  The structured $\varepsilon$-stability radius is introduced as a quantity to
assess the robustness of transient bounds of solutions to linear differential
equations under structured perturbations of the matrix. This applies to general
linear structures such as complex or real matrices with a given sparsity
pattern or with restricted range and corange, or special classes such as
Toeplitz matrices. The notion conceptually combines unstructured and structured
pseudospectra in a joint pseudospectrum, allowing for the use of resolvent
bounds as with unstructured pseudospectra and for structured perturbations as
with structured pseudospectra. We propose and study an algorithm for computing
the structured $\varepsilon$-stability radius. This algorithm solves eigenvalue
optimization problems via suitably discretized rank-1 matrix differential
equations that originate from a gradient system. The proposed algorithm has
essentially the same computational cost as the known rank-1 algorithms for
computing unstructured and structured stability radii. Numerical experiments
illustrate the behavior of the algorithm.
