the deflated initial guess for system $\hat A$ by applying (\[eq:initCG\]) on $P^{1/2}\tilde x_0$, then the initial guess $x_0$ to be given to CG is: $$\begin{aligned}
x_0 =  P^{-1/2} \hat x_0 
    &=& P^{-1/2}(P^{1/2}\tilde x_0 + \hat U \hat \Lambda^{-1} \hat U^H P^{-1/2} 
                      (P^{-1/2}b-P^{-1/2}A\tilde x_0)) \\
    &=& \tilde x_0 + P^{-1} RY \hat \Lambda^{-1} Y^HR^H P^{-1} 
                            (b - A\tilde x_0)\\ 
    &=& \tilde x_0 + VY \hat \Lambda^{-1} Y^HV^H (b-A\tilde x_0).\end{aligned}$$ The $Y$ are the eigenvectors corresponding to the eigenvalues $\hat \Lambda$ of the Lanczos tridiagonal matrix $T_m = V^H A V = (P^{-1/2}R)^H \hat A (P^{-1/2}R)$. The $T_m$ is obtained at no extra cost from the scalars $\alpha_j = r_{j-1}^Hz_j / (p_j^HAp_j)$ and $\beta_j = r^H_{j-1}z_{j}/(r_{j-2}^Hz_{j-1})$ computed during the CG iteration [@Saad_linear p.194]: $$\begin{aligned}
\hspace{12pt}
T_m = \left[ \begin{array}{ccccc}
 1/\alpha_1       &  \sqrt{\beta_2}/\alpha_1\\
\sqrt{\beta_2}/\alpha_1 & 1/\alpha_2 +\beta_2/\alpha_1 \\
& & \ddots & & \sqrt{\beta_{m+1}}/\alpha_m\\
& & & \sqrt{\beta_{m+1}}/\alpha_m & 
                1/\alpha_m + \beta_m/\alpha_{m-1} \\
    \end{array}
    \right].
\label{eq:tridiagonal}\end{aligned}$$

Traditionally, the Lanczos Ritz values and vectors are computed from $T_m$ at the end of CG. This, however, requires the storage of all $z_j$ or a second CG run to recompute them on the fly, effectively doubling the cost. Moreover, if the number of CG iterations 