Let $A=I-B$ be positive definite and $s_v=B^v s_0$. It is sometimes suggested to take $x_v{ }^*$ to be a mean of $x_0, x_1, \cdots, x_v$, and hence to form
$$
\begin{aligned}
x_v{ }^* & =\alpha_{v 0} x_0+\alpha_{v 1} x_1+\cdots+\alpha_{v v} x_v, \\
1 & =\alpha_{v 0}+\alpha_{v 1}+\cdots+\alpha_v .
\end{aligned}
$$
Show that this is equivalent to the process indicated by (4.3.32), or, alternatively, that
$$
s_v{ }^*=\psi_v(B) s_0, \quad \psi_v(1)=1 .
$$
If $\rho(B)=\rho<1$ the optimal choice is
where
$$
\psi_v(\lambda)=\cos v \theta / \cosh v \omega_0
$$
$$
\lambda=\rho \cos \theta, \quad 1=\rho \cosh \omega_0 .
$$
Then
$$
\begin{aligned}
x_{v+1}^* & =\beta_{v+1}\left(B x_v{ }^*+h-x_{v-1}^*\right)+x_{v-1}^*, \\
\beta_{v+1} & =2 \rho^{-1} \cosh v \omega_0 / \cosh (v+1) \omega_0 .
\end{aligned}
$$
Show also that
$$
\beta_1=1, \quad \beta_2=2 /\left(2-\rho^2\right), \quad \beta_{v+1}=\left[1-\left(\rho^2 \beta_v / 4\right)\right]^{-1}
$$
(Golub and Varga).