fix: align S[...] notation with address formula in Tile Layout section

[xush6528]

The dimension names were listed as (tile_row, row_in_tile, tile_col, col_in_tile) but the address formula below computes in the order (tile_row, tile_col, row_in_tile, col_in_tile) with strides (16, 8, 4, 1).

Updated S[(4, 2, 2, 4) : (16, 4, 8, 1)] to S[(4, 2, 2, 4) : (16, 8, 4, 1)] and reordered the dimension names to match.

[gemini-code-assist]

Code Review

This pull request corrects the coordinate order and corresponding strides for a 4-D layout representing tiled matrices in the documentation. Specifically, it changes the coordinate order from (tile_row, row_in_tile, tile_col, col_in_tile) to (tile_row, tile_col, row_in_tile, col_in_tile) and updates the strides accordingly to ensure the tiles remain contiguous. There are no review comments, so I have no feedback to provide.

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[rock42069]

You should make the change here as well.

modern-gpu-programming-for-mlsys/_extra/demo/tiled_layout.html

Line 23 in b04879c

<h1>S[(4, 2, 2, 4) : (16, 4, 8, 1)]</h1>

[tlopex]

@xush6528 @rock42069 Thanks so much for catching this, we really appreciate it!
However, we think we should fix slightly different.

The intended convention follows the default tiling rule(like numpy,PyTorch and TIRx): each original dimension is split into an outer and inner coordinate before moving on to the next dimension, i.e.

(outer_dim0, inner_dim0, outer_dim1, inner_dim1, ...)

For a 2D matrix tiled by 2x4, this gives:

(tile_row, row_in_tile, tile_col, col_in_tile)

So the layout should remain:

S[(4, 2, 2, 4) : (16, 4, 8, 1)]

The issue is that the address demo currently displays the terms in a different order, which makes it look as if the layout were using:

(tile_row, tile_col, row_in_tile, col_in_tile)

Thanks again for pointing this out! We’ll fix it in the upcoming Chinese-version PR for the layout chapter.