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

License | MIT |

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

Stability | Provisional |

Portability | Non-portable (GHC extensions) |

Safe Haskell | None |

Language | Haskell2010 |

Calculating integer roots, modular powers and related things. This module reexports the most needed functions from the implementation modules. The implementation modules provide some additional functions, in particular some unsafe functions which omit some tests for performance reasons.

- integerSquareRoot :: Integral a => a -> a
- isSquare :: Integral a => a -> Bool
- exactSquareRoot :: Integral a => a -> Maybe a
- integerCubeRoot :: Integral a => a -> a
- isCube :: Integral a => a -> Bool
- exactCubeRoot :: Integral a => a -> Maybe a
- integerFourthRoot :: Integral a => a -> a
- isFourthPower :: Integral a => a -> Bool
- exactFourthRoot :: Integral a => a -> Maybe a
- integerRoot :: (Integral a, Integral b) => b -> a -> a
- isKthPower :: (Integral a, Integral b) => b -> a -> Bool
- exactRoot :: (Integral a, Integral b) => b -> a -> Maybe a
- isPerfectPower :: Integral a => a -> Bool
- highestPower :: Integral a => a -> (a, Int)
- powerMod :: (Integral a, Bits a) => Integer -> a -> Integer -> Integer

# Integer Roots

## Square roots

integerSquareRoot :: Integral a => a -> a Source

Calculate the integer square root of a nonnegative number `n`

,
that is, the largest integer `r`

with `r*r <= n`

.
Throws an error on negative input.

isSquare :: Integral a => a -> Bool Source

Test whether the argument is a square.
After a number is found to be positive, first `isPossibleSquare`

is checked, if it is, the integer square root is calculated.

exactSquareRoot :: Integral a => a -> Maybe a Source

Returns `Nothing`

if the argument is not a square,

if `Just`

r`r*r == n`

and `r >= 0`

. Avoids the expensive calculation
of the square root if `n`

is recognized as a non-square
before, prevents repeated calculation of the square root
if only the roots of perfect squares are needed.
Checks for negativity and `isPossibleSquare`

.

## Cube roots

integerCubeRoot :: Integral a => a -> a Source

Calculate the integer cube root of an integer `n`

,
that is the largest integer `r`

such that `r^3 <= n`

.
Note that this is not symmetric about `0`

, for example
`integerCubeRoot (-2) = (-2)`

while `integerCubeRoot 2 = 1`

.

exactCubeRoot :: Integral a => a -> Maybe a Source

Returns `Nothing`

if the argument is not a cube,
`Just r`

if `n == r^3`

.

## Fourth roots

integerFourthRoot :: Integral a => a -> a Source

Calculate the integer fourth root of a nonnegative number,
that is, the largest integer `r`

with `r^4 <= n`

.
Throws an error on negaitve input.

isFourthPower :: Integral a => a -> Bool Source

Test whether an integer is a fourth power. First nonnegativity is checked, then the unchecked test is called.

exactFourthRoot :: Integral a => a -> Maybe a Source

Returns `Nothing`

if `n`

is not a fourth power,
`Just r`

if `n == r^4`

and `r >= 0`

.

## General roots

integerRoot :: (Integral a, Integral b) => b -> a -> a Source

Calculate an integer root,

computes the (floor of) the `integerRoot`

k n`k`

-th
root of `n`

, where `k`

must be positive.
`r = `

means `integerRoot`

k n`r^k <= n < (r+1)^k`

if that is possible at all.
It is impossible if `k`

is even and `n < 0`

, since then `r^k >= 0`

for all `r`

,
then, and if `k <= 0`

,

raises an error. For `integerRoot`

`k < 5`

, a specialised
version is called which should be more efficient than the general algorithm.
However, it is not guaranteed that the rewrite rules for those fire, so if `k`

is
known in advance, it is safer to directly call the specialised versions.

isKthPower :: (Integral a, Integral b) => b -> a -> Bool Source

checks whether `isKthPower`

k n`n`

is a `k`

-th power.

isPerfectPower :: Integral a => a -> Bool Source

checks whether `isPerfectPower`

n`n == r^k`

for some `k > 1`

.

highestPower :: Integral a => a -> (a, Int) Source

produces the pair `highestPower`

n`(b,k)`

with the largest
exponent `k`

such that `n == b^k`

, except for

,
in which case arbitrarily large exponents exist, and by an
arbitrary decision `abs`

n <= 1`(n,3)`

is returned.

First, by trial division with small primes, the range of possible
exponents is reduced (if `p^e`

exactly divides `n`

, then `k`

must
be a divisor of `e`

, if several small primes divide `n`

, `k`

must
divide the greatest common divisor of their exponents, which mostly
will be `1`

, generally small; if none of the small primes divides
`n`

, the range of possible exponents is reduced since the base is
necessarily large), if that has not yet determined the result, the
remaining factor is examined by trying the divisors of the `gcd`

of the prime exponents if some have been found, otherwise by trying
prime exponents recursively.

powerMod :: (Integral a, Bits a) => Integer -> a -> Integer -> Integer Source

Modular power.

powerMod base exponent modulus

calculates `(base ^ exponent) `mod` modulus`

by repeated squaring and reduction.
If `exponent < 0`

and `base`

is invertible modulo `modulus`

, `(inverse ^ |exponent|) `mod` modulus`

is calculated. This function does some input checking and sanitation before calling the unsafe worker.