← All parts of this equation

Equation 17 · Part 3 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

Symbol N^3

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 .
N3N^3

What this part means

N3N^3 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Its job in the formula

N3N^3 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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

Read this part in the article →

Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

Open the illustrated exponents: repeated multiplication and powers guide →

See this notation across published equations →

Sources cited in the article section

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