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[Documentation] Fixes the confusion between concatenated vs. composed layout in CuTe documentation (#1498)

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* Update 02_layout_algebra.md

* Update 02_layout_algebra.md
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Manish Gupta 2024-05-02 12:35:12 -07:00 committed by GitHub
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@ -191,21 +191,21 @@ Clearly, CuTe does not use arrays to store shapes or strides and the above code
#### Example 1 -- Reshape a layout into a matrix
`20:2 o (5,4):(4,1)`.
`20:2 o (5,4):(4,1)`. Composition formulation.
This describes interpreting the layout `20:2`
as a 5x4 matrix in a row-major order.
1. ` = 20:2 o (5:4,4:1)`. Concatenation of sublayouts.
1. ` = 20:2 o (5:4,4:1)`. Layout `(5,4):(4,1)` as concatenation of sublayouts.
2. ` = (20:2 o 5:4, 20:2 o 4:1)`. Left distributivity.
* `20:2 o 5:4 => 5:8`. Trivial case.
* `20:2 o 4:1 => 4:2`. Trivial case.
3. ` = (5:8, 4:2)`.
3. ` = (5:8, 4:2)`. Composed Layout as concatenation of sublayouts.
4. ` = (5,4):(8,2)`. Concatenation of sublayouts.
4. ` = (5,4):(8,2)`. Final composed layout.
#### Example 2 -- Reshape a layout into a matrix
@ -214,18 +214,18 @@ as a 5x4 matrix in a row-major order.
This describes interpreting the layout `(10,2):(16,4)`
as a 5x4 matrix in a column-major order.
1. ` = (10,2):(16,4) o (5:1,4:5)`. Concatenation of sublayouts.
1. ` = (10,2):(16,4) o (5:1,4:5)`. Layout `(5,4):(1,5)` as concatenation of sublayouts.
2. ` = ((10,2):(16,4) o 5:1, (10,2):(16,4) o 4:5)`. Left distributivity.
* `(10,2):(16,4) o 5:1 => (5,1):(16,4)`. Mod out the shape `5`.
* `(10,2):(16,4) o 4:5 => (2,2):(80,4)`. Div out the stride `5`.
3. ` = ((5,1):(16,4), (2,2):(80,4))`. Collect results.
3. ` = ((5,1):(16,4), (2,2):(80,4))`. Composed Layout as concatenation of sublayouts.
4. ` = (5:16, (2,2):(80,4))`. By-mode coalesce.
5. ` = (5,(2,2))):(16,(80,4))`. Concatenation of sublayouts.
5. ` = (5,(2,2))):(16,(80,4))`. Final composed layout.
We get exactly this result with CuTe
if we use compile-time shapes and strides.