  Spectral computations of infinite-dimensional operators are notoriously
difficult, yet ubiquitous in the sciences. Indeed, despite more than half a
century of research, it is still unknown which classes of operators allow for
computation of spectra and eigenvectors with convergence rates and error
control. Recent progress in classifying the difficulty of spectral problems
into complexity hierarchies has revealed that the most difficult spectral
problems are so hard that one needs three limits in the computation, and no
convergence rates nor error control is possible. This begs the question: which
classes of operators allow for computations with convergence rates and error
control? In this paper we address this basic question, and the algorithm used
is an infinite-dimensional version of the QR algorithm. Indeed, we generalise
the QR algorithm to infinite-dimensional operators. We prove that not only is
the algorithm executable on a finite machine, but one can also recover the
extremal parts of the spectrum and corresponding eigenvectors, with convergence
rates and error control. This allows for new classification results in the
hierarchy of computational problems that existing algorithms have not been able
to capture. The algorithm and convergence theorems are demonstrated on a wealth
of examples with comparisons to standard approaches (that are notorious for
providing false solutions).We also find that in some cases the IQR algorithm
performs better than predicted by theory and make conjectures for future study.
