% 1. sparse matrices
a = sparse(4,6,3,10,10)

a =

   (4,6)        3

spy(a)
% a more interesting plot is generated by
% the so-called Pascal matrices:
p32 = pascal(32);
s32 = rem(p32,2); % creates a sparse 0/1 matrix
spy(s32)
% to reduce this matrix to a sparse matrix :
[i,j,s] = find(s32);
sp32 = sparse(i,j,s)

sp32 =

   (1,1)        1
   (2,1)        1
   (3,1)        1
   (4,1)        1
   (5,1)        1
   (6,1)        1
   (7,1)        1
   (8,1)        1
   (9,1)        1
  (10,1)        1
  (11,1)        1
  (12,1)        1
  (13,1)        1
  (14,1)        1
  (15,1)        1
  (16,1)        1
  (17,1)        1
  (18,1)        1
  (19,1)        1
  (20,1)        1
  (21,1)        1
  (22,1)        1
  (23,1)        1
  (24,1)        1
  (25,1)        1
  (26,1)        1
  (27,1)        1
  (28,1)        1
  (29,1)        1
  (30,1)        1
  (31,1)        1
  (32,1)        1
   (1,2)        1
   (3,2)        1
   (5,2)        1
   (7,2)        1
   (9,2)        1
  (11,2)        1
  (13,2)        1
  (15,2)        1
  (17,2)        1
  (19,2)        1
  (21,2)        1
  (23,2)        1
  (25,2)        1
  (27,2)        1
  (29,2)        1
  (31,2)        1
   (1,3)        1
   (2,3)        1
   (5,3)        1
   (6,3)        1
   (9,3)        1
  (10,3)        1
  (13,3)        1
  (14,3)        1
  (17,3)        1
  (18,3)        1
  (21,3)        1
  (22,3)        1
  (25,3)        1
  (26,3)        1
  (29,3)        1
  (30,3)        1
   (1,4)        1
   (5,4)        1
   (9,4)        1
  (13,4)        1
  (17,4)        1
  (21,4)        1
  (25,4)        1
  (29,4)        1
   (1,5)        1
   (2,5)        1
   (3,5)        1
   (4,5)        1
   (9,5)        1
  (10,5)        1
  (11,5)        1
  (12,5)        1
  (17,5)        1
  (18,5)        1
  (19,5)        1
  (20,5)        1
  (25,5)        1
  (26,5)        1
  (27,5)        1
  (28,5)        1
   (1,6)        1
   (3,6)        1
   (9,6)        1
  (11,6)        1
  (17,6)        1
  (19,6)        1
  (25,6)        1
  (27,6)        1
   (1,7)        1
   (2,7)        1
   (9,7)        1
  (10,7)        1
  (17,7)        1
  (18,7)        1
  (25,7)        1
  (26,7)        1
   (1,8)        1
   (9,8)        1
  (17,8)        1
  (25,8)        1
   (1,9)        1
   (2,9)        1
   (3,9)        1
   (4,9)        1
   (5,9)        1
   (6,9)        1
   (7,9)        1
   (8,9)        1
  (17,9)        1
  (18,9)        1
  (19,9)        1
  (20,9)        1
  (21,9)        1
  (22,9)        1
  (23,9)        1
  (24,9)        1
   (1,10)       1
   (3,10)       1
   (5,10)       1
   (7,10)       1
  (17,10)       1
  (19,10)       1
  (21,10)       1
  (23,10)       1
   (1,11)       1
   (2,11)       1
   (5,11)       1
   (6,11)       1
  (17,11)       1
  (18,11)       1
  (21,11)       1
  (22,11)       1
   (1,12)       1
   (5,12)       1
  (17,12)       1
  (21,12)       1
   (1,13)       1
   (2,13)       1
   (3,13)       1
   (4,13)       1
  (17,13)       1
  (18,13)       1
  (19,13)       1
  (20,13)       1
   (1,14)       1
   (3,14)       1
  (17,14)       1
  (19,14)       1
   (1,15)       1
   (2,15)       1
  (17,15)       1
  (18,15)       1
   (1,16)       1
  (17,16)       1
   (1,17)       1
   (2,17)       1
   (3,17)       1
   (4,17)       1
   (5,17)       1
   (6,17)       1
   (7,17)       1
   (8,17)       1
   (9,17)       1
  (10,17)       1
  (11,17)       1
  (12,17)       1
  (13,17)       1
  (14,17)       1
  (15,17)       1
  (16,17)       1
   (1,18)       1
   (3,18)       1
   (5,18)       1
   (7,18)       1
   (9,18)       1
  (11,18)       1
  (13,18)       1
  (15,18)       1
   (1,19)       1
   (2,19)       1
   (5,19)       1
   (6,19)       1
   (9,19)       1
  (10,19)       1
  (13,19)       1
  (14,19)       1
   (1,20)       1
   (5,20)       1
   (9,20)       1
  (13,20)       1
   (1,21)       1
   (2,21)       1
   (3,21)       1
   (4,21)       1
   (9,21)       1
  (10,21)       1
  (11,21)       1
  (12,21)       1
   (1,22)       1
   (3,22)       1
   (9,22)       1
  (11,22)       1
   (1,23)       1
   (2,23)       1
   (9,23)       1
  (10,23)       1
   (1,24)       1
   (9,24)       1
   (1,25)       1
   (2,25)       1
   (3,25)       1
   (4,25)       1
   (5,25)       1
   (6,25)       1
   (7,25)       1
   (8,25)       1
   (1,26)       1
   (3,26)       1
   (5,26)       1
   (7,26)       1
   (1,27)       1
   (2,27)       1
   (5,27)       1
   (6,27)       1
   (1,28)       1
   (5,28)       1
   (1,29)       1
   (2,29)       1
   (3,29)       1
   (4,29)       1
   (1,30)       1
   (3,30)       1
   (1,31)       1
   (2,31)       1
   (1,32)       1

