integer-roots
Integer roots and perfect powers
https://github.com/Bodigrim/integer-roots
| LTS Haskell 22.37: | 1.0.2.0@rev:1 |
| Stackage Nightly 2024-10-08: | 1.0.2.0@rev:1 |
| Latest on Hackage: | 1.0.2.0@rev:1 |
integer-roots-1.0.2.0@sha256:67a8b36c783337cb9f51a83adfc657eb8d7724a12c7b3ba186ba70ff7ce2c3b9,2476Module documentation for 1.0.2.0
- Math
- Math.NumberTheory
integer-roots 
Calculating integer roots and testing perfect powers of arbitrary precision.
Integer square root
The integer square root
(integerSquareRoot)
of a non-negative integer
n
is the greatest integer
m
such that
.
Alternatively, in terms of the
floor function,
.
For example,
> integerSquareRoot 99
9
> integerSquareRoot 101
10
It is tempting to implement integerSquareRoot via sqrt :: Double -> Double:
integerSquareRoot :: Integer -> Integer
integerSquareRoot = truncate . sqrt . fromInteger
However, this implementation is faulty:
> integerSquareRoot (3037000502^2)
3037000501
> integerSquareRoot (2^1024) == 2^1024
True
The problem here is that Double can represent only
a limited subset of integers without precision loss.
Once we encounter larger integers, we lose precision
and obtain all kinds of wrong results.
This library features a polymorphic, efficient and robust routine
integerSquareRoot :: Integral a => a -> a,
which computes integer square roots by
Karatsuba square root algorithm
without intermediate Doubles.
Integer cube roots
The integer cube root
(integerCubeRoot)
of an integer
n
equals to
.
Again, a naive approach is to implement integerCubeRoot
via Double-typed computations:
integerCubeRoot :: Integer -> Integer
integerCubeRoot = truncate . (** (1/3)) . fromInteger
Here the precision loss is even worse than for integerSquareRoot:
> integerCubeRoot (4^3)
3
> integerCubeRoot (5^3)
4
That is why we provide a robust implementation of
integerCubeRoot :: Integral a => a -> a,
which computes roots by
generalized Heron algorithm.
Higher powers
In spirit of integerSquareRoot and integerCubeRoot this library
covers the general case as well, providing
integerRoot :: (Integral a, Integral b) => b -> a -> a
to compute integer k-th roots of arbitrary precision.
There is also highestPower routine, which tries hard to represent
its input as a power with as large exponent as possible. This is a useful function
in number theory, e. g., elliptic curve factorisation.
> map highestPower [2..10]
[(2,1),(3,1),(2,2),(5,1),(6,1),(7,1),(2,3),(3,2),(10,1)]