**3. Smoothers Based on the HSS and the STS Iteration Methods**

A particular problem when using the MGM is the choice of smoothers. There are a number of iteration methods that can be used as smoothers, but not all of them are effective for solving strongly non-Hermitian SLAEs. The behavior of the HSS and the STS iteration methods is similar to the behavior of the Gauss–Seidel method, which quickly damps the high-frequency harmonics of the error, slowing down in the future. We give the formulas of these iteration methods.

**The HSS iteration method** [1]: Given an initial guess *<sup>v</sup>*(0), for *<sup>k</sup>* <sup>=</sup> 0, 1, 2, ... until {*v*(*k*)} convergence, compute

$$\begin{cases} (\varkappa I + A\_0)\upsilon^{(k+\frac{1}{2})} = (\varkappa I - A\_1)\upsilon^{(k)} + b\_\prime\\ (\varkappa I + A\_1)\upsilon^{(k+1)} = (\varkappa I - A\_0)\upsilon^{(k+\frac{1}{2})} + b\_\prime \end{cases}$$

where *α* is a given positive constant and *I* is an identity matrix.

Bai, Golub, and Ng [1] proved that the HSS iteration method converges unconditionally to the exact solution of the SLAE (2). Moreover, the upper bound of the contraction factor depends on the spectrum of *A*<sup>0</sup> but is independent of the spectrum of *A*1.

We can rewrite the HSS iteration method in the following form:

$$
\upsilon^{(k+1)} = G(\alpha)\upsilon^{(k)} + B(\alpha)^{-1}b\_{\prime}
$$

where

$$G(\alpha) = B(\alpha)^{-1} (B(\alpha) - A)^{\alpha}$$

and

$$B(\alpha) = \frac{1}{2\alpha} (\alpha I + A\_0)(\alpha I + A\_1).$$

**The STS iteration method** [6,8]: Given an initial guess *v*(0) and two positive parameters *ω* and *τ*. For *<sup>k</sup>* <sup>=</sup> 0, 1, 2, ... until {*v*(*k*)} convergence, compute

$$v^{(k+1)} = G(\omega, \mathfrak{r}) v^{(k)} + \mathfrak{r}B(\omega)^{-1}b\_{\mathfrak{r}}$$

where

$$G(\omega, \tau) = B(\omega)^{-1} (B(\omega) - \tau A)\_{\tau}$$

*ω* and *τ* are two acceleration parameters, and *B*(*ω*) is defined by

$$B(\omega) = B\_{\mathcal{E}} + \omega((1+j)K\_L + (1-j)K\_{\mathcal{U}}), \quad j = \pm 1$$

with *Bc* <sup>∈</sup> <sup>C</sup>*n*×*<sup>n</sup>* a prescribed Hermitian matrix.

For the STS method a convergence analysis, optimal choice of parameters and an accelerating procedure have presented in [8]. As it was mentioned above, smoothers in the MGMs should have a smoothing effect on the error of approximation. It was shown in [14] that the skew-Hermitian triangular iteration methods have such properties. Therefore, these methods can be used as smoothers in the MGMs.
