Module: MoreMath::Lambert

Defined in:
lib/more_math/lambert.rb

Overview

Provides the Lambert W function implementation for solving equations of the form x * e^x = y

The Lambert W function is a special function that solves the implicit equation x * e^x = y for x in terms of y. It has applications in various fields including combinatorics, physics, and engineering problems involving exponential growth and decay.

Instance Method Summary collapse

Instance Method Details

#lambert_w(y, epsilon = 1E-16) ⇒ Float

Calculates the principal branch of the Lambert W function (W₀).

The Lambert W function solves the equation: x * e^x = y

Examples:

Calculate W(1)

MoreMath::Lambert.lambert_w(1)  # => 0.5671432904097838

Calculate W(0)

MoreMath::Lambert.lambert_w(0)  # => 0.0

Calculate W(-1/e)

MoreMath::Lambert.lambert_w(-1.0 / Math::E)  # => -1.0

Calculate W(100)

MoreMath::Lambert.lambert_w(100)  # => 3.432480170522278

Parameters:

  • y (Numeric) —

    The input value for which to calculate W(y)

  • epsilon (Float) (defaults to: 1E-16) —

    Convergence threshold for the root finding algorithm

Returns:

  • (Float) —

    The principal branch of the Lambert W function W(y)

Raises:

  • (Math::DomainError) —

    When y < -1/e (function is undefined for real numbers)



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# File 'lib/more_math/lambert.rb', line 30

def lambert_w(y, epsilon = 1E-16)
  # Handle special cases
  return 0.0 if y.zero?
  return Float::INFINITY if y.infinite?
  return -1.0 if (y - -1.0 / Math::E).abs < epsilon
  return 0 / 0.0 if y <= -1.0 / Math::E

  # Define the function f(x) = x * e^x - y
  func = ->(x) { x * Math.exp(x) - y }

  solver = MoreMath::NewtonBisection.new(&func)
  solver.solve(nil, 1 << 16, epsilon)
end