lib/distribution/math_extension.rb
require 'bigdecimal'
require 'bigdecimal/math'
require 'prime'
# The next few requires eventually probably need to go in their own gem. They're all functions and constants used by
# GSL-adapted pure Ruby math functions.
require 'distribution/math_extension/chebyshev_series'
require 'distribution/math_extension/erfc'
require 'distribution/math_extension/exponential_integral'
require 'distribution/math_extension/gammastar'
require 'distribution/math_extension/gsl_utilities'
require 'distribution/math_extension/incomplete_gamma'
require 'distribution/math_extension/incomplete_beta'
require 'distribution/math_extension/log_utilities'
module Distribution
# Useful additions to Math
module MathExtension
# Factorization based on Prime Swing algorithm, by Luschny (the king of factorial numbers analysis :P )
# == Reference
# * The Homepage of Factorial Algorithms. (C) Peter Luschny, 2000-2010
# == URL: http://www.luschny.de/math/factorial/csharp/FactorialPrimeSwing.cs.html
class SwingFactorial
SmallOddSwing = [1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003,
429, 6435, 6435, 109_395, 12_155, 230_945, 46_189,
969_969, 88_179, 2_028_117, 676_039, 16_900_975,
1_300_075, 35_102_025, 5_014_575, 145_422_675,
9_694_845, 300_540_195, 300_540_195]
SmallFactorial = [1, 1, 2, 6, 24, 120, 720, 5040, 40_320, 362_880,
3_628_800, 39_916_800, 479_001_600, 6_227_020_800,
87_178_291_200, 1_307_674_368_000, 20_922_789_888_000,
355_687_428_096_000, 6_402_373_705_728_000,
121_645_100_408_832_000, 2_432_902_008_176_640_000]
attr_reader :result
def bitcount(n)
bc = n - ((n >> 1) & 0x55555555)
bc = (bc & 0x33333333) + ((bc >> 2) & 0x33333333)
bc = (bc + (bc >> 4)) & 0x0f0f0f0f
bc += bc >> 8
bc += bc >> 16
bc &= 0x3f
bc
end
def initialize(n)
if n < 20
@result = SmallFactorial[n]
# naive_factorial(n)
else
@prime_list = []
exp2 = n - bitcount(n)
@result = recfactorial(n) << exp2
end
end
def recfactorial(n)
return 1 if n < 2
(recfactorial(n / 2)**2) * swing(n)
end
def swing(n)
return SmallOddSwing[n] if n < 33
sqrtN = Math.sqrt(n).floor
count = 0
Prime.each(n / 3) do |prime|
next if prime < 3
if (prime <= sqrtN)
q = n
_p = 1
while (q = (q / prime).truncate) > 0
_p *= prime if q.odd?
end
if _p > 1
@prime_list[count] = _p
count += 1
end
else
if (n / prime).truncate.odd?
@prime_list[count] = prime
count += 1
end
end
end
prod = get_primorial((n / 2).truncate + 1, n)
prod * @prime_list[0, count].inject(1) { |ac, v| ac * v }
end
def get_primorial(low, up)
prod = 1
Prime.each(up) do |prime|
next if prime < low
prod *= prime
end
prod
end
def naive_factorial(n)
@result = (self.class).naive_factorial(n)
end
def self.naive_factorial(n)
(2..n).inject(1) { |f, nn| f * nn }
end
end
# Module to calculate approximated factorial
# Based (again) on Luschny formula, with 16 digits of precision
# == Reference
# * http://www.luschny.de/math/factorial/approx/SimpleCases.html
module ApproxFactorial
class << self
def stieltjes_ln_factorial(z)
a0 = 1.quo(12); a1 = 1.quo(30); a2 = 53.quo(210); a3 = 195.quo(371)
a4 = 22_999.quo(22_737); a5 = 29_944_523.quo(19_733_142)
a6 = 109_535_241_009.quo(48_264_275_462)
zz = z + 1
(1.quo(2)) * Math.log(2 * Math::PI) + (zz - 1.quo(2)) * Math.log(zz) - zz +
a0.quo(zz + a1.quo(zz + a2.quo(zz + a3.quo(zz + a4.quo(zz + a5.quo(zz + a6.quo(zz)))))))
end
def stieltjes_ln_factorial_big(z)
a0 = 1 / 12.0; a1 = 1 / 30.0; a2 = 53 / 210.0; a3 = 195 / 371.0
a4 = 22_999 / 22_737.0; a5 = 29_944_523 / 19_733_142.0
a6 = 109_535_241_009 / 48_264_275_462.0
zz = z + 1
BigDecimal('0.5') * BigMath.log(BigDecimal('2') * BigMath::PI(20), 20) + BigDecimal((zz - 0.5).to_s) * BigMath.log(BigDecimal(zz.to_s), 20) - BigDecimal(zz.to_s) + BigDecimal((
a0 / (zz + a1 / (zz + a2 / (zz + a3 / (zz + a4 / (zz + a5 / (zz + a6 / zz))))))
).to_s)
end
# Valid upto 11 digits
def stieltjes_factorial(x)
y = x
_p = 1
_p *= y; y += 1 while y < 8
lr = stieltjes_ln_factorial(y)
r = Math.exp(lr)
if r.infinite?