i = [1:10 2:10 1:9]; % row indices with nonzeros
j = [1:10 1:9  2:10]; % corresponding columns
s = [2*ones(1,10) ones(1,9) ones(1,9)];
b = sparse(i,j,s);
spy(b)
eig(b)

ans =

    0.0810
    0.3175
    0.6903
    1.1692
    1.7154
    2.2846
    2.8308
    3.3097
    3.6825
    3.9190

% 2. communication networks
% a graph is represented by its adjency matrix:
% A(i,j) = 1 or 0 depending whether there an
% edge between i and j
dt = 2*pi/10;
t = dt:dt:2*pi;
x = cos(t); y = sin(t);
% (x,y) define the nodes on the unit circle
size(x)

ans =

     1    10

plot(x,y,'r+'); axis([-1.5 1.5 -1.5 1.5]);
% assuming that every node is connected to
% every other node
A = ones(10);
size(A)

ans =

    10    10

gplot(A,[x' y'])
A = spalloc(10,10,9);
for j = 1:9
     if mod(j,2) == 1
        A(5,j) = 1;
     else
        A(10,j) = 1;
     end;
end;
A(5,5) = 0; A(5,10) = 1;
gplot(A,[x' y'])
% note that A is now the adjacency matrix
% of a directed graph, because it is not symmetric
spy(A)
B = A + A'

B =

   (5,1)        1
  (10,2)        1
   (5,3)        1
  (10,4)        1
   (1,5)        1
   (3,5)        1
   (7,5)        1
   (9,5)        1
  (10,5)        1
  (10,6)        1
   (5,7)        1
  (10,8)        1
   (5,9)        1
   (2,10)       1
   (4,10)       1
   (5,10)       1
   (6,10)       1
   (8,10)       1

spy(B)
gplot(B,[x' y']);
C = B^2;
spy(C)
D = B^3;
spy(D)
% C tells which nodes are connected to each other
% using two links, as C = B^2;
% D indicates which nodes are connected via 3 edges
E = A + C + D;
spy(E)
spy(B+C)
% we see that we need 3 edges to make the graph
% completely connected...
% 3. Four dimensional matrices
% consider the temperature distribution in a room
r = -2:0.1:+2; % defines a range
[x,y,z] = ndgrid(r,r,r); % sampling grid
% we defined a grid using the same range
% for x, y, and z, to see how many samples we used:
size(x)

ans =

    41    41    41

t = y.*exp(-x.^2 - y.^2 - z.^2);
slice(y,x,z,t,[],[-1.2 .8],-0.2);
hold on
xlabel('x'); ylabel('y'); zlabel('z');
hold off
cplxroot(5)
exit