Matthew J. Colbrook
Chapter 9 develops computational foundations for spectral and pseudospectral radii (largest modulus), spectral and pseudospectral abscissas (largest real part), spectral gaps, and related properties. These quantities control long-time behaviour, transient amplification, and stability of discrete- and continuous-time dynamics. Counterexamples for finite sections and domain truncation show why naïve discretisation can fail: limits in time (powers/exponentials) and truncation size need not commute. We then construct provably convergent algorithms, including a computational Gelfand formula that yields optimal two-limit procedures for spectral radii and Yamamoto-type singular-value schemes for essential spectral radii. All tasks and associated decision problems are positioned within the Solvability Complexity Index (SCI) hierarchy. For self-adjoint operators, we compute the spectral gap at the bottom of the spectrum via the Rayleigh–Ritz method, and we study the essential spectral gap, which identifies annuli containing dominant eigenvalues. Applications in spectral graph theory and systems/control illustrate the theory in practice.