# PALM

Padé Approximate Linearization Methods to solve Quadratic Eigenvalue Problems (QEP) with low–rank damping.

## Description

PALM implements the Padé Approximate Linearization algorithm: A structure respecting algorithm for solving QEP with low-rank damping

$(\lambda^2 M + \lambda C + K) x = 0, \qquad C = EF^{T},$

where $$M$$, $$C$$ and $$K$$ are $$n \times n$$ matrices, referred to as mass, damping and stiffness matrices. The damping matrix $$C$$ is of rank $$\ell \ll n$$, which admits the rank-revealing decomposition above with $$E$$ and $$F$$ to be $$n\times \ell$$ full column rank matrices.

The advantage of PALM is that it converts the QEP to a linear eigenvalue problem of dimension $$n_{\rm L} = n+ \ell m$$ with $$m$$ being the Padé approximation order

$\left(\left[\begin{array}{cc} {K}_\sigma +\sigma d C & {E}_{\sigma_1} \\ {F}_{\sigma_2}^{\rm T} & I_{\ell m}\end{array}\right] - \mu \left[\begin{array}{cc} {M}_\sigma & 0 \\ 0 & I_\ell \otimes D_m\end{array}\right]\right) x_{\rm L} = 0,$

which is significantly smaller than the standard linearization of size $$2n$$. (See Ref. [1] for details.)

PALM finds application in solving QEPs arising from analysis of structural dynamics, and acoustic analysis. It runs 33 – 47% faster than the direct linearization (DLIN) approach for solving modest size QEPs.

## Contents

### MATLAB codes and data

• Main files: pal.m  —  PAL algorithm

• Auxiliary files: Data and demo files for the numerical examples in Ref. [1].

## Examples

See Ref. [1] for details.

### Distribution of eigenvalues

Left: damped_beam problem. Right: QEP from car body design.

### Timing statistics

Left: damped_beam, n=20K. Middle: acoustic_wave_2d, n= 0.2M. Right: car, n=0.7M.

## References

1. A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low-rank damping
by Ding Lu, Xin Huang, Zhaojun Bai, and Yangfeng Su
Int. J. Numer. Methods Eng., 2015. 103(11): 840–858. (paper)

## Contact

Email: Ding.Lu@unige.ch
Homepage: http://www.unige.ch/~dlu