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=====  Matlab - Dialog window  =====
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Try these commands in Matlab and see, what they are doing:

a = 5
b = 6;  % semicolon inhibits Matlab output
b       % shows the value of the variable b
c = (a+b)*3/(20-a)

sqrt(5-3*i)  % square root can be computed from a complex number, too

a=sqrt(2) + sin(c+1);

help sin

========= arrays - vectors =============

y = [ 2 0 1 -2 -3 4 1 ];
y(3)          % the third element of the vector y
y(3) = 7;     % changing just the third element of y
y'            % transposition of the vector
x = -1:4    
x = 2 : 0.5 : 5;
format rat
x
format
x
help format
c = pi : 0.5 : 2*pi;

---- the difference y-x of vectors (of the same size) ---------

size(x)
size(y)
d = y-x

---- the maximal element of the vector d (absolute value of it) ---------

max(abs(d))


------ function of the vector element by element -----

y = x.*exp(x)+x.^2;


==========  graphs ==============

plot(x, y)
plot(x,y, 'r')
plot(x,y, 'go')
plot(x,y, 'r*')

help plot

  % parametric curve:
t = 0 : 0.2 : 2*pi;
x = sin(t);
y = cos(t);	
plot(x, y)

axis equal
axis normal


=========== 2D arrays - matrices ================


A = [ 1 2 3; 4 5 6];

size(A) % number of rows, number of columns

A(2,3)  % the element at 2. row and 3. column
A(2,:)  % the second row
A(:,3)  % the third column 

--- adding the matrices:

B = [ 0 -2 1; 1 0 1];
C = A + B;
Z = 0.3 * A

--- multiplying the matrices:

X = A*A'
Y = [1 2; -3 2] * A

--- solution of the system of the equations (with regular matrix):

A = [ 3 2 1; 0 2 0; 3 6 7];
b = [ 11; 2; 33];  % or b = [ 11 2 33]';

x = A \ b

--- function of a matrix as an entity --------

B = A^2 % notice the difference from B = A.^2
det(A)  % determinant
inv(A)  % matrix inversion

  % eigenvalues
eig(A)
[U, D]=eig(A) % eigenvalues and eigenvectors

  % check the matrix equation A*U - D*U = 0 :
A*U - D*U

  % for k = 1, 2, 3 : 
k=1;
U(:,k) % eigenvector
D(k,k) % the corresponding eigenvalue

  % so it holds  A*U(:,k)-D(k,k)*U(:,k) = 0  for k = 1,2,3 :
k=1;
A*U(:,k)-D(k,k)*U(:,k)

------ function of a matrix element by element -----

B = A.*A;
C = A.^2;
S = sin(A);


=========== APPENDIX ===============

===========  ezplot  ==============

ezplot('x*sin(x)')  % in the interval < -2*pi, 2*pi >

   or:

f = 'x*sin(x)';
ezplot(f)

grid on

ezplot(f, [-2, 10])   % in the interval < -2, 10 >

% or

ezplot(f, -2, 10) 


==================== ezplot 2D ============

ezplot('x^2 + y^2 - 4')

a = 9;
ezplot(['x^2 + y^2 - ', int2str(a) ])

f = 'x^2 - y^2-5';
ezplot(f, [-10, 10])        % in a square
ezplot(f, [-10, 10, -5, 5]) % in a rectangle


% graphical solution of equations
ezplot('x^2 + 1')        % the first function
hold on                  % plotting into the same graph
ezplot('x^2 + y^2 - 4')  % the second function
axis([-2 2 0 3])         % zoom in at the intersection

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