Higher roots and r-means

The previous post looked at a simple method of finding square roots that amounts to a special case of Newton's method, though it is much older than Newton's method.

We can extend Newton's method to find cube roots and nth roots in general. And when we do, we begin to see a connection to r-means. I've written about these means several times, most recently in connection with finding the perimeter of an ellipse and the surface area of an ellipsoid.

To find the nth root of y, we apply Newton's root-finding method to find where the function

is zero. We start with an initial estimate x₀ and our updated estimate is

When n = 2, this reduces to the method in our previous post: the updated estimate x₁ equals the average of our initial estimate x₀ and y/x₀. That is, our updated estimate is the arithmetic mean of x₀ and y/x₀, and the geometric mean of the two terms is the square root of y.

For n in general, we have two terms whose geometric mean is the nth root of y, and we take their weighted arithmetic mean. Said another way, our updated estimate is a convex combination of these two terms. The rest of the post will explore this further and point out some connections.

Weighted means and geometric mean

The ellipse and ellipsoid posts mention above make use of the means

as defined in Hardy, Littlewood, and Polya. More generally the authors define

where the weights p_i are positive numbers that sum to 1. The unweighted mean corresponds to the special case where all p_i equal 1/n.

The authors also define the geometric mean

and weighted extensions which we will not need here.

Connections between means

We note two connections between the geometric mean and the r-mean. First, the geometric mean is the exponential of the arithmetic mean of the logarithms.

Second, the geometric mean is the limit of the r-means as r goes to 0, and so you could define the r-mean with r = 0 to be the geometric mean.

Newton's method

Reading Newton's method for nth roots in terms of r means, we take an initial estimate x₀ and an auxiliary estimate x₀ such that their geometric mean is the nth root we're after. Then we take the arithmetic mean of x₀ and x₀ with weights (n-1)/n and 1/n.

That is, let

and solve for x_0' such that

Then

where

If we write the geometric mean as the r-mean with r = 0, we could describe Newton's method entirely in terms of r-means.

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