# TOAR

A Two-level Orthogonal ARnoldi procedure.

## Description

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.

## Contents

### MATLAB codes and data

• Main files

• Auxiliary files

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

## Examples

See Ref. [1] for details.

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

## References

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.

## Contact

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