We study the numerical solution of infinite matrix equations Au=f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasi-diagonalization of pseudo-differential operators in the weighted Sjöstrand class. The proposed algorithm has two main features: firstly, it converges to the solution with quasi-optimal order and complexity with respect to classes of localized vectors; secondly, in addition to ℓ²-convergence, the algorithm converges automatically in some stronger norms of weighted ℓ^p-spaces. As an application we approximate the canonical dual frame of a localized frame and show that this approximation is again a frame, and even an atomic decomposition for a class of associated Banach spaces. The main tools are taken from adaptive algorithms, from the theory of localized frames, and the special Banach algebra properties of the Jaffard algebra.