# Stability analysis of improved Two-level orthogonal Arnoldi procedure

## TL;DR

This paper introduces the I-TOAR method, an improved version of the Two-level orthogonal Arnoldi procedure, and provides a stability analysis addressing an open problem related to the numerical stability of second-order Krylov subspace computations.

## Contribution

The paper proposes the I-TOAR method and solves the open problem of its stability analysis with respect to quadratic problem coefficients.

## Key findings

- I-TOAR enhances numerical stability over previous methods.
- The stability analysis confirms robustness of I-TOAR.
- Results demonstrate improved accuracy in second-order Krylov subspace computations.

## Abstract

The SOAR method for computing an orthonormal basis of a second-order Krylov subspace can be numerically unstable (see Lu et al. (2016)). In the Two-level orthogonal Arnoldi(TOAR) procedure, an alternative to SOAR, the problem of instability had circumvented. A stability analysis of the second-order Krylov subspace's orthonormal basis in TOAR with respect to the coefficient matrices of a quadratic problem remain open; see Lu et al. (2016). This paper proposes the Improved-TOAR method(I-TOAR) and solves the said open problem for I-TOAR.