$\left(\begin{array}{cc}{\left[X_0^{\text{T}}{\left(I_N-M^{-1}{\mathcal{V}}_m{\mathcal{V}}_m^{\text{T}}{M}^{-\text{T}}{M}^{-1}{\mathcal{V}}_m\right)}^{-1}{\mathcal{V}}_m^{\text{T}}{M}^{-\text{T}}{X}_0+I_s\right]}^{\frac{1}{2}}& 0\\ C{\left({\mathcal{V}}_m^{\text{T}}{M}^{-\text{T}}{M}^{-1}{\mathcal{V}}_m\right)}^{-1}{\mathcal{V}}_m^{\text{T}}{M}^{-\text{T}}{X}_0& C\end{array}\right){\left(L_m\right)}^{-1}{z}_i={\xi }_i{w}_i\mathrm{.}$ (3.2)

normalise the last s component of $u_i={\left(L_m\right)}^{-1}{z}_i$ following (2.5) and form the approximate solution $X_m=X_0+Z_m{Y}_m$ .

4: Set ${\Vert {\Delta }_{A,B,m}\Vert }_F^2={\displaystyle \sum _{i=ms+1}^{\left(m+1\right)s}}{\xi }_i^2$ if satisfied, stop, else set $X_0:=X_m$ , compute $R_0:=B-A{X}_0$ , $V_1{B}_1:=R_0$ and set ${\mathcal{V}}_1:=V_1$ and go to step 2.