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Equation 19 · Part 9 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

Denominator: sqrtM

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

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

sqrtM 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 fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.