# 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

## Examples

See Ref. [1] for details.

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

**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)

**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