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module AufgabeFFP2
where
-- 1
primes :: [Integer]
primes = sieve [2 ..]
-- sieve of eratosthenes (according to lecture slides)
sieve :: [Integer] -> [Integer]
sieve (x:xs) = x : sieve [ y | y <- xs, mod y x > 0]
-- generate all pairs with map and then filter only the valid ones
-- TODO: filtering should be optimized, still a bit slow
pps :: [(Integer, Integer)]
pps = filter validPair $ map (\x -> (x,x+2)) primes
where
-- pair is valid if the second component n is a prime
-- since simply checking if the Integer is element of the prime-list
-- does not terminate for non-primes, we take the first n primes
validPair :: (Integer,Integer) -> Bool
validPair (_,maybePrime) = elem maybePrime $ take (fromInteger maybePrime) primes
-------------------------------------------------------------------------------
-- 2
-- generates powers of 2
pof2s :: [Integer]
pof2s = [1] ++ (map (2*) pof2s)
pow :: Int -> Integer
pow 0 = 1
pow n = pow (n-1) + pow (n-1)
-- uses memtable for power-calculation
powFast :: Int -> Integer
powFast 0 = 1
powFast n = pof2s !! (n-1) + pof2s !! (n-1)
-------------------------------------------------------------------------------
-- 3
-- power series of N
pofNs :: Integer -> [Integer]
pofNs n = [1] ++ (map (n*) $ pofNs n)
-- faculty function
fac :: Integer -> Integer
fac n = product [1..n]
-- stream of faculties
facGen :: [Integer]
facGen = map (fac) [1..]
-- function g with memoization (using hFast)
fMT :: Int -> Int -> Float
fMT z k = g z k hMT
-- function g without memoization (uning hSlow)
f :: Int -> Int -> Float
f z k = g z k h
-- actual function g (converts Int to Integer for more precision)
g :: Int -> Int -> (Integer -> Integer -> Float) -> Float
g z k h = sum $ map (h $ fromIntegral z) [1..(fromIntegral k)]
-- helper function h using mem-table for the power-series (z^i) and for faculty (i!)
hMT :: Integer -> Integer -> Float
hMT z i = (fromInteger $ pofNs z !! (fromInteger i)) / (fromInteger $ facGen !! ((fromIntegral i)-1))
-- helper function h without memoization
h :: Integer -> Integer -> Float
h z i = (fromInteger $ z^i) / (fromInteger $ fac i)
-------------------------------------------------------------------------------
-- 4
-- gets the digits of an integer as a list
digits :: Integer -> [Integer]
digits x
| x<=0 = []
| otherwise = (digits $ x `div` 10)++[x `mod` 10]
-- calculates the goedel-number for the given integer
-- returns 0 for non-positive numbers
gz :: Integer -> Integer
gz n
| n<=0 = 0
| otherwise = product $ zipWith (^) primes (digits n)
-- goedel-number generator
gzs :: [Integer]
gzs = map (gz) [1..]
|
