  The numerical solution of spectral fractional diffusion problems in the form
${\mathcal A}^\alpha u = f$ is studied, where $\mathcal A$ is a selfadjoint
elliptic operator in a bounded domain $\Omega\subset {\mathbb R}^d$, and
$\alpha \in (0,1]$. The finite difference approximation of the problem leads to
the system ${\mathbb A}^\alpha {\mathbf u} = {\mathbf f}$, where ${\mathbb A}$
is a sparse, symmetric and positive definite (SPD) matrix, and ${\mathbb
A}^\alpha$ is defined by its spectral decomposition. In the case of finite
element approximation, ${\mathbb A}$ is SPD with respect to the dot product
associated with the mass matrix. The BURA method is introduced by the best
uniform rational approximation of degree $k$ of $t^{\alpha}$ in $[0,1]$,
denoted by $r_{\alpha,k}$. Then the approximation ${\bf u}_k\approx {\bf u}$
has the form ${\bf u}_k = c_0 {\mathbf f} +\sum_{i=1}^k c_i({\mathbb A} -
{\widetilde{d}}_i {\mathbb I})^{-1}{\mathbf f}$, ${\widetilde{d}}_i<0$, thus
requiring the solving of $k$ auxiliary linear systems with sparse SPD matrices.
The BURA method has almost optimal computational complexity, assuming that an
optimal PCG iterative solution method is applied to the involved auxiliary
linear systems. The presented analysis shows that the absolute values of first
%${\widetilde{d}}_i$ $\left\{{\widetilde{d}}_i\right\}_{i=1}^{k'}$ can be
extremely large. In such a case the condition number of ${\mathbb A} -
{\widetilde{d}}_i {\mathbb I}$ is practically equal to one. Obviously, such
systems do not need preconditioning. The next question is if we can replace
their solution by directly multiplying ${\mathbf f}$ with
$-c_i/{\widetilde{d}}_i$. Comparative analysis of numerical results is
presented as a proof-of-concept for the proposed RS-BURA method.
