An Efficient Bound for the Condition Number of the Matrix Exponential

A new bound for the condition number of the matrix exponential is presented. Using the bound, we propose an efficient approximation to the condition number, denoted by $\kappa_g(s,X)$, that \emph{avoids} the computation of the Fr\'echet derivative of the matrix exponential that underlies condition number estimation in the existing algorithms. We exploit the identity $e^X=(e^{X/2^s})^{2^s}$ for a nonnegative integer $s$ with the properties of the Fr\'echet derivative operator to obtain the bound. Our cost analysis reveals that considerable computational savings are possible since estimating the condition number by the existing algorithms requires several invocation of the Fr\'echet derivative of the matrix exponential whose single invocation costs as twice as the cost of the matrix exponential itself. The bound and hence $\kappa_g(s,X)$ only involve Fr\'echet derivative of a monomial of degree $2^s$, which can be computed exactly in $2s$ matrix multiplications. We propose two versions of the scaling and squaring algorithm that implement $\kappa_g(s,X)$. %that represents the squaring phase of %the scaling and squaring method. Our numerical experiments show that $\kappa_g(s,X)$ captures the behavior of the condition number and moreover outperforms the condition number in the estimation of relative forward errors for a wide range of problems