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The phrase "nearly linear time" come from the paper.

However, the paper gives a better definition:

When we refer to a nearly linear time SDD system solver we mean an an algorithm that computes such a x in time O(mlog^c(n)logε⁻¹) where m is the number of nonzero entries in A and c ≥ 0 ∈ ℝ is a fixed constant.

The term "nearly linear time" is fine. The next part of that sentence is where the problem arises: it suggests that the opposite of "nearly linear", and the status quo, is that running time "increase[s] exponentially". But the existing standard algorithms already scale polynomially, not exponentially.

This article is probably not written by the CS department!

Does everything have to turn into a dick-waving contest around here? Just send a correction if you care.

Looks like somebody else doesn't know what near-linear means.

In colloquial terminology, x^2 is an exponential. This isn't actually an important part of the article. I don't feel that a paragraph explaining why "Polynomial time" is bad (but not too bad) would improve the article.