  There exist two major problems in application of the conventional block
BiCGSTAB method to the O(a)-improved Wilson-Dirac equation with multiple
right-hand-sides: One is the deviation between the true and the recursive
residuals. The other is the convergence failure observed at smaller quark
masses for enlarged number of the right-hand-sides. The block BiCGGR algorithm
which was recently proposed by the authors succeeds in solving the former
problem. In this article we show that a preconditioning technique allows us to
improve the convergence behavior for increasing number of the right-hand-sides.
