Symbol N^3
is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
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Why this formula appears here
Matrix multiplication is the canonical demonstration because it has enormous latent reuse and a naive schedule that throws almost all of it away. Multiplying two N N matrices performs about 2 operations on 3 words — an intensity proportional to N , which for large N is spectacularly compute-bound in principle. Executed as three nested loops in the obvious order, though, each element of one input is re-read from main memory on every pass, and the realised intensity collapses toward a small constant. The available reuse and the exploited reuse are different quantities, and only the second appears in the roofline.
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Equation 11 · AI Hardware & Semiconductors
Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Matrix multiplication is the canonical demonstration because it has enormous latent reuse and a naive schedule that throws almost all of it away. Multiplying two N N matrices performs about 2 operations on 3 words — an intensity proportional to N , which for large N is spectacularly compute-bound in principle. Executed as three nested loops in the obvious order, though, each element of one input is re-read from main memory on every pass, and the realised intensity collapses toward a small constant. The available reuse and the exploited reuse are different quantities, and only the second appears in the roofline.
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