Bernoulli numbers: Difference between revisions
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=={{header|Haskell}}== |
=={{header|Haskell}}== |
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====Task algorithm==== |
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This program works as a command line utility, that reads from stdin the number of elements to compute (default 60) and prints them in stdout. |
This program works as a command line utility, that reads from stdin the number of elements to compute (default 60) and prints them in stdout. |
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The implementation of the algorithm is in the function bernoullis. The rest is for printing the results. |
The implementation of the algorithm is in the function bernoullis. The rest is for printing the results. |
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{{Out}} |
{{Out}} |
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<pre>B(0) = 1/1 |
<pre>B(0) = 1/1 |
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B(1) = 1/2 |
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B(2) = 1/6 |
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B(4) = -1/30 |
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B(6) = 1/42 |
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B(8) = -1/30 |
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B(10) = 5/66 |
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B(12) = -691/2730 |
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B(14) = 7/6 |
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B(16) = -3617/510 |
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B(18) = 43867/798 |
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B(20) = -174611/330 |
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B(22) = 854513/138 |
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B(24) = -236364091/2730 |
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B(26) = 8553103/6 |
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B(28) = -23749461029/870 |
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B(30) = 8615841276005/14322 |
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B(32) = -7709321041217/510 |
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B(34) = 2577687858367/6 |
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B(36) = -26315271553053477373/1919190 |
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B(38) = 2929993913841559/6 |
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B(40) = -261082718496449122051/13530 |
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B(42) = 1520097643918070802691/1806 |
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B(44) = -27833269579301024235023/690 |
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B(46) = 596451111593912163277961/282 |
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B(48) = -5609403368997817686249127547/46410 |
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B(50) = 495057205241079648212477525/66 |
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B(52) = -801165718135489957347924991853/1590 |
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B(54) = 29149963634884862421418123812691/798 |
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B(56) = -2479392929313226753685415739663229/870 |
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B(58) = 84483613348880041862046775994036021/354 |
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B(60) = -1215233140483755572040304994079820246041491/56786730</pre> |
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====Derivation from Faulhaber's triangle==== |
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<lang haskell>import Data.Ratio (Ratio, numerator, denominator, (%)) |
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bernouilliNumbers :: Integer -> [Rational] |
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bernouilliNumbers = |
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fmap head . |
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tail . |
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scanl |
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(\rs n -> |
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let xs = zipWith ((*) . (n %)) [2 ..] rs |
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in 1 - sum xs : xs) |
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[] . |
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enumFromTo 0 |
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main :: IO () |
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main = |
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(putStrLn . unlines) |
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(concat $ |
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zipWith |
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(\i x -> |
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let n = numerator x |
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in [ concat ["B(", show i, ") = ", show n, "/", show (denominator x)] |
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| n /= 0 ]) |
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[0 ..] |
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(bernouilliNumbers 60))</lang> |
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{{Out}} |
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<pre> |
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maybe Monad |
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identity.scpt |
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js-001.scpt |
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js-002.scpt |
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js-003.scpt |
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js-004.scpt |
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jsMaybe.scpt |
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maybeAS-001.scpt |
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maybeFunction-001.scpt |
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maybeJS-005.scpt |
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setComprehension.scpt |
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setQuery-001.scpt |
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maybeFunction-001.scpt |
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TP2OG-022.scpt |
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TP2OG-024.scpt |
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bitPopulationJS-002.scpt |
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Untitled |
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Main.hs |
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tmp.hs |
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Settings |
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faulHaberES6-010.scpt |
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1 |
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13 |
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14 |
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15 |
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16 |
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17 |
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18 |
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19 |
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20 |
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21 |
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22 |
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23 |
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24 |
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7 |
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8 |
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9 |
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10 |
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11 |
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12 |
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1 |
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2 |
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3 |
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4 |
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5 |
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6 |
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main :: IO () |
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main = |
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(putStrLn . unlines) |
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(concat $ |
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zipWith |
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(\i x -> |
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let n = numerator x |
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in [ concat ["B(", show i, ") = ", show n, "/", show (denominator x)] |
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| n /= 0 ]) |
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[0 ..] |
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(bernouilliNumbers 60)) |
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scanl |
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(\rs n -> |
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let xs = zipWith ((*) . (n %)) [2 ..] rs |
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in 1 - sum xs : xs) |
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[] . |
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enumFromTo 0 |
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import Data.Ratio (Ratio, numerator, denominator, (%)) |
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bernouilliNumbers :: Integer -> [Rational] |
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bernouilliNumbers = |
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fmap head . |
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tail . |
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Haskell - Main.hs:22 |
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B(0) = 1/1 |
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B(1) = 1/2 |
B(1) = 1/2 |
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B(2) = 1/6 |
B(2) = 1/6 |
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