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


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.


MATLAB codes and data

C++ routines


See Ref. [1] for details.

Distribution of eigenvalues

Distribution of computed eigenvalues of the damped beam problem 

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

Timing statistics

Distribution of computed eigenvalues of the damped beam problem 

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


  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)