Algorithmization and Programming

=======================================
=========    FUNCTIONS    =============
=======================================

each function accessible from the Command Window 
has to be stored at a file with the same name


=========================================

### Example 1 

---- write a matlab function for f(x) = (x^3-7x+6)/7 ---


==================
==  file f1.m  ===
==================

function y = f1(x)
%
% function y = f1(x)
% input:  x ... an array
% output: y = (x^3-7x+6)/7 
%
  y = (x.^3-7.*x+6)/7;
end


----- using the function f1 -------

>> help f1  

>> f1(5)

>> u = 0:0.1:4;
>> v = f1(u);
>> plot(u,v)


=========================================

### Example 2 

---- function computing volume and surface of a cuboid ---

=======================
==  file VolSurf.m  ===
=======================

function [V, S] = VolSurf(a, b, c)
%
% [V, S] = VolSurf(a, b, c)
% computes volume and surface of a cuboid
% input: a, b, c ... size of cuboid edges
% output: V ... the volume
%         S ... the surface
%
  V = a*b*c;
  S = 2*(a*b + b*c + a*c);
end


--------- calling the function -------------

>> [p,q] = VolSurf(1,3,4)

p =
    12
q =
    38

>> VolSurf(1,3,4)   % returns the first value only

ans =
    12

  --- parameters can be expressions

>> a = 5;
>> [u,v] = VolSurf(1/8, 3*a, a*sin(4));

  --- Matlab command "help" can be used

>> help VolSurf 




==========================================================
SCOPE OF THE VARIABLES

 function parameters are passed by VALUE, not by REFERENCE
==========================================================

### Example 3 


=================
==  file t.m  ===
=================

% illustration of scope of the variables
%
clear all          % all - clears also all functions

a = 1; 
b = 2; 

fprintf('before f2: a=%d, b=%d\n', a, b);

c = f2(b);   % b ... actual parameter

fprintf(' after f2: a=%d, b=%d, c=%d\n', a, b, c);
disp(p)

==================
==  file f2.m  ===
==================

function y = f2(a)    % a ... formal parameter
%
% input:  a ... a number
% output: y = a^2 
%
%
  y = a^2;
  b = 7;
  a = a+11;
  p = 26;
  fprintf('inside f2: a=%d, b=%d, p=%d, y=%d\n', a, b, p, y);
end



==========================================================

### Example 4


************************************************
Explicit Euler's method
for solving  y' = f(x, y)
************************************************


  f(x,y) = y / x^2,  x_0 = 1,  y_0 = 2


Solving in the Dialog window:
=============================

>> h = 0.2;          % choose a step-size
>> x = 1; y = 2;     % initial condition - starting point

Repeat the following 2 steps:

>> y = y + y / x^2 * h   % new value of y (on the tangent line)
>> x = x + h;            % new value of x



The same, using Anonymous function:
===================================
  definition of function f(x,y) 
- in the variable f will be so called Function Handle:

>> f = @(x,y) y / x^2;  

>> h = 0.2;          % choose a step-size
>> x = 1; y = 2;     % initial condition - starting point

Repeat the following 2 steps:

>> y = y + f(x,y) * h    % new value of y (on the tangent line)
>> x = x + h;            % new value of x



The same, using  a script and a function:
=========================================

===================
==  file Eul.m  ===
===================

clear all

h = 0.2;          % choose a step-size
x = 1; y = 2;     % initial condition - starting point

for k=1:5
  y = y + f(x,y) * h;    % new value of y (on the tangent line)
  x = x + h;             % new value of x
  disp(y);               % print the new value on the screen
end

=================
==  file f.m  ===
=================

function z = f(x,y)
% f(x,y) = y / x^2; 
  z = y / x^2; 
end



===================================
you can store auxiliary functions in the file with the main one
(however, only the main function is available from outside):

===================
==  file Eul.m  ===
===================

function y = Eul(x0, y0, h, n)
%
% input: 
%   x0, y0  - initial condition (starting point)
%   h       - a step-size
%   n       - number of steps 

  x = x0;
  y = zeros(1,n);
  y(1) = y0;
  for k=2:n
    y(k) = y(k-1) + f(x,y(k-1)) * h; 
    x = x + h;       
  end
end

function z = f(x,y)
% f(x,y) = y / x^2; 
  z = y / x^2; 
end

---------------

  --- calling the function Eul:

>> p = Eul(1, 2, 0.2, 5)

>> f(2,6)    % error - the function is not accessible outside Eul


====================================
====================================

Learn from Matlab/Octave programmers

warning: DO NOT CHANGE Matlab FUNCTIONS 
- make a copy of your own, under different name

>> edit trace
>> edit range
>> edit date

====================================