We develop a framework for proving rapid convergence of shifted QR algorithms
which use Ritz values as shifts, in finite arithmetic. Our key contribution is
a dichotomy result which addresses the known forward-instability issues
surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure
which provably either computes a set of approximate Ritz values of a Hessenberg
matrix with good forward stability properties, or leads to early decoupling of
the matrix via a small number of QR steps.
Using this framework, we show that the shifting strategy introduced in Part I
of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in
finite arithmetic with a polylogarithmic bound on the number of bits of
precision required, when invoked on matrices of controlled eigenvector
condition number and minimum eigenvalue gap