Copyright | (c) 2011 Daniel Fischer |
---|---|

License | MIT |

Maintainer | Daniel Fischer <daniel.is.fischer@googlemail.com> |

Stability | Provisional |

Portability | Non-portable (GHC extensions) |

Safe Haskell | Safe |

Language | Haskell2010 |

Efficient calculation of Lucas sequences.

# Documentation

fibonacci :: Int -> Integer Source #

calculates the `fibonacci`

k`k`

-th Fibonacci number in
*O*(`log (abs k)`

) steps. The index may be negative. This
is efficient for calculating single Fibonacci numbers (with
large index), but for computing many Fibonacci numbers in
close proximity, it is better to use the simple addition
formula starting from an appropriate pair of successive
Fibonacci numbers.

fibonacciPair :: Int -> (Integer, Integer) Source #

returns the pair `fibonacciPair`

k`(F(k), F(k+1))`

of the `k`

-th
Fibonacci number and its successor, thus it can be used to calculate
the Fibonacci numbers from some index on without needing to compute
the previous. The pair is efficiently calculated
in *O*(`log (abs k)`

) steps. The index may be negative.

lucasPair :: Int -> (Integer, Integer) Source #

computes the pair `lucasPair`

k`(L(k), L(k+1))`

of the `k`

-th
Lucas number and its successor. Very similar to

.`fibonacciPair`

generalLucas :: Integer -> Integer -> Int -> (Integer, Integer, Integer, Integer) Source #

calculates the quadruple `generalLucas`

p q k`(U(k), U(k+1), V(k), V(k+1))`

where `U(i)`

is the Lucas sequence of the first kind and `V(i)`

the Lucas
sequence of the second kind for the parameters `p`

and `q`

, where `p^2-4q /= 0`

.
Both sequences satisfy the recurrence relation `A(j+2) = p*A(j+1) - q*A(j)`

,
the starting values are `U(0) = 0, U(1) = 1`

and `V(0) = 2, V(1) = p`

.
The Fibonacci numbers form the Lucas sequence of the first kind for the
parameters `p = 1, q = -1`

and the Lucas numbers form the Lucas sequence of
the second kind for these parameters.
Here, the index must be non-negative, since the terms of the sequence for
negative indices are in general not integers.