MathObj.FactoredRational

A representation of rational numbers as lists of prime powers, allowing efficient representation, multiplication and division of large numbers, especially of the sort occurring in combinatorial computations.

The module also includes a method for generating factorials in factored form directly, and for generating all divisors of factored integers, or computing Euler's totient (phi) function and the Möbius (mu) function.

Type

data T

The type of factored rationals.

Instances are provided for Eq, Ord, Additive, Ring, ZeroTestable, Real, ToRational, Integral, RealIntegral, ToInteger, and Field.

Note that currently, addition is performed on factored rationals by converting them to normal rationals, performing the addition, and factoring. This could probably be made more efficient by finding a common denominator, pulling out common factors from the numerators, and performing the addition and factoring only on the relatively prime parts.

Instances

Eq T

Defined in MathObj.FactoredRational

Methods

(==) :: T -> T -> Bool

(/=) :: T -> T -> Bool

Ord T

Defined in MathObj.FactoredRational

Methods

compare :: T -> T -> Ordering

(<) :: T -> T -> Bool

(<=) :: T -> T -> Bool

(>) :: T -> T -> Bool

(>=) :: T -> T -> Bool

max :: T -> T -> T

min :: T -> T -> T

Show T

Defined in MathObj.FactoredRational

Methods

showsPrec :: Int -> T -> ShowS

show :: T -> String

showList :: [T] -> ShowS

C T

Defined in MathObj.FactoredRational

Methods

toInteger :: T -> Integer

C T

Defined in MathObj.FactoredRational

Methods

quot :: T -> T -> T

rem :: T -> T -> T

quotRem :: T -> T -> (T, T)

C T

Defined in MathObj.FactoredRational

Methods

toRational :: T -> Rational

C T

Defined in MathObj.FactoredRational

Methods

(/) :: T -> T -> T

recip :: T -> T

fromRational' :: Rational -> T

(^-) :: T -> Integer -> T

C T

Defined in MathObj.FactoredRational

Methods

abs :: T -> T

signum :: T -> T

C T

Defined in MathObj.FactoredRational

Methods

div :: T -> T -> T

mod :: T -> T -> T

divMod :: T -> T -> (T, T)

C T

Defined in MathObj.FactoredRational

Methods

(*) :: T -> T -> T

one :: T

fromInteger :: Integer -> T

(^) :: T -> Integer -> T

C T

Defined in MathObj.FactoredRational

Methods

isZero :: T -> Bool

C T

Defined in MathObj.FactoredRational

Methods

zero :: T

(+) :: T -> T -> T

(-) :: T -> T -> T

negate :: T -> T

Utilities

factorial :: Integer -> T

Efficiently compute n! directly as a factored rational.

eulerPhi :: T -> Integer

Compute Euler's totient function (eulerPhi n is the number of integers less than and relatively prime to n). Only makes sense for (nonnegative) integers.

divisors :: T -> [T]

List of the divisors of n. Only makes sense for integers.

mu :: T -> Integer

Möbius (mu) function of a positive integer: mu(n) = 0 if one or more repeated prime factors, 1 if n = 1, and (-1)^k if n is a product of k distinct primes.