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

3: Solve the generalized eigenvalue problem

(2.8) $L_m^{\text{T}}{L}_m{z}_i=\lambda G_m^{\text{T}}{G}_m{z}_i,$

to get the s smallest generalized eigenvalues, where $L_m$ and $G_m$ is defined in (2.7).

4: Scale block components of Z as shown in theorem 2.1 and form $X_m=X_0+M^{-1}{\mathcal{V}}_m{Y}_m$ .