Roots of Matrices in the Study of GMRES Convergence and Crouzeix's Conjecture
Daeshik Choi, Anne Greenbaum · SIAM Journal on Matrix Analysis and Applications · 2015
For any nonsingular matrix $A$ and any positive integer $m$, the $m$th root, $A^{1/m}$, can be defined using any branch cut of the $m$th root function that does not pass through an eigenvalue of $A$. Cleary, $A^{1/m}$ approaches the identity as $m \rightarrow \infty$, but we are interested in how it approaches the identity. It is shown that $\lim_{m \rightarrow \infty} [W( A^{1/m} ) ]^m = \exp [ W( \log A ) ]$, where $W$ denotes the field of values and $\log A$ is defined using the same branch cut. It is also shown that $\| A^{1/m} \|^m$ approaches $\exp( \alpha ( \log A ) )$, where $\| \cdot \|$ denotes the spectral norm and $\alpha ( \cdot )$ is the numerical abscissa. Implications for the convergence rate of the GMRES algorithm are discussed, especially when $W(A)$ contains the origin. Additionally, it is shown that if $A$ is a strict contraction and $\rho \in ( \| A \| , 1 )$, then the matrices $[ ( I + \rho A )^{-1} (A + \rho I ) ]^{1/m}$, $m=1,2, \ldots,$ are all strict contractions. The significance of results of this sort in a possible approach to proving Crouzeix's conjecture is discussed.