An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems

作者：Hammarling Sven^*; Munro Christopher J; Tisseur Francoise

来源：ACM Transactions on Mathematical Software, 2013, 39(3): 18.

DOI：10.1145/2450153.2450156

• Algorithms
• Performance
• Reliability
• Quadratic eigenvalue problem
• deflation
• linearization
• companion form
• backward error
• condition number
• scaling
• eigenvector

We develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials. It incorporates scaling of the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable conditioning and backward stability properties, and a preprocessing step that reveals and deflates the zero and infinite eigenvalues contributed by singular leading and trailing matrix coefficients. The algorithm is backward-stable for quadratics that are not too heavily damped. Numerical experiments show that our MATLAB implementation of the algorithm, quadeig, outperforms the MATLAB function polyeig in terms of both stability and efficiency.

• 出版日期2013-4