  The adaptive $s$-step CG algorithm is a solver for sparse, symmetric positive
definite linear systems designed to reduce the synchronization cost per
iteration while still achieving a user-specified accuracy requirement. In this
work, we improve the adaptive $s$-step conjugate gradient algorithm by use of
iteratively updated estimates of the largest and smallest Ritz values, which
give approximations of the largest and smallest eigenvalues of $A$, using a
technique due to Meurant and Tich{\' y} [G. Meurant and P. Tich{\' y}, Numer.
Algs. (2018), pp.~1--32]. The Ritz value estimates are used to dynamically
update parameters for constructing Newton or Chebyshev polynomials so that the
conditioning of the $s$-step bases can be continuously improved throughout the
iterations. These estimates are also used to automatically set a variable
related to the ratio of the sizes of the error and residual, which was
previously treated as an input parameter. We show through numerical experiments
that in many cases the new algorithm improves upon the previous adaptive
$s$-step approach both in terms of numerical behavior and reduction in number
of synchronizations.
