Equation 6 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

What does this equation mean?

I^{*} = P_{\max}/B

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Inputs and operationsP_max/B

Result or conditionI^*

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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I^{*}

Symbol I^*

$I^*$ is the quantity selected or evaluated by the optimization written on the right.

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P_{\max}

Symbol P_max

the peak arithmetic rate.

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B

Symbol B

the achievable bandwidth at the level that dominates.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subscript

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.

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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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How to interpret it

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What the article says around this equation

with $P_{\max}$ the peak arithmetic rate, I the arithmetic intensity in operations per byte, and B the achievable bandwidth at the level that dominates. The two terms cross at a ridge point $I^{*}$ = $P_{\max}$ /B . Below it a kernel is memory-bound and its runtime is set by traffic; above it the kernel is compute-bound and traffic is hidden.

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