Now suppose that I get one new data point, y_k+1. The new value of u₁ is:

(28)

Here I'm using the '+' superscript to indicate the updated version of u₁. Looking at the two values, it's clear that:

(29)

You can implement this algorithm very efficiently by keeping the y's in a circular buffer. Subtract the oldest value, replace it with the newest one, and add that one to the sum.

Computing u₂, the second term in U, is trickier because of our need to deal with the new offset in the x-values. Before the update, u₂ will be:

(30)

After processing the new data point, it's going to be:

(31)

Subtracting the "before" and "after" forms tells us what we must do to update the summation. We get:

(32)

Now look at the sum inside the parentheses in Equation 32. It looks suspiciously like the sum in u₁⁺. In fact, we can write:

(33)

Therefore, I can write:

(34)