Page 139 - Numerical methods for chemical engineering

Numerical calculation of eigenvalues and eigenvectors 125

e = eigs(A,3,‘SM’),
e=
0.1300
0.0581
0.0146

and
e = eigs(A,4,1.0),
e=
1.2908
1.0706
0.8639
0.6738
In addition to this output, eigs writes information about its internal calculations to the
screen. To turn this off, use
OPTS.disp = 0;
e = eigs(A,5,‘LM’,OPTS);
Other fields in OPTS allow us to modify the behavior of the algorithm, e.g. by decreasing
the tolerances. Type help eigs for more information.
With two arguments, eigs returns a matrix W of eigenvectors and a diagonal matrix D of
eigenvalues such that AW = WD. The following code computes and plots the eigenvectors
of largest and smallest magnitudes,

[W,D] = eigs(A,2,‘BE’,OPTS);
figure; plot(W(:,1),‘–’);
hold on; plot(W(:,2));
xlabel(‘component’); ylabel(‘w(k)’);
title(‘Eigenvectors of 1-D diffusion matrix’);
phrase1 = [‘ \ lambda = ’, num2str(D(1,1))];
phrase2 = [‘ \ lambda = ’, num2str(D(2,2))];
legend(phrase1, phrase2, ‘Location’, ’Best’);

The graph generated by this code is shown in Figure 3.5.
Above, we have called eigs for a sparse-format matrix A. A matrix stored in full matrix
format can also be used. Additionally, we can merely supply a routine that returns for each
input v, the corresponding vector Av. For the matrix above, this is done by

function Av = diff matrix 1D mult(v);
N = length(v);
Av = zeros(N,1);
Av(1) = 2*v(1) - v(2);
for k = 2:(N-1)
Av(k) = -v(k-1) +2*v(k)-v(k+1);