Equation 12 · Part 1 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read
Symbol N^2
What this part means
the square of N; spectacularly compute-bound in principle.
Its job in the formula
is squared: multiply the underlying quantity by itself. The square is a mathematical operation, not a second independent variable.
Full expression→Symbol N^2→Article meaning
Where the article explains it
Multiplying two N N matrices performs about 2 operations on 3 words — an intensity proportional to N , which for large N is spectacularly compute-bound in principle.
The passage around this formula
Matrix multiplication is the canonical demonstration because it has enormous latent reuse and a naive schedule that throws almost all of it away. Multiplying two N N matrices performs about 2 operations on 3 words — an intensity proportional to N , which for large N is spectacularly compute-bound in principle. Executed as three nested loops in the obvious order, though, each element of one input is re-read from main memory on every pass, and the realised intensity collapses toward a small constant. The available reuse and the exploited reuse are different quantities, and only the second appears in the roofline.
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 →
The article lists its research sources here.