Module: Polynomial::Ultraspherical

Defined in:
lib/polynomial/ultraspherical.rb

Overview

Generate ultraspherical (or Gegenbauer) polynomials. For the mathematics, see Wikipedia or Mathworld entry.

Class Method Summary collapse

Class Method Details

.ultraspherical(degree, alpha) ⇒ Object Also known as: us

Generates the ultraspherical (or Gegenbauer) polynomial of given degree and parameter using an interactive implementation of the recurrence relation which takes O(r^2) time.



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# File 'lib/polynomial/ultraspherical.rb', line 17

def self.ultraspherical(degree, alpha)
  degree >= 0 or raise RangeError, 'degree should be non-negative'
  case alpha
  when 0   then Polynomial::Chebyshev.first_kind(degree)
  when 0.5 then Polynomial::Legendre.legendre(degree)
  when 1   then Polynomial::Chebyshev.second_kind(degree)
  else
    case degree
    when 0
      Polynomial.new(1)
    when 1
      Polynomial.new(0, 2*alpha)
    else
      prev2, prev1 = self.ultraspherical(0, alpha), self.ultraspherical(1, alpha)
      curr = nil # scope reasons
      2.upto(degree) do |m|
        a = Polynomial.new(0, 2*(m+alpha-1)) * prev1
        b = Polynomial.new(m+2*alpha-2) * prev2
        curr = (a - b).quo(m)
        raise RangeError, "alpha #{alpha} is degenerate for specified degree #{degree}" if curr.degree < m
        prev1, prev2 = curr, prev1
      end
      curr
    end
  end
end