Copyright | (c) 2016 Andrew Lelechenko |
---|---|

License | MIT |

Maintainer | Andrew Lelechenko <andrew.lelechenko@gmail.com> |

Stability | Provisional |

Portability | Non-portable (GHC extensions) |

Safe Haskell | Safe |

Language | Haskell2010 |

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]] Source #

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]] Source #

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]] Source #

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]] Source #

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]] Source #

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]] Source #

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] Source #

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.