Module: MoreMath::Functions

Extended by:
Math
Included in:
ChiSquareDistribution, NormalDistribution, TDistribution
Defined in:
lib/more_math/functions.rb

Class Method Summary collapse

Instance Method Summary collapse

Class Method Details

.beta(a, b) ⇒ Object

Returns the value of the beta function for (a, b), +a > 0, b > 0’.



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# File 'lib/more_math/functions.rb', line 54

def beta(a, b)
  if a > 0 && b > 0
    exp(log_beta(a, b))
  else
    0.0 / 0
  end
end

.beta_regularized(x, a, b, epsilon: 1E-16, max_iterations: 1 << 16) ⇒ Object

Return an approximation value of Euler’s regularized beta function for x, a, and b with an error <= epsilon, but only iterate max_iterations-times.



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# File 'lib/more_math/functions.rb', line 65

def beta_regularized(x, a, b, epsilon: 1E-16, max_iterations: 1 << 16)
  x, a, b = x.to_f, a.to_f, b.to_f
  case
  when a.nan? || b.nan? || x.nan? || a <= 0 || b <= 0 || x < 0 || x > 1
    0 / 0.0
  when x > (a + 1) / (a + b + 2)
    1 - beta_regularized(1 - x, b, a, epsilon: epsilon, max_iterations: max_iterations)
  else
    fraction = ContinuedFraction.for_b do |n, y|
      if n % 2 == 0
        m = n / 2.0
        (m * (b - m) * y) / ((a + (2 * m) - 1) * (a + (2 * m)))
      else
        m = (n - 1) / 2.0
        -((a + m) * (a + b + m) * y) / ((a + 2 * m) * (a + 2 * m + 1))
      end
    end
    exp(a * log(x) + b * log(1.0 - x) - log(a) - log_beta(a, b)) /
      fraction[x, epsilon: epsilon, max_iterations: max_iterations]
  end
rescue Errno::ERANGE, Errno::EDOM
  0 / 0.0
end

.cantor_pairing(*xs) ⇒ Object

Returns Cantor’s tuple function for the tuple *xs (the size must be at least 2).



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# File 'lib/more_math/functions.rb', line 190

def cantor_pairing(*xs)
  CantorPairingFunction.cantor_pairing(*xs)
end

.cantor_pairing_inv(c, n = 2) ⇒ Object

Returns the inverse of Cantor’s tuple function for the value c. n is the length of the tuple (defaults to 2, a pair).



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# File 'lib/more_math/functions.rb', line 196

def cantor_pairing_inv(c, n = 2)
  CantorPairingFunction.cantor_pairing_inv(c, n)
end

.gamma(x) ⇒ Object

Returns the value of the gamma function, extended to a negative domain.



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# File 'lib/more_math/functions.rb', line 38

def gamma(x)
  if x < 0.0
    return PI / (sin(PI * x) * exp(log_gamma(1 - x)))
  else
    exp(log_gamma(x))
  end
end

.gammaP_regularized(x, a, epsilon: 1E-16, max_iterations: 1 << 16) ⇒ Object

Return an approximation of the regularized gammaP function for x and a with an error of <= epsilon, but only iterate max_iterations-times.



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# File 'lib/more_math/functions.rb', line 92

def gammaP_regularized(x, a, epsilon: 1E-16, max_iterations: 1 << 16)
  x, a = x.to_f, a.to_f
  case
  when a.nan? || x.nan? || a <= 0 || x < 0
    0 / 0.0
  when x == 0
    0.0
  when 1 <= a && a < x
    1 - gammaQ_regularized(x, a, epsilon: epsilon, max_iterations: max_iterations)
  else
    n = 0
    an = 1 / a
    sum = an
    while an.abs > epsilon && n < max_iterations
      n += 1
      an *= x / (a + n)
      sum += an
    end
    if n >= max_iterations
      raise Errno::ERANGE
    else
      exp(-x + a * log(x) - log_gamma(a)) * sum
    end
  end
rescue Errno::ERANGE, Errno::EDOM
  0 / 0.0
end

.gammaQ_regularized(x, a, epsilon: 1E-16, max_iterations: 1 << 16) ⇒ Object

Return an approximation of the regularized gammaQ function for x and a with an error of <= epsilon, but only iterate max_iterations-times.