r = BigMath.exp(BigDecimal(lr.to_s), 20)
r = (r * x) / (_p * y) if x < 8
r = r.to_i
else
r = (r * x) / (_p * y) if x < 8
end
r
end
end
end
# Exact factorial.
# Use lookup on a Hash table on n<20
# and Prime Swing algorithm for higher values.
def factorial(n)
SwingFactorial.new(n).result
end
# Approximate factorial, up to 16 digits
# Based of Luschy algorithm
def fast_factorial(n)
ApproxFactorial.stieltjes_factorial(n)
end
# Beta function.
# Source:
# * http://mathworld.wolfram.com/BetaFunction.html
def beta(x, y)
(gamma(x) * gamma(y)).quo(gamma(x + y))
end
# Get pure-Ruby logbeta
def logbeta(x, y)
Beta.log_beta(x, y).first
end
# Log beta function conforming to style of lgamma (returns sign in second array index)
def lbeta(x, y)
Beta.log_beta(x, y)
end
# I_x(a,b): Regularized incomplete beta function
# Fast version. For a exact calculation, based on factorial
# use exact_regularized_beta_function
def regularized_beta(x, a, b)
return 1 if x == 1
IncompleteBeta.evaluate(a, b, x)
end
# I_x(a,b): Regularized incomplete beta function
# TODO: Find a faster version.
# Source:
# * http://dlmf.nist.gov/8.17
def exact_regularized_beta(x, a, b)
return 1 if x == 1
m = a.to_i
n = (b + a - 1).to_i
(m..n).inject(0) do|sum, j|
sum + (binomial_coefficient(n, j) * x**j * (1 - x)**(n - j))
end
end
# Incomplete beta function: B(x;a,b)
# +a+ and +b+ are parameters and +x+ is
# integration upper limit.
def incomplete_beta(x, a, b)
IncompleteBeta.evaluate(a, b, x) * beta(a, b)
# Math::IncompleteBeta.axpy(1.0, 0.0, a,b,x)
end
# Rising factorial
def rising_factorial(x, n)
factorial(x + n - 1).quo(factorial(x - 1))
end
# Ln of gamma
def loggamma(x)
Math.lgamma(x).first
end
def incomplete_gamma(a, x = 0, with_error = false)
IncompleteGamma.p(a, x, with_error)
end
alias_method :gammp, :incomplete_gamma
def gammq(a, x, with_error = false)
IncompleteGamma.q(a, x, with_error)
end
def unnormalized_incomplete_gamma(a, x, with_error = false)
IncompleteGamma.unnormalized(a, x, with_error)
end
# Not the same as erfc. This is the GSL version, which may have slightly different results.
def erfc_e(x, with_error = false)
Erfc.evaluate(x, with_error)
end
# Sequences without repetition. n^k'
# Also called 'failing factorial'
def permutations(n, k)
return 1 if k == 0
return n if k == 1
return factorial(n) if k == n
(((n - k + 1)..n).inject(1) { |ac, v| ac * v })
# factorial(x).quo(factorial(x-n))
end
# Binomial coeffients, or:
# ( n )
# ( k )
#
# Gives the number of *different* k size subsets of a set size n
#
# Uses:
#
# (n) n^k' (n)..(n-k+1)
# ( ) = ---- = ------------
# (k) k! k!
#
def binomial_coefficient(n, k)
return 1 if k == 0 || k == n
k = [k, n - k].min
permutations(n, k).quo(factorial(k))
# The factorial way is
# factorial(n).quo(factorial(k)*(factorial(n-k)))
# The multiplicative way is
# (1..k).inject(1) {|ac, i| (ac*(n-k+i).quo(i))}
end
# Binomial coefficient using multiplicative algorithm
# On benchmarks, is faster that raising factorial method
# when k is little. Use only when you're sure of that.
def binomial_coefficient_multiplicative(n, k)
return 1 if k == 0 || k == n
k = [k, n - k].min
(1..k).inject(1) { |ac, i| (ac * (n - k + i).quo(i)) }
end
# Approximate binomial coefficient, using gamma function.
# The fastest method, until we fall on BigDecimal!
def binomial_coefficient_gamma(n, k)
return 1 if k == 0 || k == n
k = [k, n - k].min
# First, we try direct gamma calculation for max precission
val = gamma(n + 1).quo(gamma(k + 1) * gamma(n - k + 1))
# Ups. Outside float point range. We try with logs
if val.nan?
# puts "nan"
lg = loggamma(n + 1) - (loggamma(k + 1) + loggamma(n - k + 1))
val = Math.exp(lg)
# Crash again! We require BigDecimals
if val.infinite?
# puts "infinite"
val = BigMath.exp(BigDecimal(lg.to_s), 16)
end
end
val
end
alias_method :combinations, :binomial_coefficient
end
end
module Math
include Distribution::MathExtension
module_function :factorial, :beta, :loggamma, :erfc_e, :unnormalized_incomplete_gamma, :incomplete_gamma, :gammp, :gammq, :binomial_coefficient, :binomial_coefficient_gamma, :exact_regularized_beta, :incomplete_beta, :regularized_beta, :permutations, :rising_factorial, :fast_factorial, :combinations, :logbeta, :lbeta
end
# Necessary on Ruby 1.9
module CMath # :nodoc:
include Distribution::MathExtension
module_function :factorial, :beta, :loggamma, :unnormalized_incomplete_gamma,
:incomplete_gamma, :gammp, :gammq, :erfc_e, :binomial_coefficient,
:binomial_coefficient_gamma, :incomplete_beta, :exact_regularized_beta,
:regularized_beta, :permutations, :rising_factorial, :fast_factorial,
:combinations, :logbeta, :lbeta
end