Consider the following eigenvalue problem of an ordinary differential equation

with the boundary conditions

.

It can be shown that the eigenvalues are given by

,

which are complex. The solutions of this equation are of the form

for , where .

The eigenproblem of (3) can be approximated by finite differences as follows. Let y_{i} denote the approximate solution at the point . Replacing the second derivatives in (3) with a centered difference operators to obtain the generalized matrix eigenvalue problem

,

for , where with 1's on off-diagonals, -2 on diagonal, and an additional row appended with values (4,-1,... gamma, -4gamma, 3gamma)" >

and . Problem (4) can be recast as the standard eigenvalue problem

,

where .

The matrix-vector products Y = CX can be formed by solving the linear system AY = BX for Y using the banded Gaussian elimination. Fortran calling sequence for Y = CX.