1: Initialize: Choose $X_0$ , compute $R_0:=B-A{X}_0$ , $V_1{B}_1:=R_0$ and set ${\mathcal{V}}_1:=V_1$ .

2: Flexible Arnoldi process: Perform m steps of the block Arnoldi process to compute ${\mathcal{V}}_{m+1}$ and ${\tilde{H}}_m$

for $j:=1,\cdots ,m$ , compute $Z_j:=M^{-1}{V}_j$ ,

accumulate ${\mathcal{Z}}_j\mathrm{<mo>\; :\; </mo>}=\left[Z_1\mathrm{,}{Z}_2\mathrm{,}\cdots \mathrm{,}{Z}_j\right]$ ,

compute $W:=A{Z}_j$ ,

for $i:=1,\cdots ,j$ ,

$H_{i,j}:=V_i\ast W,W:=W-V_i\ast H_{i,j}$ .

3: PBMinPert solution:

If $H_{j+\mathrm{1,}j}=0$ , then

compute the exact solution $X_j=X_0+Z_j{H}_j^{-1}{E}_1{B}_1$ ; stop.

If $H_{j+\mathrm{1,}j}\ne 0$ , then

compute the Cholesky decomposition $Q^{\text{T}}Q=Z_m^{\text{T}}{Z}_m$ and update $L_m^{-1}$ ,

compute the largest singular value and the associated right singular vector

for