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Equation 19 · Part 7 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

subscript

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

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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 subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

Sources cited in the article section

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