40 lines
2 KiB
Text
40 lines
2 KiB
Text
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In mathematics, the arithmetic geometric mean of two positive real numbers is
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computed by repeatedly taking half their sum and the square root of their
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product until the two numbers converge. For instance, the arithmetic geometric
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mean of 24 and 6 is 13.456171…, with the iterative steps computed as follows:
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0 24 6
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1 15 12
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2 13.5 13.416407864998738175455042
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3 13.458203932499369089227521 13.458139030990984877207090
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4 13.458171481745176983217305 13.458171481706053858316334
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5 13.458171481725615420766820 13.458171481725615420766806
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The arithmetic geometric mean was invented by Lagrange and studied by Gauss, and
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is used today to compute various transcendental functions because it converges
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so quickly.
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In the world of Randall Munroe, the geothmetic meandian of any set of positive
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numbers is computed by iterating three sequences — the arithmetic mean, the
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geometric mean, and the median — until they converge. For instance, the
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geothmetic meandian of the set (1,1,2,3,5) is 2.089, computed as follows:
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1 2.4 1.9743504858348200 2
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2 2.1247834952782734 2.1161924605448084 2
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3 2.0803253186076938 2.0795368194795802 2.1161924605448084
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4 2.0920181995440275 2.0919486049152223 2.0803253186076938
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5 2.0880973743556477 2.0880901331209600 2.0919486049152223
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6 2.0893787041306098 2.0893779142184865 2.0880973743556477
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7 2.0889513309015815 2.0889512436159920 2.0893779142184865
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8 2.0890934962453533 2.0890934865653277 2.0889513309015815
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9 2.0890461045707540 2.0890461034958396 2.0890934865653277
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[ I hate the damned new editor at WordPress; I struggle with it every time I
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post an exercise. I could not figure out how to embed the image from XKCD in
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this blog post. You can see it here. ]
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Your task is to write programs that compute the arithmetic geometric mean and
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geothmetic meandian. When you are finished, you are welcome to read or run a
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suggested solution, or to post your own solution or discuss the exercise in the
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comments below.
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