15. Cuboid formula

• Notations :

  • R = rank
  • index: (i0,…,i_{R-1})
  • Lengths: (L0, … , L_{R-1})
  • Strides: (S0, … , S_{R-1})
  • position = start_shift + sum_{j=0}^{R-1} i_j S_j

• Strides for compact cuboid:

If \(\sigma(i)\) is the i-th index in memory (0 the first, R-1 the last one), we have

\begin{align*} S_{\sigma(0)} &= 1\\ S_{\sigma(1)} &= L_{\sigma(0)} S_{\sigma(0)} = L_{\sigma(0)}\\ S_{\sigma(i)} &= L_{\sigma(i-1)} S_{\sigma(i-1)} = \prod_{j=i-1}^{0} L_{\sigma(j)} \end{align*}

• Slicing the cuboid:

• Argument of the slice: (R0, R1, …, R_{R-1}), with R_i = range: (start,end,step) for the moment, if R_i = E_i, a number, consider it as a special range of length 1.
• new index (k0,…., k_{R-1}). i_u = R_u.start + k_u * R_u.step
• Compute the new strides: Sn_j

\def\offset{\text{offset}} \begin{align*} position &= \offset + \sum_{j=0}^{R-1} i_j S_j \\ &= \offset + \sum_{j=0}^{R-1} (k_j* R_j.step + R_j.start) S_j \\ &= (\offset + \sum_{j=0}^{R-1} (k_j* R_j.step + R_j.start) S_j ) + \sum_{j=0}^{R-1} k_j (S_j * R_j.start) \\ \offset_\text{new} &= \offset + \sum_{j=0}^{R-1} (k_j* R_j.step + R_j.start) S_j \\ &= position( R_j.start) \\ Sn_j &= S_j * R_j.start \end{align*}

• Now: for the cases R_i = E_i, just drop the index.