Math.NumberTheory.Recurrencies.Bilinear

Bilinear recurrent sequences and Bernoulli numbers, roughly covering Ch. 5-6 of Concrete Mathematics by R. L. Graham, D. E. Knuth and O. Patashnik.

Note on memory leaks and memoization. Top-level definitions in this module are polymorphic, so the results of computations are not retained in memory. Make them monomorphic to take advantages of memoization. Compare

> :set +s
> binomial !! 1000 !! 1000 :: Integer
1
(0.01 secs, 1,385,512 bytes)
> binomial !! 1000 !! 1000 :: Integer
1
(0.01 secs, 1,381,616 bytes)

against

> let binomial' = binomial :: [[Integer]]
> binomial' !! 1000 !! 1000 :: Integer
1
(0.01 secs, 1,381,696 bytes)
> binomial' !! 1000 !! 1000 :: Integer
1
(0.01 secs, 391,152 bytes)

Documentation

binomial :: Integral a => [[a]]

Infinite zero-based table of binomial coefficients (also known as Pascal triangle): binomial !! n !! k == n! / k! / (n - k)!.

> take 5 (map (take 5) binomial)
[[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1]]

Complexity: binomial !! n !! k is O(n) bits long, its computation takes O(k n) time and forces thunks binomial !! n !! i for 0 <= i <= k. Use the symmetry of Pascal triangle binomial !! n !! k == binomial !! n !! (n - k) to speed up computations.

One could also consider binomial to compute stand-alone values.

stirling1 :: (Num a, Enum a) => [[a]]

Infinite zero-based table of Stirling numbers of the first kind.

> take 5 (map (take 5) stirling1)
[[1],[0,1],[0,1,1],[0,2,3,1],[0,6,11,6,1]]

Complexity: stirling1 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks stirling1 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

One could also consider unsignedStirling1st to compute stand-alone values.

stirling2 :: (Num a, Enum a) => [[a]]

Infinite zero-based table of Stirling numbers of the second kind.

> take 5 (map (take 5) stirling2)
[[1],[0,1],[0,1,1],[0,1,3,1],[0,1,7,6,1]]

Complexity: stirling2 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks stirling2 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

One could also consider stirling2nd to compute stand-alone values.

lah :: Integral a => [[a]]

Infinite one-based table of Lah numbers. lah !! n !! k equals to lah(n + 1, k + 1).

> take 5 (map (take 5) lah)
[[1],[2,1],[6,6,1],[24,36,12,1],[120,240,120,20,1]]

Complexity: lah !! n !! k is O(n ln n) bits long, its computation takes O(k n ln n) time and forces thunks lah !! n !! i for 0 <= i <= k.

eulerian1 :: (Num a, Enum a) => [[a]]

Infinite zero-based table of Eulerian numbers of the first kind.

> take 5 (map (take 5) eulerian1)
[[],[1],[1,1],[1,4,1],[1,11,11,1]]

Complexity: eulerian1 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks eulerian1 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

eulerian2 :: (Num a, Enum a) => [[a]]

Infinite zero-based table of Eulerian numbers of the second kind.

> take 5 (map (take 5) eulerian2)
[[],[1],[1,2],[1,8,6],[1,22,58,24]]

Complexity: eulerian2 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks eulerian2 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

bernoulli :: Integral a => [Ratio a]

Infinite zero-based sequence of Bernoulli numbers, computed via connection with stirling2.

> take 5 bernoulli
[1 % 1,(-1) % 2,1 % 6,0 % 1,(-1) % 30]

Complexity: bernoulli !! n is O(n ln n) bits long, its computation takes O(n^3 ln n) time and forces thunks stirling2 !! i !! j for 0 <= i <= n and 0 <= j <= i.

One could also consider bernoulli to compute stand-alone values.