% Bare essentials of MATLAB for Math 23 Diff Eq - Alex Barnett 9/25/07

% To start: in windows or Mac, click on the Matlab icon.
% In linux I recommend starting it from command line without
% a command window, since less buggy, ie:  `matlab -nodesktop'

% Anything after a % is a `comment' (ignored by matlab).

% You can type or paste each line to the command line,
% or save it to your workspace as intro.m and
% run the whole thing as a program by typing `intro' (not so enlightening)

a = 5                                 % a symbol can be a number (scalar)
a = 5;                                % end command with ";" if want silent
sa = sin(a)                           % sa now contains sin of a, ie sin(5)
b = [1 2 3]                           % a row vector (row vec)
c = [3; -2; 4.6]                      % ; separators = column vector (col vec)
c(3)                                  % look up the 3rd element of c
b*c                                   % dot product. Careful with order!
% NOTE: generally * computes matrix product.
% So here b*c computes dot product (a scalar),
% but here c*b would compute `outer product' (a 3-by-3 matrix) - watch out.
dot(b,c)                              % = dot(c,b). Better way for dot prod
d = c'                                % d is now transpose of c (row vec)
b.*d                                  % element-by-element multiply, a 1-by-3
% NOTE this means b and d must be the same size (shape), here 1-by-3.
sin(b)                                % sin of all elements at once (fast!)
% NOTE we didn't have to write a loop to compute sin of many numbers at once.

% 2D GRAPHS
plot(b, c, '+');                      % plot graph of c vs b using + symbols
% Note: this means plot the 3 points (b(1),c(1)), (b(2),c(2)) and (b(3),c(3))
% b is interpreted as `x values' (horizontal), c as `y values' (vertical).
% Number of elements in b and c must match up (unless want multiple lines) 
axis([0 4 -3 5]);                     % manually choose axes range
plot(b, c, 'g-');                     % plot same using instead green lines

help                                  % list of help menus
help plot                             % give help on eg plot command
doc                                   % if you prefer graphical help window
whos                                  % show all currently-defined variables
save mydata                           % save all variables as file mydata.mat
quit                                  % close matlab
load mydata                           % load vars (can overwrite variables)
clear                                 % clear all variables (careful!)
% Use up and down arrows to scroll through recent commands, enter to execute
% I will postpone matrices & linear algebra for later...

% DIFF EQs
% Solving general 1st-order ODE y' = f(t,y) numerically ----------------------
f = @(t,y) 1-y;                       % set up func of 2 vars f(t,y) = 1-y
f(2,3)                                % check it evaluates correctly

yo = 0;                               % initial condition at t=0 is yo=0
[ts, ys] = ode45(f, [0, 3], yo);      % solve ODE (using 'Runge-Kutta')
                                      % over the t-range [0,3]
% ts is now a col vec of t values, ys is col vec of corresponding y values
% ----------------------------------------------------------------------------

plot(ts, ys, '+-');                   % plot solution with + symbols and lines

yex = 1 - exp(-ts);                   % compute analytic (exact) y values
                                      % at each already-existing t value
% Note that we subtracted exp(ts) which is a col vec, from 1 which is a scalar,
% but it figured out to do the right thing (make 1 an appropriate VECTOR of 1s)

hold on;                              % plot the next graph over this one
plot(ts, yex, 'ro');                  % compare exact solution as red o symbols
hold off;                             % stop overlaying future graphs

figure(2);                            % start a new figure
plot(ts, ys-yex, '+-');               % Subtract the two - best way to compare!
xlabel('t'); ylabel('y error');       % label axes: important to communicate!
title('dy/dt = 1-y: error in numerical solution ys-yex');

% See course website links for more complete introductions. Enjoy!