Chapter 3 develops provably convergent algorithms for spectra with certified error bounds. We prove that even for tridiagonal or bounded self-adjoint operators, the spectrum cannot, in general, be obtained in a single limit: lack of global information and resolvent blow-up force multi-limit procedures. The chapter overcomes both via approximating the injection modulus (smallest singular values) and conditioning functions that turn resolvent growth into explicit distance-to-spectrum estimates. This yields two practical algorithms: PseudoSpec, producing verified inner approximations of pseudospectra, and CompSpec, outputting certified spectral points with local error bars—designed to eliminate spectral pollution and invisibility. The framework is implemented for infinite matrices using rectangular truncations, and for differential operators through operator folding. A general coefficient-sampling and quadrature theorem gives spectral computation for differential operators on unbounded domains (a generalised Schwinger problem). Examples include quasicrystals, Maxwell eigenproblems (pollution-free finite elements), and the imaginary cubic oscillator in non-Hermitian quantum mechanics.