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Equation 19 · Part 3 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

Symbol M

Qmin⁡  ≈  23 N3M.Q_{\min} \;\approx\; 2\sqrt{3}\,\frac{N^3}{\sqrt{M}} .
MM

What this part means

M occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Its job in the formula

M occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Taking b as large as the constraint allows gives Qmin⁡  ≈  23 N3MQ_{\min} \;\approx\; 2\sqrt{3}\,\frac{N^3}{\sqrt{M}} . Two things in that expression matter more than the constant. First, traffic falls linearly in b : doubling the tile halves the bytes moved, which is why tiling is the single highest-leverage transformation in dense linear algebra. Second, and less often noticed, traffic falls only as M−1/2M^{-1/2} . Quadrupling the fast store buys a factor of two in traffic, not a factor of four. That square root is the reason architects cannot simply spend their way out of the problem with more SRAM, and it is worth holding on to when reading any claim that a larger cache will fix a bandwidth-bound kernel.

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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