Page 66 - Process Modelling and Simulation With Finite Element Methods

FEMLAB and the Basics of Numerical Analysis 53

The eigs ( function is a variant of eig ( ) which computes a specific number
of eigenvalues/eigenvector pairs for sparse matrices. Its use will be
demonstrated in the next subsection in conjuction with FEMLAB.
The matrix A has a determinant that is little different from zero and a single
eigenvalue that is effectively zero. The eigenvector associated with it is
effectively the null space of A - the direction that gets mapped to zero:
>> A*V(:, 1)
ans =
1.0e-007 *
0.2669
0.1633
0.0327
-0.2112
0.0943
All the other eigenvectors can be verified by the property that they map onto
themselves, scaled by the eigenvalue, for instance:
>> A*V(: ,2) ./ V(: ,2)
ans =
0.7000
0.7000
0.7000
0.7000
0.7000
In MATLAB, the ./ division operator is element-by-element division. The colon
above refers to the whole of the column.
Because the system is nearly singular, we should not be surprised that the
solutions to any matrix equation involving it are poorly conditioned. For
instance,
>> B=[O; 1; 0; 1; 01;
>> A\B

ans =
1.0e+006 *
2.1487
1.3142
0.2631
-1.7001
0.7593
Since the elements of A are of order one, the forcing vector B is of order one,
one would expect the solution to (1.25) to be order one, not order one million.
For chemical engineers, this is like being told that a mass balance involves input
flow rates of about 1 kgh, constraints on mass balances with appreciable