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# File 'lib/more_math/functions.rb', line 123

def gammaQ_regularized(x, a, epsilon: 1E-16, max_iterations: 1 << 16)
  x, a = x.to_f, a.to_f
  case
  when a.nan? || x.nan? || a <= 0 || x < 0
    0 / 0.0
  when x == 0
    1.0
  when a > x || a < 1
    1 - gammaP_regularized(x, a, epsilon: epsilon, max_iterations: max_iterations)
  else
    fraction = ContinuedFraction.for_a do |n, y|
      (2 * n + 1) - a + y
    end.for_b do |n, y|
      n * (a - n)
    end
    exp(-x + a * log(x) - log_gamma(a)) *
      fraction[x, epsilon: epsilon, max_iterations: max_iterations] ** -1
  end
rescue Errno::ERANGE, Errno::EDOM
  0 / 0.0
end

.log_beta(a, b) ⇒ Object

Returns the natural logarithm of the beta function value for (a, b).



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# File 'lib/more_math/functions.rb', line 47

def log_beta(a, b)
  log_gamma(a) + log_gamma(b) - log_gamma(a + b)
rescue Errno::ERANGE, Errno::EDOM
  0 / 0.0
end

.log_ceil(n, b = 2) ⇒ Object

Raises:

  • (ArgumentError) 


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# File 'lib/more_math/functions.rb', line 160

def log_ceil(n, b = 2)
  raise ArgumentError, "n is required to be > 0" unless n > 0
  raise ArgumentError, "b is required to be > 1" unless b > 1
  e, result = 1, 0
  until e >= n
    e *= b
    result += 1
  end
  result
end

.log_floor(n, b = 2) ⇒ Object

Raises:

  • (ArgumentError) 


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# File 'lib/more_math/functions.rb', line 171

def log_floor(n, b = 2)
  raise ArgumentError, "n is required to be > 0" unless n > 0
  raise ArgumentError, "b is required to be > 1" unless b > 1
  e, result = 1, 0
  until e * b > n
    e *= b
    result += 1
  end
  result
end

.logb(x, b = 2) ⇒ Object

Returns the base b logarithm of the number x. b defaults to base 2, binary logarithm.



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# File 'lib/more_math/functions.rb', line 184

def logb(x, b = 2)
  Math.log(x) / Math.log(b)
end

.numberify_string(string, alphabet = 'a'..'z') ⇒ Object

Computes a Gödel number from string in the alphabet and returns it.



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# File 'lib/more_math/functions.rb', line 201

def numberify_string(string, alphabet = 'a'..'z')
  NumberifyStringFunction.numberify_string(string, alphabet)
end

.stringify_number(number, alphabet = 'a'..'z') ⇒ Object

Computes the string in the alphabet from a Gödel number number and returns it. This is the inverse function of numberify_string.



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# File 'lib/more_math/functions.rb', line 207

def stringify_number(number, alphabet = 'a'..'z')
  NumberifyStringFunction.stringify_number(number, alphabet)
end

Instance Method Details

#erf(x) ⇒ Object

Returns an approximate value for the error function’s value for x.



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# File 'lib/more_math/functions.rb', line 147

def erf(x)
  erf_a = MoreMath::Constants::FunctionsConstants::ERF_A
  r = sqrt(1 - exp(-x ** 2 * (4 / Math::PI + erf_a * x ** 2) / (1 + erf_a * x ** 2)))
  x < 0 ? -r : r
end

#erfc(x) ⇒ Object 



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# File 'lib/more_math/functions.rb', line 155

def erfc(x)
  1.0 - erf(x)
end

#log_gamma(x) ⇒ Object 



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# File 'lib/more_math/functions.rb', line 13

def log_gamma(x)
  lgamma(x).first
end