% 1 sparse matrices
% a matrix is sparse if it has few nonzero entries
a = sparse(4,6,3,10,10)

a =

   (4,6)        3

spy(a)
% with spy we can see patterns
p32 = pascal(32);
s32 = rem(p32,2);
spy(s32)
% an application of sparse matrices
% turns up in structured linear systems
% We defined a tridiagonal matrix
% with 2 on the diagonal and ones
% above and below the diagonal.
rows = [1:10 2:10 1:9];
% rows contain the rows for the 28
% nonzero elements
cols = [1:10 1:9 2:10];
% for the column indices we have to
% make sure they match the row indices
data = [2*ones(1,10) ones(1,9) ones(1,9)];
% the data consists of 10 two's on the diagonal,
% nine ones below the diagonal and nine ones
% above the diagonal
d3 = sparse(rows,cols,data);
spy(d3)
% 2. model networks
% The main tool to model a graph is the adjacency matrix
% These matrices are also sparse.
% let us draw a graph
dt = 2*pi/10;
t = dt:dt:2*pi;
x = cos(t); y = sin(t);
% we just defined 10 vertices,
% placed on a circle
plot(x,y,'r+')
axis([-1.5 1.5 -1.5 1.5])
hold on
for i =1:4
      text(x(i),y(i)+0.1,int2str(i))
end;
text(x(5)-0.1,y(5),int2str(5))
text(x(10)+0.1,y(10),int2str(10))
for i = 6:9
      text(x(i),y(i)-0.1,int2str(i))
end;
A = ones(10);
gplot(A,[x' y'])
% gplot takes on input two arguments:
% (1) the adjacency matrix A
% (2) the coordinates of the vertices
% the prime next to x and y: x' and y'
% transpose the rows into columns.
% 3. multidimensional matrices
% A temperature distribution gives rise to a
% 4 dimensional matrix.
r = -2:0.1:2;
[x,y,z] = ndgrid(r,r,r);
t = y .* exp(-x.^2 - y.^2 - z.^2);
% we just sampled a 3 dimensional temperature function
slice(y,x,z,t,[],[-1.2 .8],-.2)
figure
slice(y,x,z,t,[],[-1.2 .8],-.2)
diary off