of functions in this category, again, with a short
% description of what each function does.

help general

% Now, you finally are interested in "clear" command, and you want to
% know more of what it does, and how it works.

help clear

% The line above will give you detailed description of "clear" command

% --------------------------------------------------------------------------------
%                           Matrices and arrays in Matlab
% --------------------------------------------------------------------------------
clear
% Create a row vector of length 5
v = [1 0 -1 3 -2]
% or 
v = [1, 0, -1, 3, -2]
% Note: in my experience, there is no difference whether you use comas to
% separate row values or whether you just use white spaces.

% Create a column array of length 3
w = [-1; 1; 2]
% Note: semicolons are important here. You have to use semicolon to
% indicate a new row.

% Create a 2x3 matrix 
A = [1 3 2; 3 2 1]

% Creating sequences with ":"
t = 0:10
% Note: t is a row vector that holds integers from 0 to 10 in the
% increasing order.

% Defining a step
s = 0:0.1:10;
% Note: s = [0 0.1 0.2 0.3 ... 10] or in other words operator works like
% this first_element:step:last_element 
%
% Note: semicolon in the end of the line prevents output

% You also can use different functions to create vectors or matrices of
% different sizes. Note, that row vector is a 1xN matrix and column
% vector is Mx1 matrix.

% let
N = 5
M = 4
% define a zero NxM matrix
B = zeros(N, M) % try Z = zeros(N) and see what the difference
% random NxM matrix
C = rand([N M]) % another equal way to define a matrix size
% or NxM matrix with ones on the diagonal
I = eye(N, M)

% Vectors and matrices are stored as one- and two-dimensional array
% correspondingly. To access array elements we use () with corresponding
% index and indices
s(51)
I(3,3)

% Matlab can operate on more then one- or two-dimensional arrays. For
% example, we now create a 3D array.
E = zeros(3, 3, 4)
% and put 5 into (2,3,1) element
E(2,3,1) = 5

% Getting dimensions and sizes
ndims(E)  % returns 3
size(B)   % returns [N M], e.g. [5 4]
length(s) % for vectors only

% Extract diagonal vector
diag(C)
% Extract lower triangular part
tril(C)
% Extract upper triangular part
triu(C)

% diag is also can be used to generate diagonal matrix from the vector
diag(w)
% or even more interesting
diag(diag(C))

% OK, now it's time to do some arithmetics
% matrix addition
B = I + C
% matrix multiplication
%D = B * C
% Note: the error will occur in the line above, because "*" is considered
% as a matrix multiplication by rules of linear algebra, so the number of
% columns of the first matrix must agree with the number of rows of the
% second one. To get element-wise multiplication we need to use ".*". For
% example:
D = B .* C
% Note that ".+" and ".-" are the same as "+" and "-" for matrices.

% Computing the action of a matrix on a vector is just a matrix-vector
% product
u = A * w % for right action
x = v * D % for left action

% Another important operation is getting transpose, which can be done by
% using unary operation "dot-prime" or ".'". For example to compute the
% squared norm of row vector v we type
v * v.'
% or we can transpose matrix A
A.'
% the "prime" character is used to get complex conjugate. Example
M = [1 + i, -i; -2 - i, i]
M.'
M'
% Note: we used built-in "i" variable to represent complex unit
% (sqrt(-1)), although we could use "j" as well.
% Note: be careful using this variable. Make sure, that it is not
% overwritten by you before you used it. For example you may use "i" and
% "j" in for-loop for array indexing. Then after the for-loop these two
% variables will hold integers. Consider the following
i = 2
j = 3
M = [1 + i, -i; -2 - i, i]
% Compare resulting matrix M with the previous result.

% --------------------------------------------------------------------------------
%                           Matlab built-in variables
% --------------------------------------------------------------------------------
clear
% Floating point bounds
% machine precision (\epsilon_{mach})
eps
% Largest positive floating point number
realmax
% Smallest positive floating point number
realmin
% Infinity
inf
% Not-a-Number
nan
% floating point functions
isnan(nan)
isinf(inf)
isfinite(1.0)

% Constants
% Imaginary unit
i
j
% pi - the ratio between circle length and its diameter
pi

% If you are not satisfied with output precision for floating point
% numbers try the following
format long e
pi
% see more with help format

% --------------------------------------------------------------------------------
%                           Numerical Linear Algebra
% --------------------------------------------------------------------------------
clear
format
% let
A = [1 3 -6; 2 -4 3; -4 6 1]
b = [3; 2; 1]

% Euclidean and matrix norms
norm(b)
norm(A)

% matrix rank
rank(A)
% rank can be applied to the vector as well
rank(b)

% determinant (only square matrices)
det(A)

% Trace
trace(A)

% Condition number
cond(A)

% Inverse matrix
inv(A)

% To solve the system Ax = b we can simply type
x = inv(A) * b
% or
x = A \ b

% more matrix functions (see help for details)
% lu     - LU factorization
% qr     - QR decomposition
% eig    - Eigenvalues and eigenvectors
% svd    - Singular value decomposition
% poly   - Characteristic polynomial
% schur  - Schur decomposition
% .........

