In minisymposium: Matrix Polynomials, Matrix Functions and Applications
Thu 11:30–12:00, Room AV 03.12Condition numbers measure the sensitivity of problems to perturbations in the data. In this talk we investigate the effect of structured perturbations in the sensitivity of matrix functions. Since the set of possible perturbations preserving the structure is smaller, we have that $\text{cond}_{\text{struc}}(f,X)\leq \text{cond}(f,X)$, where $\text{cond}_{\text{struc}}(f,X)$ and $\text{cond}(f,X)$ represent the structured and unstructured condition numbers of matrix function $f$ at a matrix $X$, respectively. If $\text{cond}_{\text{struc}}(f,X)\ll \text{cond}(f,X)$, this suggests that using structure preserving numerical methods to compute $f$ would be advantageous. This motivates the study of structured condition numbers for matrix functions.
In this work, we focus on computing the structured condition number of matrix functions defined between two smooth matrix manifolds. Kenney and Laub [1] showed that the unstructured condition number can be obtained in terms of the Fréchet derivative of $f$ at $X$. We show that when the matrix function $f$ is a smooth map between two smooth manifolds then the differential $df_X$ plays for the structured condition number the same role of the Fréchet derivative. We derive a computable expression for the structured condition number and propose algorithms to compute it or approximate it.
We analyse experimentally how the inequality $\text{cond}_{\text{struc}}(f,X)\leq \text{cond}(f,X)$ is changing with the choice of $X$ and $f$. In particular, we study the ratio of the structured condition number to the unstructured one for the matrix logarithm and matrix square root for matrices in automorphism groups, which arise in many scientific and engineering applications.
- Charles S. Kenney and Alan J. Laub, Condition Estimates for Matrix Functions, Journal of Mathematical analysis and applications 10(2) (1989), pp. 191–209.