
[index]

EuclidianRing

(G.C.D. module)

This is the module for getting G.C.D. (the greatest common divisor) from
divmod. This is included to Integer or Algebra::Polynomial.

File Name:

 ¡¦ euclidian-ring.rb

Methods:

gcd(other)
    Returns the greatest common divisor of self and other.
   
gcd_all(other0 [, other1[, ...]])
    Returns the greatest common divisor of self and other0, other1,...
   
gcd_coeff(other)
    Returns the array of the greatest common divisor of self and other and the
    coefficients for getting it.
   
    Example:
    require "polynomial"
    require "rational"
    P = Algebra.Polynomial(Rational, "x")
    x = P.var
    f = (x + 2) * (x**2 - 1)**2
    g = (x + 2)**2 * (x - 1)**3
    gcd, a, b = f.gcd_coeff(g)
    p gcd #=> 4x^3 - 12x + 8
    p a   #=> -x + 2
    p b   #=> x - 1
    p gcd == a*f + b*g #=> true
   
gcd_ext(other)
    Same as gcd_coeff
   
gcd_coeff_all(other0 [, other1[, ...]])
    Returns the array of the greatest common divisor self and other0, other1,..
    and the coefficients for getting it.
   
    Example:
    require "polynomial"
    require "rational"
    P = Algebra.Polynomial(Rational, "x")
    x = P.var
    f = (x + 2) * (x**2 - 1)**2
    g = (x + 2)**2 * (x - 1)**3
    h = (x + 2) * (x + 1) * (x - 1)
    gcd, a, b, c = f.gcd_coeff_all(g, h)
    p gcd #=> -8x^2 - 8x + 16
    p a   #=> -x + 2
    p b   #=> x - 1
    p c   #=> -4
    p gcd == a*f + b*g + c*h #=> true
   
gcd_ext_all(other0 [, other1[, ...]])
    Same as gcd_coeff_all
   
lcm(b)
    Return of the least common multiple of self and other.
   
lcm_all(other0 [, other1[, ...]])
    Return of the least common multiple of self and other0, other1,...
   

