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Equation 17 · Part 8 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

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Q(b)  ≈  2N3b+N2,subject to3b2≤M.Q(b) \;\approx\; \frac{2N^3}{b} + N^2, \qquad \text{subject to} \quad 3b^2 \le M .
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What this part means

Approximately equal to; the equality is not exact.

Its job in the formula

Approximately equal to; the equality is not exact.

The passage around this formula

The repair is to restructure the loops so that the working set of the innermost computation is small enough to sit in a fast store and is used many times before being evicted. Partition the operands into b ×\times b blocks and compute block by block. Three such blocks — two inputs and an accumulator — must be resident simultaneously, so the tile size is bounded by the fast-store capacity M in words: Q(b)  ≈  2N3b+N2,subject to3b2≤MQ(b) \;\approx\; \frac{2N^3}{b} + N^2, \qquad \text{subject to} \quad 3b^2 \le M . Taking b as large as the constraint allows gives

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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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