(section:eigen:accuracy)=
# Eigenvalue Problems

Modal solutions form the basis of much of the analysis performed in
Sierra/SD. It is essential that we understand the accuracy of the
solution computed eigenvalue pairs may have errors for a variety of
reasons, the most common is that the linear solvers all have tolerances,
and errors in these solutions feed directly into errors in eigenpairs.
It is well-known that errors in eigenvectors are typically significantly
larger than errors in eigenvalues. If the relative error in an
eigenvalue is $\varepsilon$, the relative error in the eigenvector is of
the order of $\sqrt{\varepsilon}$.

At the conclusion of a modal analysis, the Sierra/SD application reports
the eigenvalues and associated error estimates.
An example of this output is provided below.
The first column of data is the eigenvalue, which is related to the
frequency $f_j$ by,

$$
\lambda_j = (2\pi f_j)^2.
$$

The second column is an estimate of the error bound on the eigenvalue,
$\epsilon_j=|(K-\lambda_j
M)\phi_j|_2$. Generally, except for zero energy modes, the error bound
should be tiny relative to $\lambda$.

:::{output}
 Ritz values (Real, Imag) and direct residuals  
 ---------------------------------------------  
                  Col   1           Col   2  
  Row     1:     -2.16338D-06     7.34617D-07  
  Row     2:      2.07696D+07     2.25677D-06  
  Row     3:      2.07858D+07     8.73909D-07  
  Row     4:      3.56376D+08     1.48725D-06  
  Row     5:      4.84777D+08     1.69662D-06  
  Row     6:      4.84906D+08     5.01020D-06  
  Row     7:      9.59039D+08     6.06316D-06  
  Row     8:      1.11917D+09     1.22741D-05  
  Row     9:      1.11917D+09     3.30643D-06
:::

(geometric_rigid_body_modes)=
## Geometric Rigid Body Modes

An [example input](/tests/Salinas_rtest/training/exampleproblem/eigen/rigidbodymode/tied/simpleTiedCase.inp) and mesh (shown below) of the use of the Geometric Rigid Body Modes capability are provided.
This example
assumes that the reader is familiar with the parameter `num_rigid_mode`,
and has read the `GeometricRigidBodyModes Solution Case` section of the
User's manual.

:::{figure-md}

![](mesh.png)

[Example Mesh](/tests/Salinas_rtest/training/exampleproblem/eigen/rigidbodymode/tied/simpleTied.exo)
:::

:::{toctree}
:hidden:
:maxdepth: 2

LinearBuckling/main
WetModes/main
:::
