Geothmetic Meandian

April 13, 2021

In mathematics, the arithmetic geometric mean of two positive real numbers is computed by repeatedly taking half their sum and the square root of their product until the two numbers converge. For instance, the arithmetic geometric mean of 24 and 6 is 13.456171…, with the iterative steps computed as follows:

0  24                            6
1  15                           12
2  13.5                         13.416407864998738175455042
3  13.458203932499369089227521  13.458139030990984877207090
4  13.458171481745176983217305  13.458171481706053858316334
5  13.458171481725615420766820  13.458171481725615420766806

The arithmetic geometric mean was invented by Lagrange and studied by Gauss, and is used today to compute various transcendental functions because it converges so quickly.

In the world of Randall Munroe, the geothmetic meandian of any set of positive numbers is computed by iterating three sequences — the arithmetic mean, the geometric mean, and the median — until they converge. For instance, the geothmetic meandian of the set (1,1,2,3,5) is 2.089, computed as follows:

1  2.4                1.9743504858348200 2
2  2.1247834952782734 2.1161924605448084 2
3  2.0803253186076938 2.0795368194795802 2.1161924605448084
4  2.0920181995440275 2.0919486049152223 2.0803253186076938
5  2.0880973743556477 2.0880901331209600 2.0919486049152223
6  2.0893787041306098 2.0893779142184865 2.0880973743556477
7  2.0889513309015815 2.0889512436159920 2.0893779142184865
8  2.0890934962453533 2.0890934865653277 2.0889513309015815
9  2.0890461045707540 2.0890461034958396 2.0890934865653277

[ I hate the damned new editor at WordPress; I struggle with it every time I post an exercise. I could not figure out how to embed the image from XKCD in this blog post. You can see it here. ]

Your task is to write programs that compute the arithmetic geometric mean and geothmetic meandian. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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