The iterative method \(u^{(n+1)}=Gu^{(n)}+k\) with \(G=I-Q^{-1}A\), \(k=Q^{-1}b\) and a symmetric definite matrix I-G to solve the linear system \(Au=b\) is speeded up by use of Chebyshev acceleration. The adaptive procedure from L. A. Hageman and D. M. Young [Applied iterative methods (1981; Zbl 0459.65014)] for finding the necessary smallest and largest eigenvalues of G is tested in seven experiments with respect to its effectiveness. It turns out that it needs at most 35% more iterates then the optimal nonadaptive procedure, and that it is not sensitive to the starting value unless the latter is very close to the largest eigenvalue.