ritz¶

Purpose:

Compute approximative eigenvalues and eigenvectors by the Lanczos method.

Syntax:

L = ritz(K, M, f, m)
L = ritz(K, M, f, m, b)
[L, X] = ritz(K, M, f, m)
[L, X] = ritz(K, M, f, m, b)

Description:

ritz computes, by the use of the Lanczos algorithm, m approximative eigenvalues and m corresponding eigenvectors for a given pair of n-by-n matrices K and M and a given non-zero starting vector f.

If certain rows and columns in matrices \(\mathbf{K}\) and \(\mathbf{M}\) are to be eliminated in computing the eigenvalues, \(\mathbf{b}\) must be given in the command. The rows (and columns) to be eliminated are described in the vector \(\mathbf{b}\) defined as

\[\begin{split}\mathbf{b} = \begin{bmatrix} dof_1 \\ dof_2 \\ \vdots \\ dof_{nb} \end{bmatrix}\end{split}\]

Note

If the number of vectors, m, is chosen less than the total number of degrees-of-freedom, \(n\), only about the first m/2 Ritz vectors are good approximations of the true eigenvectors. Recall that the Ritz vectors satisfy the M-orthonormality condition

\[\mathbf{X}^T \mathbf{M} \mathbf{X} = \mathbf{I}\]

where \(\mathbf{I}\) is the identity matrix.