|Matrix Generator MVMODE|
|Source:||G.W. Stewart, University of Maryland|
|Discipline:||Ordinary differential equations|
|Output format:||matrix-vector multiply|
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 yi 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
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.
In the data files, .
|N||the order of the matrix|
|GAMMA||boundary condition parameter|
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Last change in this page: Wed Sep 22 13:37:31 US/Eastern 2004 [Comments: ]