Factory.Math.Implementations.Factorial

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Provides implementations of the class Algorithmic.
• Provides additional functions related to factorials, but which depends on a specific implementation, and which therefore can't be accessed throught the class-interface.
• https://en.wikipedia.org/wiki/Factorial.
• http://mathworld.wolfram.com/Factorial.html.
• http://www.luschny.de/math/factorial/FastFactorialFunctions.htm.

Types

Data-types

data Algorithm

The algorithms by which factorial has been implemented.

Constructors

Bisection

The integers from which the factorial is composed, are multiplied using Data.Interval.product'.

PrimeFactorisation

The prime factors of the factorial are extracted, then raised to the appropriate power, before multiplication.

Instances

Eq Algorithm

Methods

(==) :: Algorithm -> Algorithm -> Bool

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

Read Algorithm

Methods

readsPrec :: Int -> ReadS Algorithm

readList :: ReadS [Algorithm]

readPrec :: ReadPrec Algorithm

readListPrec :: ReadPrec [Algorithm]

Show Algorithm

Methods

showsPrec :: Int -> Algorithm -> ShowS

show :: Algorithm -> String

showList :: [Algorithm] -> ShowS

Default Algorithm

Methods

def :: Algorithm

Algorithmic Algorithm

Methods

factorial :: (Integral i, Show i) => Algorithm -> i -> i

Functions

primeFactors

Arguments

:: Integral base

=> base

The number, whose factorial is to be factorised.

-> Factors base base

The base and exponent of each prime factor in the factorial, ordered by increasing base (and decreasing exponent).

• Returns the prime factors, of the factorial of the specifed integer.
• Precisely all the primes less than or equal to the specified integer n, are included in n!; only the multiplicity of each of these known prime components need be determined.
• https://en.wikipedia.org/wiki/Factorial#Number_theory.
• CAVEAT: currently a hotspot.

risingFactorial

Arguments

:: (Integral i, Show i)

=> i

The lower bound of the integer-range, whose product is returned.

-> i

The number of integers in the range above.

-> i

The result.

Returns the rising factorial; http://mathworld.wolfram.com/RisingFactorial.html

fallingFactorial

Arguments

:: (Integral i, Show i)

=> i

The upper bound of the integer-range, whose product is returned.

-> i

The number of integers in the range beneath.

-> i

The result.

Returns the falling factorial; http://mathworld.wolfram.com/FallingFactorial.html

Operators

(!/!) infixl 7

Arguments

:: (Integral i, Fractional f, Show i)

=> i

The numerator.

-> i

The denominator.

-> f

The resulting fraction.

• Returns the ratio of two factorials.
• It is more efficient than evaluating both factorials, and then dividing.
• For more complex combinations of factorials, such as in the Binomial coefficient, extract the prime factors using primeFactors then manipulate them using the module Data.PrimeFactors, and evaluate it using by Data.PrimeFactors.product'.