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Equation 17 · Part 1 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

Symbol Q

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 .
QQ

What this part means

Q is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Its job in the formula

Q is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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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A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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