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Stephen Seo 2021-08-18 19:26:56 +09:00
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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.