Packing Blocks of B

In the cache optimized GEMM-Operation \(\beta C + \alpha A B \to C\) the matrix \(A\) gets partitioned into blocks of maximal dimension \(M_c \times K_c\). Each block \(B_{l,j}\) of \(B\) gets packed into row-major vertical panels with \(N_r\) columns.

Exercise

Assume that \(p\) is a buffer for \(K_c \cdot N_c\) elements and \(X\) a matrix block with dimension \(k \times n\) where \(k \leq K_c\) and \(n \leq N_c\).

Be honest for your own sake and derive the following algorithm (or any equivalent algorithm) for packing blocks of \(B\):

`dgepack_B(X, p)`

Input:
- matrix \(X = \left(x_{l,j}\right)\) with dimension \(k \times n\) (assume \(k \leq K_c\), \(n \leq N_c\))
- array \(p\) with length \(K_c \cdot N_c\)
On Return:
- \(p\) contains \(X\) packed in vertical row-major panels with \(N_r\) columns

for \(j_1\) with \(0 \leq j_1 < \left\lceil \frac{n}{N_r} \right\rceil\)
- for \(j_0\) with \(0 \leq j_0 < N_r\)
  - for \(l\) with \(0 \leq l < k\)
    - \(j \leftarrow j_1 \cdot N_r + j_0\)
    - \(\nu \leftarrow j_1 \cdot N_r \cdot k + l \cdot N_r + j_0\)
    - if \(j < n\)
      - \(p_\nu \leftarrow x_{l,j}\)
    - else
      - \(p_\nu \leftarrow 0\)

Exercise

Implement and test the following algorithm for packing blocks of \(A\)
Start with an empty source file!