## Reliable computation of the rightmost eigenvalues of large matrices

- Oct. 16, 2017
- 2:15 p.m.
- LeConte 312

## Abstract

Numerical computation of the rightmost eigenvalues (with largest real part) of large and typically sparse matrices is important for stability analysis. Due to the unknown imaginary parts of these eigenvalues, traditional algorithms such as the implicitly restarted Arnoldi (ARPACK) have difficulties finding them if they are significantly smaller than the dominant eigenvalues in modulus. We propose an exponential transformation $A \rightarrow exp(hA)$ (h > 0) to map the rightmost eigenvalues of $A$ to dominant eigenvalues of $exp(hA)$. The key technique is to compute the exponential matrix-vector multiplications $exp(hA)v$ without explicitly forming the dense matrix $exp(hA)$. Several polynomial and rational algorithms are compared for this purpose, and the reliability of the new approach systematically outperforms existing techniques in our numerical experiments.