Characteristic-Value Problems

In this lecture we showed how BVP with one parameter translate naturally into eigenvalue problems when using finite differences.

As example, we considered the problem 

      2
     d u(x)    2
    ------- + k  u(x) = 0,  u(0) = 0 and u(1) = 0.
         2
      d x
The following sequence of commands (for Octave or MATLAB) generates the matrix and computes its eigenvalues: 
n = 10;
h = 1/(n+1);
a = diag(2*ones(1:n))-diag(ones(1:n-1),-1)-diag(ones(1,n-1),+1);
e = eig(a);
k2 = e/h^2;
k = sqrt(k2)
k/pi
The output is 
k =

   3.1309
   6.1981
   9.1391
  11.8941
  14.4069
  16.6265
  18.5076
  20.0119
  21.1088
  21.7761

ans =

  0.99660
  1.97292
  2.90908
  3.78601
  4.58587
  5.29238
  5.89114
  6.36999
  6.71915
  6.93154
We see the first n multiples of Pi appearing.

We derived the Power method to compute the eigenvector with the dominant eigenvalue and derived some variants of this method: the inverse and shifted power method.