Figure 13

The vertical decomposition in the Outside VMT algorithm, for the case where | I | ≤ | J |. (a) A_{I, J}sustains the pre-condition (see Figure 12 for notations). (b) A_{I, J}is partitioned vertically to $A_{I,J_1}$ and $A_{I,J_2}$, where $A_{I,J_1}$ is computed recursively. (c) If μ_{i, j}is of the form ⊕ _q∈(j,n](α_{i, q}⊗ β_{j, q}), then the pre-condition for computing $A_{I,J_2}$ is established, by updating $A_{I,J_2}$ to be $\left(A_{I,L}\otimes {\left({\beta }_{J_2,L}\right)}^T\right)\oplus A_{I,J_2}$ (in this example L = J₁, since I ends before J₁ starts. The corresponding sub-matrix of β is not shown). In the case where μ_{i, j}= ⊕_q∈[0,i)(β_{q, i}⊗ α_{q, j}), the pre-condition is met without any need for performing a matrix multiplication. Then, $A_{I,J_2}$ is computed recursively (not shown).