% --------------------------------------------------------------------------------
%                           Auxiliary functions
% --------------------------------------------------------------------------------
clear

% let
t = 0:0.1:5;

% The functions in this class can operate on arrays element-wise. For
% example:
y = sin(t);
% will take sin function in all points of vector t and assign the result
% into array y. The result is always the same number of dimensions and
% size as the argument


% Matlab library has a variety of different functions, here is the list of
% categories and some function examples:
%
%  TRIGONOMETRIC
%    sin, cos, tan, asin, acos, atan, sinh, cosh, ...
%  EXPONENTIAL
%    exp, log, log10, sqrt
%  COMPLEX
%    abs         - Absolute value
%    angle       - Phase angle
%    imag        - Complex imaginary part
%    real        - Complex real part
%    isreal      - True for real array
%  ROUNDING
%    fix         - Round towards zero
%    floor       - Round towards minus infinity
%    ceil        - Round towards plus infinity
%    round       - Round towards nearest integer
%    mod         - Modulus (signed remainder after division)
%    rem         - Remainder after division
%    sign        - Signum
%  DATA FUNCTIONS
%    max         - Largest component
%    min         - Smallest component
%    sort        - Sort in ascending order
%    sortrows    - Sort rows in ascending order
%    sum         - Sum of elements

% --------------------------------------------------------------------------------
%                           Plotting data
% --------------------------------------------------------------------------------

% simple plot
plot(t, y)

% add more style
plot(t, y, '-.k')

% some more plotting functions
%  loglog    - Log-log scale plot
%  semilogx  - Semi-log scale plot
%  semilogy  - Semi-log scale plot
%  polar     - Polar coordinate plot

% type hold on to hold previous graph on a plot, and all successive
% graphs will be plotted on top
hold on
plot(t, sin(t + pi / 6.), '-r')
plot(t, sin(t + pi / 4.), '--b')
plot(t, sin(t + pi / 3.), ':m')

% you can scale picture differently using axis function
axis equal

% we can add title to the graph
title('sin function')

% labels to the axes
xlabel('t')
ylabel('sin(t + \phi)')
% Note: you can use LaTeX commands to format the text in Matlab

% and the last but not the least - legend
legend('\phi = 0', '\phi = \pi / 6', '\phi = \pi / 4', '\phi = \pi / 3')

% you also can add text to the graph
text(1, -1, 'pretty picture')
% or use gtext to add text at cursor

% and finally print your graph as a bitmap or vector drawing
print -depsc -r300 'sin.eps'  % you can use .eps file to embed it into
                              % LaTeX document of your Homework
print -dpng -r70 'sin.png'    % you can use .png to embed your nice
                              % graphs into your cool web page

% hold off command will stop holding the graphs in a frame
hold off

% subplot makes several sub plots in one frame
% Example:
x = 0:0.1:2;
subplot(1,2,1)
plot(x, x.^2, '-ro')
subplot(1,2,2)
plot(x, sqrt(x), '-ms')


% --------------------------------------------------------------------------------
%                           Matlab programming
% --------------------------------------------------------------------------------
clear
% Some Matlab programming structures, to program with .m files

% For loops

for k = 1:5
  2^k
end

% note that 1:5 = [1 2 3 4 5], so we can use any vector

x = [1.5 2 4.5 8]
for k = x
  2^k
end

% while loop

k = 0
while (k < 10)
  k = k + 1
  if k == 3
    continue
  end
  2^k
  if k == 5
    break
  end
end
% Note: we have used conditional operator "<", also other relational
% operators are available, such as ">", "==", ">=", "<=", and "~="

% conditional statement if

x = 1
y = 2
z = 1.333

if (x < y)
  k = 1;
elseif (y > z)
  k = 2;
else
  k = 3;
end

% conditional statement switch

switch k
 case 1
  disp('x < y')
 case 2
  disp('x >= y and y > z ')
 otherwise
  disp('x >= y and y <= z ')
end

% --------------------------------------------------------------------------------
%                           Polynomials and Interpolation
% --------------------------------------------------------------------------------

help polyfun

%  Spline interpolation
%    spline      - Cubic spline interpolation
%    ppval       - Evaluate piecewise polynomial
%  Polynomials
%    roots       - Find polynomial roots
%    poly        - Convert roots to polynomial
%    polyval     - Evaluate polynomial
%    polyvalm    - Evaluate polynomial with matrix argument
%    polyfit     - Fit polynomial to data
%    polyder     - Differentiate polynomial
%    polyint     - Integrate polynomial analytically
%    conv        - Multiply polynomials
%    deconv      - Divide polynomials

% --------------------------------------------------------------------------------
%                           Functions of a functions
% --------------------------------------------------------------------------------

help funfun

%  Function functions and ODE solvers
%  Optimization and root finding
%    fminbnd    - Scalar bounded nonlinear function minimization
%    fminsearch - Multidimensional unconstrained nonlinear minimization
%    fzero      - Scalar nonlinear zero finding
%  Numerical integration (quadrature)
%    quad       - Numerically evaluate integral, low order method
%    quadl      - Numerically evaluate integral, higher order method
%  Differential equation solvers
%    ode45     - Solve non-stiff differential equations, medium order method
%    ode23     - Solve non-stiff differential equations, low order method
%  Boundary value problem solver for ODEs
%    bvp4c     - Solve two-point boundary value problems for ODEs by collocation
%  1D Partial differential equation solver
%    pdepe     - Solve initial-boundary value problems for parabolic-elliptic PDEs