A Two-level Orthogonal ARnoldi procedure.


TOAR is a memory efficient, and numerically reliable algorithm to compute the compact Arnoldi decomposition:

\[ \left[\begin{array}{cc} A & B \\ I & 0 \end{array} \right] \left[\begin{array}{c} Q_kR_{k,1}\\ Q_kR_{k,2} \end{array}\right] = \left[\begin{array}{c} Q_{k+1}R_{k+1,1}\\ Q_{k+1}R_{k+1,2} \end{array}\right] \underline H_{k+1}. \]

It is also a stabilized version of the SOAR procedure to compute the orthonormal basis \(Q_k\) of the second order Krylov subspace

\[ \mathcal{G}_k( A, B; r_{-1}, r_0)\equiv \mbox{span}\{ r_{-1}, r_0, r_1, \dots, r_{k-1}\} \quad\mbox{with}\quad r_ j= A r_{ j-1}+ B r_{ j-2}\quad \mbox{for}\quad j \ge 1. \]

TOAR finds applications in nonlinear eigenvalue computation, and model order reduction. It demonstrates superior performance over the SOAR procedure.


MATLAB codes and data


See Ref. [1] for details.

Accuracy of reduced transfer function of the shaft problem 

Fig: Accuracy of reduced transfer functions of a second order dynamical system by TOAR and SOAR.


  1. Stability Analysis of the two-level orthogonal Arnoldi procedure
    by Ding Lu, Yangfeng Su and Zhaojun Bai
    SIAM J. Matrix Anal. Appl., 2016. 37(1): 195–214. (paper)

  2. SOAR: A second-order Arnoldi method for the solution of the quadratic eigenvalue problem
    by Zhaojun Bai and Yangfeng Su
    SIAM J. Matrix Anal. Appl., 2005. 26(3): 640-659.