"Hybrid block GMRES" means that the solver first runs block GMRES.It stores the resulting coefficients, which form a matrix polynomial. It then can reuse this polynomial for subsequent solves. This avoids the cost of the inner products and norms in GMRES. However, the resulting polynomial is not necessarily as effective as the equivalent number of GMRES iterations.

We call solvers that take this approach "seed solvers." Belos implements both a Block GMRES seed solver (this class) and a CG-based seed solver (Belos::PCPGSolMgr).

Here is a list of all the parameters that this solver accepts:

"Convergence Tolerance" (MagnitudeType): The level that residual norms must reach to decide convergence. Default: 1e-8.
"Block Size" (int): The block size to be used by the underlying block GMRES solver. Default: 1 (which means don't use blocks).
"Num Blocks" (int): The restart length; that is, the number of blocks allocated for the Krylov basis. Default: 300.
"Maximum Iterations" (int): The maximum number of iterations GMRES is allowed to perform, across all restarts. Default: 1000.
"Maximum Restarts" (int): The maximum number of restarts the underlying solver is allowed to perform. This does not include the first restart cycle. Default: 20.
"Orthogonalization" (std::string): The desired orthogonalization method. Default: "DGKS".
"Verbosity" (Belos::MsgType): A sum of Belos::MsgType values specifying what kinds of messages to print. Default: Belos::Errors.
"Output Style" (Belos::OutputType): The output style. Default: Belos::General.

Like all Belos solvers, parameters have relative or "delta" semantics. This means the following:

Any parameter that was never set has its default value
Any parameter not explicitly set in the input ParameterList retains its current value