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##

Example

We continue to use the example introduced in
§2.1 and
Figure 2.1.
We again consider the case with arbitrary masses ,
and zero damping constants . This simplifies the equations
of motion to
.
As in §2.3.8 we solve the equations of motion
by substituting
, where is a constant vector
and is a constant scalar to be determined. This leads
to
. Letting , where
the Cholesky factor
,
we see that we need to compute the eigenvalues of the symmetric tridiagonal
matrix
shown in equation (2.2).

Now we note that the stiffness matrix can be factored as ,
where
and

This lets us write

Therefore, the singular values of the bidiagonal matrix

are the square roots of
the eigenvalues of , and the left singular vectors of are
the eigenvectors of .
Bidiagonal matrices have particularly fast and efficient SVD algorithms.

** Next:** Non-Hermitian Eigenproblems J. Demmel
** Up:** Singular Value Decomposition J.
** Previous:** Related Singular Value Problems
** Contents**
** Index**
Susan Blackford
2000-11-20