Observing the AMGr relation with Chebyshev polynomials, we extend the concept of the AMGr method.
The AMGr method presented by MacLachlan, Manteuffel, and McCormick in  is motivated from a reduction point of view.
which results in the AMGr method with error propagation matrix
This estimate adds further insight into the two-grid convergence of AMGr methods and also leads to a proof of convergence of AMGr-based methods with full-grid smoothers, i.e., for the case where, as opposed to (3.2), the block row of M corresponding to the C variables is non-zero.
Reduction-based AMGr methods use only smoothing in the space of fine degrees of freedom, F, and, as such, can be interpreted as multiplicative hierarchical basis methods based on the space decomposition
The two-grid hierarchical basis method (a symmetric two-level AMGr method) is defined by the error propagation operator