%%
clear
clc

%%
% Here is a matrix with both lower and upper bandwidth equal to one. Such a
% matrix is called *tridiagonal*. The spdiags command creates a sparse
% matrix given its diagonal elements. The ``main'' or central
% diagonal is numbered zero, above and to the right of that is positive,
% and below and to the left is negative.
n = 50;
d = [ n*ones(n,1), ones(n,1), -(1:n)'];  % diagonal entries
pos = [-3 0 1];                          % which diagonals
A = spdiags(d, pos, n, n);
np = 12;
full( A(1:np,1:np) )
spy(A)

%%
% Without pivoting, the LU factors have the same lower and upper bandwidth
% as the orignal matrix. 
[L, U] = lufact(A);
subplot(1,2,1), spy(L), title('L')
subplot(1,2,2), spy(U), title('U')

%%
% However, if we introduce row pivoting, bandedness may be expanded or
% destroyed.
[L, U, P] = lu(A);
subplot(1,2,1), spy(L), title('L')
subplot(1,2,2), spy(U), title('U')