  In this paper, we propose some Chebyshev polynomials of the 1st-kind which
produce optimal bound for a polynomial dependent constant involved in the AMG
$V$-cycle error bound and do not require information about the spectrum of
matrices. We formulate a variant of a minimax problem already proposed in [J.
Lottes, Optimal polynomial smoothers for multigrid V-cycles, Numer. Lin. Alg.
with Appl., 30 (2023), p. e2518, https://doi.org/10.1002/nla.2518] and define
Chebyshev polynomial of the 1st-kind as acceleration for a weighted-Jacobi
smoother; we also describe efficient GPU kernels for the application of the
polynomial smoother and compare results with accelerators defined in [J.
Lottes, Optimal polynomial smoothers for multigrid V-cycles, Numer. Lin. Alg.
with Appl., 30 (2023), p. e2518, https://doi.org/10.1002/nla.2518] on usual
benchmarks at very large scales.
