In this paper, the difference between the residual and the true residual caused by the computation errors that arise in matrix multiplications for solutions generated by the Block IDR($s$) method is analyzed. Moreover, in order to reduce the difference between the residual and the true residual, a modified Block IDR($s$) method is proposed. Numerical experiments demonstrate that the difference under the proposed method is smaller than that of the conventional Block IDR($s$) method.